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Non-convex entropies for conservation laws with involutions Constantine M. Dafermos
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Research Cite this article: Dafermos CM. 2013 Non-convex entropies for conservation laws with involutions. Phil Trans R Soc A 371: 20120344. http://dx.doi.org/10.1098/rsta.2012.0344
One contribution of 11 to a Theme Issue ‘Entropy and convexity for nonlinear partial differential equations’.
Subject Areas: differential equations Keywords: conservation laws, involutions, contingent entropies, elastodynamics Author for correspondence: Constantine M. Dafermos e-mail: [email protected]
Division of Applied Mathematics, Brown University, Providence, RI 02912, USA The paper discusses systems of conservation laws endowed with involutions and contingent entropies. Under the assumption that the contingent entropy function is convex merely in the direction of a cone in state space, associated with the involution, it is shown that the Cauchy problem is locally well posed in the class of classical solutions, and that classical solutions are unique and stable even within the broader class of weak solutions that satisfy an entropy inequality. This is on a par with the classical theory of solutions to hyperbolic systems of conservation laws endowed with a convex entropy. The equations of elastodynamics provide the prototypical example for the above setting.
1. Introduction In consequence of the Second Law of Thermodynamics, systems of conservation laws arising in continuum physics are endowed with an entropy function of the state variable. Actually, in that connection, the term entropy, which has become standard in the literature, is a misnomer for free energy, which is intimately related with, but never identical to, the physical entropy. As a common occurrence, the entropy function is convex. This is notably the case for the Euler equations, governing the flow of elastic fluids, which provide the prototypical example of a hyperbolic system of conservation laws. Convexity of the entropy renders the Cauchy problem locally well posed: when the initial data lie in a Sobolev space of sufficiently high order, there exists a classical solution on a (generally finite) maximal time interval. Furthermore, this solution is L2 -stable, and thereby unique, not just in the class of classical solutions, but even within the broader class of admissible weak solutions [1,2].
2013 The Author(s) Published by the Royal Society. All rights reserved.
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We consider systems of conservation laws ∂t U +
m
∂α Gα (U) = 0.
(2.1)
α=1
To be determined is the state vector U, with values in Rn , as a function of the spatial variable x, in Rm , and the scalar temporal variable t. For α = 1, . . . , m, ∂α denotes ∂/∂xα , whereas ∂t stands for ∂/∂t. The n × m matrix-valued flux G, with column vectors Gα , is a given smooth function defined on Rn . We shall use consistently matrix notation, identifying vectors in Rs with s × 1 matrices. The symbol D will denote the differential in Rn , with respect to U, and when used in conjunction with matrix calculations, shall be regarded as a row operation: D = [∂/∂U1 , . . . , ∂/∂Un ]. We assume that, for α = 1, . . . , m, there exist constant k × n matrices Mα such that Mα Gβ (U) + Mβ Gα (U) = 0,
α, β = 1, . . . , m, U ∈ Rn .
(2.2)
This implies that the extra conservation law m
Mα ∂α U = 0,
(2.3)
α=1
called an involution [3,4], holds for any solution to the Cauchy problem for (2.1), so long as it is satisfied by the initial data. We assume further that (2.1) is endowed with a contingent entropy–entropy flux pair (η, Q), namely a scalar function η(U) and an Rm -valued function Q(U), defined on Rn and satisfying DQα (U) = Dη(U)DGα (U) + Ξ (U) Mα ,
α = 1, . . . , m, U ∈ Rn ,
(2.4)
......................................................
2. Involutions and contingent entropies
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Although common, convexity of the entropy function is by no means ubiquitous. Indeed, for a plethora of important systems of conservation laws, arising in continuum mechanics, thermomechanics and electrodynamics, convexity of the entropy is ruled out as incompatible with geometric invariance dictated by physics. Quite often, failure of convexity in the entropy function is encountered in systems equipped with involutions, namely stationary conservation laws satisfied automatically by (classical or weak) solutions of the Cauchy problem, so long as they hold for the initial data. The system of conservation laws of elastodynamics provides an illustrative example, which will serve as the paradigm in our discussions. Involutions may compensate for the loss of convexity by rendering the Cauchy problem locally well posed, provided that the entropy function be convex merely in certain directions in state space, spanning the so-called involution cone [2, §5.4, 3]. Unlike full convexity, partial convexity of the entropy in the direction of the involution cone is not incompatible with the requirements of geometric invariance. Still, the regularity and stability properties of classical solutions established in the above references are weaker than those obtained under the assumption of convexity of the entropy function. The aim of this paper is to review the theory of conservation laws endowed with involutions, and upgrade it by relaxing the requirement that an entropy be present—assuming instead the existence of a weaker, contingent entropy—while at the same time improving the regularity, and clarifying the stability properties, of solutions. In the process, technical subtleties that were glossed over in earlier treatments will be presented here in detail. The gains may be modest; however, considering the physical importance of systems of conservation laws equipped with involutions and partially convex entropies, the labour expended to establish that their solutions are on a par with solutions to systems endowed with convex entropies may well be justified.
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m
∂α Qα (U) = 0
(2.5)
α=1
holds for any classical solution of (2.1) that satisfies the involution (2.3). Introducing the notation A(U) = D2 η(U)
(2.6)
for the Hessian of the contingent entropy, we deduce from (2.4) A(U)DGα (U) + DΞ (U) Mα = DGα (U) A(U) + M α DΞ (U),
α = 1, . . . , m,
(2.7)
which generalizes the well-known property that A(U)DGα (U) is symmetric when η is a standard entropy. With any ν ∈ Sm−1 , we associate the k × n matrix N(ν) =
m
να Mα
(2.8)
α=1
and introduce the involution cone C as the union of the kernels of N(ν), over all ν ∈ Sm−1 : C = {X ∈ Rn : N(ν)X = 0, for some ν ∈ Sm−1 }.
(2.9)
We assume that the contingent entropy η is uniformly convex in the direction of C, i.e. for some μ > 0, X A(U)X ≥ μ|X|2 ,
X ∈ C, U ∈ Rn .
(2.10)
The class to which the system (2.1) belongs is encoded in the spectral properties of the matrices m
Λ(ν; U) =
να DGα (U),
(2.11)
α=1
defined for ν ∈ Sm−1 and U ∈ Rn . Upon combining (2.2) with (2.11), we obtain N(ν)Λ(ν; U) = 0,
ν ∈ Sm−1 , U ∈ Rn ,
(2.12)
which shows that zero is an eigenvalue of Λ(ν; U), with the rows of N(ν) being associated eigenvectors. We make the assumption that the family of involutions is complete, in the sense that the rank of N(ν) equals the algebraic, and thereby also the geometric, multiplicity of the zero eigenvalue of Λ(ν; U). It then follows from (2.12) that the range of Λ(ν; U) coincides with the kernel of N(ν). In particular, any eigenvector R of Λ(ν; U), associated with a non-zero eigenvalue λ, must lie in the (complexification of the) kernel of N(ν), and hence B(ν; U)R = λA(U)R,
(2.13)
B(ν, U) = A(U)Λ(ν, U) + DΞ (U) N(ν).
(2.14)
where
Note that B(ν; U) is symmetric, by virtue of (2.7). Thus, multiplying (2.13), from the left, by R∗ yields that λ, and thereby R, are real. Furthermore, upon multiplying (2.13), from the left, by S , where S is any eigenvector of Λ(ν; U) associated with any non-zero eigenvalue different from λ, we deduce S A(U)R = 0. This orthogonality relation implies that the geometric multiplicity of any non-zero eigenvalue of Λ(ν; U) equals its algebraic multiplicity. Thus, under the current assumptions, the system of conservation laws (2.1) is hyperbolic.
......................................................
∂t η(U) +
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for some Rk -valued function Ξ (U) on Rn . This notion, introduced by Serre [5], extends the standard concept of an entropy–entropy flux pair (which corresponds to the special case Ξ ≡ 0) and has been designed so that the extra conservation law
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The m(m + 1)-dimensional state vector consists of the m × m matrix-valued deformation gradient F and the Rm -valued velocity v. The m × m matrix-valued Piola–Kirchhoff stress S is determined by Sjβ (F) =
∂σ (F) , ∂Fjβ
j = 1, . . . , m, β = 1, . . . , m,
(2.16)
where σ is the Helmholtz free energy at constant temperature, for the isothermal case, or the internal energy at constant entropy, for the isentropic case. The system (2.15) is endowed with the involutions ∂β Fiα − ∂α Fiβ = 0,
i = 1, . . . , m, α, β = 1, . . . , m.
(2.17)
In fact, only solutions of (2.15) that satisfy (2.17) have physical meaning, because F must be a gradient. The involution cone associated with the involutions (2.17) is C = {(F, v) : v ∈ Rm , F = ξ ⊗ ν, ξ ∈ Rm , ν ∈ Rm },
(2.18)
i.e. it is spanned by deformation gradients of rank one. The system (2.15) is also equipped with a (standard) entropy–entropy flux pair given by 1 η = |v|2 + σ (F), 2
Qα = −
m i=1
vi
∂σ (F) , α = 1, . . . , m. ∂Fiα
(2.19)
The principle of material frame indifference dictates that σ be invariant under rigid rotations, σ (OF) = σ (F) for all rotation matrices O, and this rules out convexity of η, unless σ is quadratic. However, partial convexity of η, merely in the direction of the involution cone C, which is expressed by the condition that σ is rank-one convex [6], m m i,j=1 α,β=1
∂ 2 σ (F) ξi ξj να νβ ≥ μ|ξ |2 |ν|2 , ∂Fiα ∂Fjβ
ξ ∈ Rm , ν ∈ Rm ,
(2.20)
is not incompatible with material frame indifference and is a physically reasonable assumption. Note that (2.20) renders the system (2.15) hyperbolic, because the rank of N(ν), and the geometric and algebraic multiplicity of the zero eigenvalue of Λ(ν; U) are all equal to m(m − 1). Turning to a broader class of elastic materials, if (2.16) is replaced by the more general constitutive equation Sjβ (F) =
∂σ (F) + Σjβ (F), ∂Fjβ
j = 1, . . . , m, β = 1, . . . , m,
(2.21)
where the m × m matrix-valued function Σ(F) satisfies the differential equation m
∂β Σjβ (F) = 0,
j = 1, . . . , m,
(2.22)
β=1
as an identity, for all F satisfying the involutions (2.17), then (η, Q), defined by (2.19), becomes a contingent entropy–entropy flux pair, for the system (2.15).
......................................................
β=1
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The above conditions are met by a number of important systems arising in continuum mechanics, thermomechanics and electrodynamics, including the equations of isothermal or isentropic elastodynamics, in Lagrangian coordinates, which read ⎧ ⎪ i = 1, . . . , m, α = 1, . . . , m ⎪∂t Fiα − ∂α vi = 0, ⎨ m (2.15) ⎪ ∂β Sjβ (F) = 0, j = 1, . . . , m. ∂v − ⎪ ⎩ t j
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In addition to (2.22), and in order to comply with conservation of angular momentum and the principle of material frame indifference, Σ(F) must satisfy
Σ(OF) = OΣ(F),
(2.24)
and for all rotation matrices O. The class of Σ(F) that meet the above qualifications is not empty. For example, when m = 3, one may construct Σ(F) as Σjβ (F) =
∂φ(F) , ∂Fjβ
j = 1, 2, 3, β = 1, 2, 3,
(2.25)
where φ is a null Lagrangian, namely φ(F) = tr(EF) + tr(ΘF∗ ) + α det F + β,
(2.26)
with F∗ = (det F)F−1 , E and Θ arbitrary constant 3 × 3 matrices, and α and β arbitrary scalars. In that case, η = 12 |v|2 + σ (F) + φ(F) becomes a (standard) entropy for (2.15). Furthermore, σ (F) and σ (F) + φ(F) satisfy (2.20) for the same μ, because null Lagrangians are affine in the direction of the involution cone.
3. Stability of classical solutions We consider the system of conservation laws (2.1), equipped with the involution (2.3) and the contingent entropy–entropy flux pair (η, Q), where η is uniformly convex (2.10) in the direction of the involution cone C, defined by (2.9). A bounded measurable Rn -valued function V, defined on Rm × [0, T], is a local weak solution of (2.1) if it satisfies the equation m ∂α Gα (V) = 0 (3.1) ∂t V + α=1
× [0, T], in the sense of distributions. As (3.1) is in divergence form, V can be normalized on [7, §4.3], so that the function t → V(·, t) is continuous on [0, T] in L∞ (Rm ) weak∗ . It is in that sense that one may assign initial conditions V(x, 0) = V0 (x), with V0 ∈ L∞ (Rm ). We restrict our consideration to weak solutions satisfying the involution Rm
m
Mα ∂α V = 0,
(3.2)
α=1
in the sense of distributions. This will be the case when the initial data satisfy the involution: Mα ∂α V0 = 0, on Rm , in the sense of distributions. The contingent entropy–entropy flux pair provides a vehicle for disqualifying (at least some of) the spurious solutions. The weak solution V, satisfying the involution (3.2), shall be deemed admissible if the entropy inequality ∂t η(V) +
m
∂α Qα (V) ≤ 0
(3.3)
α=1
holds, in the sense of distributions, on Rm × [0, T]. Because the left-hand side of (3.3) is a measure, the function t → η(U(·, t)) is continuous in L∞ weak∗ on a subset F of [0, T] whose complement is at most countable (see [2, §4.5]). As is customary, we complement the admissibility condition (3.3) with the requirement 0 ∈ F , in which case η(V(·, t)) → η(V0 (·)), in L∞ weak∗ .
as t ↓ 0,
(3.4)
......................................................
(2.23)
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Σ(F)F = FΣ(F)
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Theorem 3.2. Let U be a classical C1 solution and V an admissible weak solution of the Cauchy problem for (2.1), defined on Rm × [0, T] and taking values in some open, bounded convex subset O of Rn , with initial data U0 ∈ C1 (Rm ) and V0 ∈ L∞ (Rm ) that satisfy the involution. Assume that V(·, t) − U(·, t) lies in H0 , for t ∈ [0, T], and
V(·, t) − U(·, t) 0 ≤ α,
t ∈ [0, T],
(3.5)
for some constant α. Furthermore, let the asymptotic oscillation of both U and V, as well as the local oscillation of V be sufficiently small, namely for some ρ > 0 and a small, depending solely on bounds of derivatives of η on O, |U(x, t) − U(y, t)| < aμ, |V(x, t) − V(y, t)| < aμ,
|x| > ρ, |y| > ρ, t ∈ [0, T]
(3.6)
and |V(x, t) − V(y, τ )| < aμ,
lim sup
y ∈ Rm , τ ∈ [0, T],
(3.7)
x→y, t→τ
where μ is the constant in (2.10). Under the above assumptions,
V(·, t) − U(·, t) 0 ≤ ceλt V0 (·) − U0 (·) 0 ,
t ∈ [0, T],
(3.8)
where c depends on O, while λ depends on O, on T, and on the maximum of |∇U| over Rm × [0, T]. In particular, U is the unique solution of the Cauchy problem for (2.1), with initial data U0 , within the class of admissible weak solutions with sufficiently small local oscillation (3.7). The following proposition will play a pivotal role in the proof of the above theorem, as well as in §4 of this paper. Accordingly, its proof, which is found in [2, §5.4], will be reproduced here, adapted to the current assumptions and notations. Lemma 3.3. Let P be a bounded measurable symmetric n × n matrix-valued function on Rm , such that X P(x)X ≥ 12 μ|X|2 ,
X ∈ C, x ∈ Rm ,
(3.9)
where C is the involution cone (2.9). Assume further that there is a finite covering of Rm by the union of open sets Ω0 , Ω1 , . . . , ΩK with the property that for J = 0, 1, . . . , K, |P(y) − P(x)| < 14 μ,
x, y ∈ ΩJ .
Then, there is b, depending solely on the covering, such that 1 Z(x) P(x)Z(x) dx ≥ μ Z 20 − b Z 2−1 , m 8 R
(3.10)
(3.11)
holds for any Z ∈ H0 that satisfies the involution m α=1
in the sense of distributions on Rm .
Mα ∂α Z = 0,
(3.12)
......................................................
Notation 3.1. For = −1, 0, 1, . . . , H will denote the Sobolev space [W ,2 (Rm )]n . The norm in H will be denoted by · .
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In particular, any classical C1 solution U of (2.1) on Rm × [0, T], satisfying the involution (2.3), is admissible, as (2.5) holds, by virtue of (2.4). In what follows, it will be shown that classical solutions to the Cauchy problem for (2.1) are L2 -stable, and thereby unique, not just in the class of classical solutions, but even within a broader class of admissible weak solutions. Variants of this, with sketched proofs, are found in [3] and in successive editions of [2]. Here, we present a more general version, with detailed proof.
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Proof. Fix U ∈ Rn and consider the linear differential operator (3.13)
α=1
We construct Φ ∈ H1 such that
LΦ + Φ = Z.
(3.14)
We solve this equation by use of Fourier transform. Recalling (2.11), ˆ ). ˆ ) = Z(ξ {i|ξ |Λ(|ξ |−1 ξ ; U) + I}Φ(ξ
(3.15)
The above linear system is solvable, because all the eigenvalues of Λ(ν; U) are real. Furthermore, ˆ ) lies in the kernel of N(|ξ |−1 ξ ). This, in turn, implies that Φ(ξ ˆ ) also by virtue of (3.12) and (2.8), Z(ξ −1 −1 lies in the kernel of N(|ξ | ξ ), on which all eigenvalues of Λ(|ξ | ξ ; U) are non-zero. Therefore, ˆ )|2 , ˆ )|2 ≤ b2 (1 + |ξ |2 )−1 |Z(ξ |Φ(ξ 1 for some constant b1 , whence
ξ ∈ Rm ,
Φ 0 ≤ b1 Z −1 .
(3.16) (3.17)
Next, we fix a partition of unity ψ0 , ψ1 , . . . , ψK subordinate to the covering Ω0 , Ω1 , . . . , ΩK , i.e. for J = 0, 1, . . . , K, ψJ ∈ C∞ (Rm ), sptψJ ⊂ ΩJ and K
ψJ2 (x) = 1,
x ∈ Rm .
(3.18)
J=0
We also fix yJ ∈ ΩJ , J = 0, 1, . . . , K, and write Rm
Z(x) P(x)Z(x) dx =
=
K m J=0 R
K m J=0 R
+
ψJ2 (x)Z(x) P(x)Z(x) dx ψJ2 (x)Z(x) P(yJ )Z(x) dx
K m J=0 R
ψJ2 (x)Z(x) [P(x) − P(yJ )]Z(x) dx.
(3.19)
By virtue of (3.10), K J=0
Rm
1 ψJ2 (x)Z(x) [P(x) − P(yJ )]Z(x) dx ≥ − μ Z 20 . 4
(3.20)
For each J = 0, 1, . . . , K, we split ψJ Z into ψJ Z = SJ + TJ ,
(3.21)
SJ = L(ψJ Φ)
(3.22)
where and
TJ = ψJ I −
m
∂α ψJ DGα (U) Φ.
(3.23)
α=1
Note that
N(|ξ |−1 ξ )SˆJ (ξ ) = i|ξ |N(|ξ |−1 ξ )Λ(|ξ |−1 ξ ; U)(ψ
J Φ) = 0,
(3.24)
so both the real and the imaginary parts of SˆJ (ξ ) are in C, for any ξ ∈ Rm and J = 0, 1, . . . , K. Thus, applying Parseval’s relation and using (3.9) results in 1 SJ (x) P(yJ )SJ (x) dx = |SJ (x)|2 dx. (3.25) SˆJ (ξ )∗ P(yJ )SˆJ (ξ ) dξ ≥ μ 2 Rm Rm Rm
......................................................
DGα (U)∂α .
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L=
m
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Furthermore, from (3.23) and (3.17) we infer, for J = 0, 1, . . . , K, |TJ (x)|2 dx ≤ b2 Z 2−1 .
8
Rm
≥
7 8
≥
7 μ 16
Rm
SJ (x) P(yJ )SJ (x) dx − 8
Rm
Rm
TJ (X) P(yJ )TJ (x) dx
|SJ (x)|2 dx − b3 Z 2−1 .
Again by (3.21) and (3.26), Rm
|SJ (x)|2 dx ≥
7 8
(3.27)
Rm
ψJ2 (x)|Z(x)|2 dx − b4 Z 2−1 .
Combining (3.18)–(3.20), (3.27) and (3.28), we arrive at (3.11). The proof is complete.
(3.28)
Proof of theorem 3.2. On Rn × Rn , we define the functions h(V, U) = η(V) − η(U) − Dη(U)[V − U],
(3.29)
Yα (V, U) = Qα (V) − Qα (U) − Dη(U)[Gα (V) − Gα (U)] − Ξ (U) Mα [V − U]
(3.30)
Γα (V, U) = Dη(U)[Gα (V) − Gα (U) − DGα (U)[V − U]],
(3.31)
and
all of quadratic order in V − U, by virtue of (2.4). We evaluate the above functions along the classical solution U and the weak solution V. Recalling that U satisfies (2.1), (2.3) and (2.5), whereas V satisfies (3.1)–(3.3), we deduce ∂t h(V, U) +
m
∂α Yα (V, U) = −∂t Dη(U)[V − U]
α=1
−
m
∂α Dη(U)[Gα (V) − Gα (U)]
α=1
−
m
∂α Ξ (U) Mα [V − U],
(3.32)
α−1
in the sense of distributions. On account of (2.1), (2.6), (2.7) and (2.3), ∂t Dη(U) = ∂t U A(U) = −
m
∂α U DGα (U) A(U)
α=1
=−
m
∂α U A(U)DGα (U) −
α=1
m
∂α Ξ (U) Mα .
(3.33)
α=1
Therefore, (3.32) and (3.31) yield ∂t h(V, U) +
m α=1
∂α Yα (V, U) ≤ −
m
∂α U Γα (V, U).
(3.34)
α=1
We fix any t ∈ F and integrate (3.34) over Rm × [0, t] (see [7]). Recalling (3.5), we obtain t h(V(x, t), U(x, t)) dx ≤ h(V0 (x), U0 (x)) dx − ∂α U Γα (V, U) dx dτ . (3.35) Rm
Rm
0 Rm
From (3.29), h(V, U) = (V − U) H(V, U)(V − U),
(3.36)
......................................................
We now return to (3.19). From (3.21), (3.25) and (3.26), it follows that ψJ2 (x)Z(x) P(yJ )Z(x) dx
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(3.26)
Rm
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where H(V, U) =
1
(3.37)
At the same time, by (2.1) and (3.1),
V(·, t) − U(·, t) −1 ≤ V0 (·) − U0 (·) −1 + ≤ V0 (·) − U0 (·) 0 +
t 0
∂τ [V(·, τ ) − U(·, τ )] −1 dτ
t m 0 α=1
≤ V0 (·) − U0 (·) 0 + c0
t
0
Gα (V(·, τ )) − Gα (U(·, τ )) 0 dτ
V(·, τ ) − U(·, τ ) 0 dτ ,
(3.39)
where c0 depends solely on O. Hence
V(·, t) − U(·, t) 2−1 ≤ 2 V0 (·) − U0 (·) 20 + 2c20 T
t 0
V(·, τ ) − U(·, τ ) 20 dτ .
Setting ω(t) = V(·, t) − U(·, t) 20 and combining (3.35), (3.38), (3.40) and (3.31) yields t ω(t) ≤ c2 ω(0) + 2λ ω(τ ) dτ ,
(3.40)
(3.41)
0
for any t ∈ F . However, as t → ω(t) is lower semicontinuous, (3.41) holds for all t ∈ [0, T]. Then, (3.8) follows by Gronwall’s lemma. The proof is complete.
4. Existence and regularity of solutions Here, we establish the existence of locally defined classical solutions to the Cauchy problem for systems of conservation laws equipped with involutions and contingent entropies, convex in the direction of the involution cone. We follow closely the treatment of this problem in [2, §5.4], but the assumptions are here slightly weaker, whereas the conclusion is improved. In fact, we demonstrate that the solutions are as regular as the solutions to systems endowed with convex entropies. Theorem 4.1. Assume that the system (2.1) is equipped with the involution (2.3) and the contingent entropy η, that is uniformly convex (2.10) in the direction of the involution cone C, defined by (2.9). Given U0 ∈ H with > m/2 + 1, satisfying the involution, there exists a unique C1 classical solution U to the Cauchy problem for (2.1), with initial data U0 , defined on a time interval [0, T∞ ), with 0 < T∞ ≤ ∞, and U(·, t) ∈
Ck ([0, T∞ ); H−k ).
(4.1)
k=0
The interval [0, T∞ ) is maximal in that if T∞ < ∞ then lim sup ∇U(·, t) L∞ = ∞. t↑T∞
(4.2)
......................................................
for any X ∈ C. In particular, X H(V, U)X ≥ On account of (3.6), there is ρ > 0 such that, for each fixed t ∈ [0, T], the oscillation of H(V(·, t), U(·, t)) over the set Ω0 = {x ∈ Rm : |x| > ρ} is less than 14 μ. Furthermore, when (3.7) holds with a sufficiently small, there exists δ > 0 such that, for each t ∈ [0, T], the oscillation of H(V(·, t), U(·, t)) over any ball of radius δ in Rm is less than 14 μ. Therefore, we may cover Rm by the union Ω0 ∪ Ω1 ∪ · · · ∪ ΩK , where, for J = 1, . . . , K, ΩJ is a ball of radius δ, so that, for each fixed t ∈ [0, T], H(V(·, t), U(·, t)) satisfies the assumptions for P in lemma 3.3. Consequently, 1 h(V(x, t), U(x, t)) dx ≥ μ V(·, t) − U(·, t) 20 − b V(·, t) − U(·, t) 2−1 . (3.38) m 8 R 1 2 2 μ|X| ,
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0
9 sA(sU + (1 − s)V) ds.
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Proof. As in [2], the solution U will be constructed by the vanishing viscosity method, namely as the ε ↓ 0 limit of solutions Uε to the parabolic system ∂α Gα (U) = εU,
(4.3)
α=1
under the same initial condition Uε (x, 0) = U0 (x), x ∈ Rm . This method is effective because (4.3) inherits the involution (2.3) from (2.1). As shown in [2, lemma 5.1.2], for any ω > 0 and U0 ∈ H with U0 < ω, the Cauchy problem for (4.3) admits a solution Uε on [0, Tε ) such that Uε (·, t) ∈ C0 ([0, Tε ); H ) ∩ L2 ([0, Tε ); H+1 )
(4.4)
and
Uε (·, t) < ω,
t ∈ [0, Tε ),
(4.5)
and [0, Tε ) is maximal in that if Tε < ∞ then Uε (·, t) → ω, as t ↑ Tε . The aim is to show that if
U0 < cω, with c sufficiently small, then, for all ε > 0, Tε > T > 0, in which case one may derive the solution of (2.1) on [0, T] by passing to the limit ε ↓ 0. We thus fix ε > 0, sufficiently small, and simplify the notation by dropping the subscript ε and denoting the solution of (4.3) simply by U. Once ω has been fixed, the range of U will be confined in a ball B of Rn . In what follows, c will denote a generic constant that may depend on bounds of G(U), η(U), Ξ (U), and their derivatives, on B. In order to secure sufficient regularity for our calculations, we assume temporarily that U0 ∈ H+2 , and therefore U(·, t) ∈ H+2 . We fix any multi-index r = (r1 , . . . , rm ), of order |r| ≤ , and apply ∂ r to (4.3). Upon setting Ur = ∂ r U, ∂t Ur +
m
∂ r ∂α Gα (U) = εUr .
(4.6)
α=1
We multiply (4.6) by ∂t Ur and integrate over Rm × [0, t], 0 < t < Tε , which yields t ε |∂t Ur |2 dx dτ + |∇Ur (x, t)|2 dx 2 Rm 0 Rm t m ε |∇U0r (x)|2 dx − ∂t Ur ∂ r ∂α Gα (U) dx dτ . = 2 Rm 0 Rm
(4.7)
α=1
By standard Sobolev space estimates,
∂ r ∂α Gα (U) 0 ≤ c U ≤ cω,
(4.8)
for any r of order |r| ≤ − 1, and τ ∈ [0, t]. Hence, applying the Cauchy–Schwarz inequality on the right-hand side of (4.7) and summing over all r of order 0 ≤ |r| ≤ − 1, we deduce t
∂t U 2−1 dτ ≤ (ε + ct)ω2 . (4.9) 0
In turn, (4.9) implies
U(·, t) − U0 (·) 2−1 ≤ t(ε + ct)ω2
(4.10)
max |U(x, t) − U0 (x)| ≤ c(ε + t)1/2 t1/2 ω.
(4.11)
and x∈Rm
Next, we rewrite (4.6) as ∂t Ur +
m α=1
DGα (U)∂α Ur =
m α=1
{DGα (U)∂α Ur − ∂ r [DGα (U)∂α U]} + εUr ,
(4.12)
......................................................
m
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multiply by Ur A(U) and integrate over Rm × [0, t]. We note the identities
and m
Ur A(U)Ur =
∂α [Ur A(U)∂α Ur ] −
α=1
m
∂α Ur A(U)∂α Ur −
α=1
m
Ur ∂α A(U)∂α Ur .
(4.14)
α=1
Furthermore, recalling (2.3) and (2.7), m
2Ur A(U)DGα (U)∂α Ur =
α=1
m
2Ur [A(U)DGα (U) + DΞ (U) Mα ]∂α Ur
α=1
=
m
∂α {Ur [A(U)DGα (U) + DΞ (U) Mα ]Ur }
α=1
−
m
Ur ∂α [A(U)DGα (U) + DΞ (U) Mα ]Ur .
(4.15)
α=1
Therefore, 1 2
Rm
Ur (x, t)A(U(x, t))Ur (x, t) dx + ε
+ε
m Rm α=1
1 2
=
−
+
m Rm α=1
∂α Ur (x, t)A(U(x, t))∂α Ur (x, t) dx
Ur (x, t)∂α A(U(x, t))∂α Ur (x, t) dx
U0r (x)A(U0 (x))U0r (x) dx +
Rm
1 2
t
m
0 Rm α=1
t
0 Rm
1 2
t 0 Rm
Ur ∂α A(U)Ur dx dτ
Ur ∂α [A(U)DGα (U) + DΞ (U) Mα ]Ur dx dτ
Ur A(U)
m
{DGα (U)∂α Ur − ∂ r [DGα (U)∂α U]} dx dτ .
(4.16)
α=1
We fix r of order |r| = . By standard Sobolev space estimates,
DGα (U)∂α Ur − ∂ r [DGα (U)∂α U] 0 ≤ c ∇U L∞ U .
(4.17)
From (4.11) and since |U0 (x)| → 0 as |x| → ∞, we deduce that when ρ is sufficiently large and T is sufficiently small, the oscillation of A(U(·, t)) over the set Ω0 = {x ∈ Rm : |x| > ρ} is less than 12 μ, for any t ∈ [0, T]. Furthermore, as ∇U L∞ is bounded by cω, the oscillation of A(U(·, t)) over any ball of sufficiently small radius δ is likewise less than 12 μ. These, together with (2.10) imply that 1 2 A(U(·, t)) satisfies the assumptions for P(·) in lemma 3.3. Therefore, 1 2
Rm
1 Ur (x, t)A(U(x, t))Ur (x, t) dx ≥ μ Ur (·, t) 20 − b U(·, t) 2−1 , 8
(4.18)
and
m Rm
α=1
1 ∂α Ur (x, t)A(U(x, t))∂α Ur (x, t) dx ≥ μ ∇Ur (·, t) 20 − 2b U(·)t 2 . 4
(4.19)
......................................................
(4.13)
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t ∈ [0, T].
(4.20)
Thus, if c U0 2 < ω2 and T is sufficiently small, then (4.20) yields U(·, t) < ω, for t ∈ [0, T] and any ε 1. We have now laid the preparation for constructing the solution to the Cauchy problem for (2.1), under initial data U0 ∈ H , on the time interval [0, T]. For that purpose, we fix a sequence {U0ν } in H+2 and a sequence {εν }, such that U0ν → U0 in H and εν → 0, as ν → ∞. We let Uν denote the solution of (4.3), with ε = εν and initial data U0ν . As shown earlier, {Uν } is contained in a bounded set of L∞ ([0, T]; H ) and {∂t Uν } is contained in a bounded set of L2 ([0, T]; H−1 ). Thus, {Uν } is uniformly equicontinuous on Rm × [0, T]. Hence, we may extract a subsequence, again denoted by {Uν }, that converges to a continuous function U, uniformly on compact subsets of Rm × [0, T]. Clearly, U is a distributional solution of (2.1) on Rm × [0, T], with initial value U0 . Furthermore, U(·, t) ∈ L∞ ([0, T]; H ), and this together with (2.1) implies U(·, t) ∈
W k,∞ ([0, T]; H−k ).
(4.21)
k=0
In fact, t → U(·, t) is weakly continuous in H on [0, T]. We may extend the domain of this solution beyond [0, T], by restarting at t = T, with initial data U(·, T), and repeating the above process. On account of (4.20), this may proceed for as long as ∇U L∞ stays bounded. Therefore, the maximal interval [0, T∞ ) of existence will be finite when ∇U(·, t) blows up as t ↑ T∞ , i.e. (4.2) holds. It remains to prove that the solution U is endowed with regularity (4.1). As (2.1) is invariant under time translations and reflections, it will suffice to show that t → U(·, t) is right-continuous in H at t = 0. We begin with a sequence {U0ν } in H+2 , such that U0ν → U0 in H , as ν → ∞, and consider the solution Uν of the Cauchy problem for (2.1), with initial data U0ν , defined on a time interval [0, T], independent of ν. By virtue of (4.11) and the uniform boundedness of ∇Uν , for T small, both the asymptotic and local oscillations of A(U(·, t)) and A(Uν (·, t)) are uniformly small. We may thus apply theorem 3.2 to deduce that Uν (·, t) → U(·, t) strongly in H0 , for t ∈ [0, T]. On the other hand, Uν (·, t) → U(·, t) weakly in H . It follows that Uν (·, t) → U(·, t),
strongly in H−1 , t ∈ [0, T].
(4.22)
We now fix r of order |r| ≤ , and apply ∂ r to (2.1) thus arriving at (4.12), with ε = 0. Since Uν (·, t) ∈ H+2 , there is sufficient regularity to justify (4.13) and (4.15), so that Uν satisfies (4.16), with ε = 0. In particular, Uνr (x, t)A(Uν (x, t))Uνr (x, t) dx = U0νr (x)A(U0ν (x))U0νr (x) dx + O(t). (4.23) Rm
Rm
Next, we write the identity [Uνr (x, t) − Ur (x, t)] A(U(x, t))[Uνr (x, t) − Ur (x, t)] dx Rm
=
Uνr (x, t)A(Uν (x, t))Uνr (x, t) dx −
Rm
−
Rm
Rm
Ur (x, t)A(U(x, t))Ur (x, t) dx
Uνr (x, t)[A(Uν (x, t)) − A(U(x, t))]Uνr (x, t) dx
−2
Rm
[Uνr (x, t) − Ur (x, t)]A(U(x, t))Ur (x, t) dx
(4.24)
......................................................
U(·, t) 2 ≤ c U0 (·) 2 + ct[ε + t + ∇U L∞ ]ω2 ,
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Next, we sum (4.16) over all r of order |r| = , use (4.18) and (4.19) to estimate the left-hand side, use (4.17) and Schwarz’s inequality to estimate the right-hand side, and combine the result with (4.10) to conclude that for ε 1,
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1 ≥ μ Uνr (·, t) − Ur (·, t) 20 − 2b Uν (·, t) − U(·, t) 2−1 . 4
(4.25)
As ν → ∞, the last term on the right-hand side of (4.25) tends to zero, by virtue of (4.22). We thus conclude Ur (x, t)A(U(x, t))Ur (x, t) dx ≤ lim inf Uνr (x, t)A(Uν (x, t))Uνr (x, t) dx. (4.26) ν→∞
Rm
Combining (4.23) with (4.26) yields Ur (x, t)A(U(x, t))Ur (x, t) dx ≤ Rm
Rm
Rm
U0r (x)A(U0 (x))U0r (x) dx + O(t).
(4.27)
Finally, we write the identity [Ur (x, t) − U0r (x)] A(U0 (x))[Ur (x, t) − U0r (x)] dx Rm
=
Ur (x, t)A(U(x, t))Ur (x, t) dx −
Rm
−
Rm
Rm
U0r (x)A(U0 (x))U0r (x) dx
Ur (x, t)[A(U(x, t)) − A(U0 (x))]Ur (x, t) dx
−2
Rm
[Ur (x, t) − U0r (x)] A(U0 (x))U0r (x) dx
(4.28)
and let t ↓ 0. On the right-hand side, the third term tends to zero, by the Lebesgue dominated convergence theorem, as U(x, t) → U0 (x); and the last term also tends to zero, as Ur (·, t) → U0r (·) weakly in H0 ; furthermore, the limit superior of the difference of the first two terms is non-positive, because of (4.27). For the left-hand side, lemma 3.3 yields [Ur (x, t) − U0r (x)] A(U0 (x))[Ur (x, t) − U0r (x)] dx Rm
1 ≥ μ Ur (·, t) − U0r (·) 20 − 2b U(·, t) − U0 (·) 2−1 . 4
(4.29)
Recalling (4.10), we conclude that Ur (·, t) − U0r (·) 0 → 0, as t ↓ 0, i.e. U(·, t) → U0 (·), strongly in H . The proof is complete.
References 1. Benzoni-Gavage S, Serre D. 2007 Multi-dimensional hyperbolic partial differential equations. Oxford, UK: Oxford University Press. 2. Dafermos CM. 2010 Hyperbolic conservation laws in continuum physics, 3rd edn. Heidelberg, Germany: Springer. 3. Dafermos CM. 1986 Quasilinear hyperbolic systems with involutions. Arch. Ration. Mech. Anal. 94, 373–389. (doi:10.1007/BF00280911) 4. Boillat G. 1988 Involutions des systèmes conservatifs. C.R. Acad. Sci. Paris, Série I 307, 891–894. 5. Serre D. 2004 Hyperbolicity of the non-linear models of Maxwell’s equations. Arch. Ration. Mech. Anal. 172, 309–331. (doi:10.1007/s00205-003-0303-4) 6. Ball JM. 1977 Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63, 337–403. (doi:10.1007/BF00279992) 7. Chen G-Q, Frid H. 1999 Divergence measure fields and conservation laws. Arch. Ration. Mech. Anal. 147, 89–118. (doi:10.1007/s002050050146)
......................................................
Rm
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and let ν → ∞. On the right-hand side, the third term tends to zero, by the Lebesgue dominated convergence theorem, as Uν → U; and the last term also tends to zero, as Uνr (·, t) → Ur (·, t), weakly in H0 . On the left-hand side, by lemma 3.3, [Uνr (x, t) − Ur (x, t)] A(U(x, t))[Uνr (x, t) − Ur (x, t)] dx
Non-convex entropies for conservation laws with involutions Constantine M. Dafermos
rsta.royalsocietypublishing.org
Research Cite this article: Dafermos CM. 2013 Non-convex entropies for conservation laws with involutions. Phil Trans R Soc A 371: 20120344. http://dx.doi.org/10.1098/rsta.2012.0344
One contribution of 11 to a Theme Issue ‘Entropy and convexity for nonlinear partial differential equations’.
Subject Areas: differential equations Keywords: conservation laws, involutions, contingent entropies, elastodynamics Author for correspondence: Constantine M. Dafermos e-mail: [email protected]
Division of Applied Mathematics, Brown University, Providence, RI 02912, USA The paper discusses systems of conservation laws endowed with involutions and contingent entropies. Under the assumption that the contingent entropy function is convex merely in the direction of a cone in state space, associated with the involution, it is shown that the Cauchy problem is locally well posed in the class of classical solutions, and that classical solutions are unique and stable even within the broader class of weak solutions that satisfy an entropy inequality. This is on a par with the classical theory of solutions to hyperbolic systems of conservation laws endowed with a convex entropy. The equations of elastodynamics provide the prototypical example for the above setting.
1. Introduction In consequence of the Second Law of Thermodynamics, systems of conservation laws arising in continuum physics are endowed with an entropy function of the state variable. Actually, in that connection, the term entropy, which has become standard in the literature, is a misnomer for free energy, which is intimately related with, but never identical to, the physical entropy. As a common occurrence, the entropy function is convex. This is notably the case for the Euler equations, governing the flow of elastic fluids, which provide the prototypical example of a hyperbolic system of conservation laws. Convexity of the entropy renders the Cauchy problem locally well posed: when the initial data lie in a Sobolev space of sufficiently high order, there exists a classical solution on a (generally finite) maximal time interval. Furthermore, this solution is L2 -stable, and thereby unique, not just in the class of classical solutions, but even within the broader class of admissible weak solutions [1,2].
2013 The Author(s) Published by the Royal Society. All rights reserved.
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We consider systems of conservation laws ∂t U +
m
∂α Gα (U) = 0.
(2.1)
α=1
To be determined is the state vector U, with values in Rn , as a function of the spatial variable x, in Rm , and the scalar temporal variable t. For α = 1, . . . , m, ∂α denotes ∂/∂xα , whereas ∂t stands for ∂/∂t. The n × m matrix-valued flux G, with column vectors Gα , is a given smooth function defined on Rn . We shall use consistently matrix notation, identifying vectors in Rs with s × 1 matrices. The symbol D will denote the differential in Rn , with respect to U, and when used in conjunction with matrix calculations, shall be regarded as a row operation: D = [∂/∂U1 , . . . , ∂/∂Un ]. We assume that, for α = 1, . . . , m, there exist constant k × n matrices Mα such that Mα Gβ (U) + Mβ Gα (U) = 0,
α, β = 1, . . . , m, U ∈ Rn .
(2.2)
This implies that the extra conservation law m
Mα ∂α U = 0,
(2.3)
α=1
called an involution [3,4], holds for any solution to the Cauchy problem for (2.1), so long as it is satisfied by the initial data. We assume further that (2.1) is endowed with a contingent entropy–entropy flux pair (η, Q), namely a scalar function η(U) and an Rm -valued function Q(U), defined on Rn and satisfying DQα (U) = Dη(U)DGα (U) + Ξ (U) Mα ,
α = 1, . . . , m, U ∈ Rn ,
(2.4)
......................................................
2. Involutions and contingent entropies
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Although common, convexity of the entropy function is by no means ubiquitous. Indeed, for a plethora of important systems of conservation laws, arising in continuum mechanics, thermomechanics and electrodynamics, convexity of the entropy is ruled out as incompatible with geometric invariance dictated by physics. Quite often, failure of convexity in the entropy function is encountered in systems equipped with involutions, namely stationary conservation laws satisfied automatically by (classical or weak) solutions of the Cauchy problem, so long as they hold for the initial data. The system of conservation laws of elastodynamics provides an illustrative example, which will serve as the paradigm in our discussions. Involutions may compensate for the loss of convexity by rendering the Cauchy problem locally well posed, provided that the entropy function be convex merely in certain directions in state space, spanning the so-called involution cone [2, §5.4, 3]. Unlike full convexity, partial convexity of the entropy in the direction of the involution cone is not incompatible with the requirements of geometric invariance. Still, the regularity and stability properties of classical solutions established in the above references are weaker than those obtained under the assumption of convexity of the entropy function. The aim of this paper is to review the theory of conservation laws endowed with involutions, and upgrade it by relaxing the requirement that an entropy be present—assuming instead the existence of a weaker, contingent entropy—while at the same time improving the regularity, and clarifying the stability properties, of solutions. In the process, technical subtleties that were glossed over in earlier treatments will be presented here in detail. The gains may be modest; however, considering the physical importance of systems of conservation laws equipped with involutions and partially convex entropies, the labour expended to establish that their solutions are on a par with solutions to systems endowed with convex entropies may well be justified.
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m
∂α Qα (U) = 0
(2.5)
α=1
holds for any classical solution of (2.1) that satisfies the involution (2.3). Introducing the notation A(U) = D2 η(U)
(2.6)
for the Hessian of the contingent entropy, we deduce from (2.4) A(U)DGα (U) + DΞ (U) Mα = DGα (U) A(U) + M α DΞ (U),
α = 1, . . . , m,
(2.7)
which generalizes the well-known property that A(U)DGα (U) is symmetric when η is a standard entropy. With any ν ∈ Sm−1 , we associate the k × n matrix N(ν) =
m
να Mα
(2.8)
α=1
and introduce the involution cone C as the union of the kernels of N(ν), over all ν ∈ Sm−1 : C = {X ∈ Rn : N(ν)X = 0, for some ν ∈ Sm−1 }.
(2.9)
We assume that the contingent entropy η is uniformly convex in the direction of C, i.e. for some μ > 0, X A(U)X ≥ μ|X|2 ,
X ∈ C, U ∈ Rn .
(2.10)
The class to which the system (2.1) belongs is encoded in the spectral properties of the matrices m
Λ(ν; U) =
να DGα (U),
(2.11)
α=1
defined for ν ∈ Sm−1 and U ∈ Rn . Upon combining (2.2) with (2.11), we obtain N(ν)Λ(ν; U) = 0,
ν ∈ Sm−1 , U ∈ Rn ,
(2.12)
which shows that zero is an eigenvalue of Λ(ν; U), with the rows of N(ν) being associated eigenvectors. We make the assumption that the family of involutions is complete, in the sense that the rank of N(ν) equals the algebraic, and thereby also the geometric, multiplicity of the zero eigenvalue of Λ(ν; U). It then follows from (2.12) that the range of Λ(ν; U) coincides with the kernel of N(ν). In particular, any eigenvector R of Λ(ν; U), associated with a non-zero eigenvalue λ, must lie in the (complexification of the) kernel of N(ν), and hence B(ν; U)R = λA(U)R,
(2.13)
B(ν, U) = A(U)Λ(ν, U) + DΞ (U) N(ν).
(2.14)
where
Note that B(ν; U) is symmetric, by virtue of (2.7). Thus, multiplying (2.13), from the left, by R∗ yields that λ, and thereby R, are real. Furthermore, upon multiplying (2.13), from the left, by S , where S is any eigenvector of Λ(ν; U) associated with any non-zero eigenvalue different from λ, we deduce S A(U)R = 0. This orthogonality relation implies that the geometric multiplicity of any non-zero eigenvalue of Λ(ν; U) equals its algebraic multiplicity. Thus, under the current assumptions, the system of conservation laws (2.1) is hyperbolic.
......................................................
∂t η(U) +
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for some Rk -valued function Ξ (U) on Rn . This notion, introduced by Serre [5], extends the standard concept of an entropy–entropy flux pair (which corresponds to the special case Ξ ≡ 0) and has been designed so that the extra conservation law
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The m(m + 1)-dimensional state vector consists of the m × m matrix-valued deformation gradient F and the Rm -valued velocity v. The m × m matrix-valued Piola–Kirchhoff stress S is determined by Sjβ (F) =
∂σ (F) , ∂Fjβ
j = 1, . . . , m, β = 1, . . . , m,
(2.16)
where σ is the Helmholtz free energy at constant temperature, for the isothermal case, or the internal energy at constant entropy, for the isentropic case. The system (2.15) is endowed with the involutions ∂β Fiα − ∂α Fiβ = 0,
i = 1, . . . , m, α, β = 1, . . . , m.
(2.17)
In fact, only solutions of (2.15) that satisfy (2.17) have physical meaning, because F must be a gradient. The involution cone associated with the involutions (2.17) is C = {(F, v) : v ∈ Rm , F = ξ ⊗ ν, ξ ∈ Rm , ν ∈ Rm },
(2.18)
i.e. it is spanned by deformation gradients of rank one. The system (2.15) is also equipped with a (standard) entropy–entropy flux pair given by 1 η = |v|2 + σ (F), 2
Qα = −
m i=1
vi
∂σ (F) , α = 1, . . . , m. ∂Fiα
(2.19)
The principle of material frame indifference dictates that σ be invariant under rigid rotations, σ (OF) = σ (F) for all rotation matrices O, and this rules out convexity of η, unless σ is quadratic. However, partial convexity of η, merely in the direction of the involution cone C, which is expressed by the condition that σ is rank-one convex [6], m m i,j=1 α,β=1
∂ 2 σ (F) ξi ξj να νβ ≥ μ|ξ |2 |ν|2 , ∂Fiα ∂Fjβ
ξ ∈ Rm , ν ∈ Rm ,
(2.20)
is not incompatible with material frame indifference and is a physically reasonable assumption. Note that (2.20) renders the system (2.15) hyperbolic, because the rank of N(ν), and the geometric and algebraic multiplicity of the zero eigenvalue of Λ(ν; U) are all equal to m(m − 1). Turning to a broader class of elastic materials, if (2.16) is replaced by the more general constitutive equation Sjβ (F) =
∂σ (F) + Σjβ (F), ∂Fjβ
j = 1, . . . , m, β = 1, . . . , m,
(2.21)
where the m × m matrix-valued function Σ(F) satisfies the differential equation m
∂β Σjβ (F) = 0,
j = 1, . . . , m,
(2.22)
β=1
as an identity, for all F satisfying the involutions (2.17), then (η, Q), defined by (2.19), becomes a contingent entropy–entropy flux pair, for the system (2.15).
......................................................
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The above conditions are met by a number of important systems arising in continuum mechanics, thermomechanics and electrodynamics, including the equations of isothermal or isentropic elastodynamics, in Lagrangian coordinates, which read ⎧ ⎪ i = 1, . . . , m, α = 1, . . . , m ⎪∂t Fiα − ∂α vi = 0, ⎨ m (2.15) ⎪ ∂β Sjβ (F) = 0, j = 1, . . . , m. ∂v − ⎪ ⎩ t j
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In addition to (2.22), and in order to comply with conservation of angular momentum and the principle of material frame indifference, Σ(F) must satisfy
Σ(OF) = OΣ(F),
(2.24)
and for all rotation matrices O. The class of Σ(F) that meet the above qualifications is not empty. For example, when m = 3, one may construct Σ(F) as Σjβ (F) =
∂φ(F) , ∂Fjβ
j = 1, 2, 3, β = 1, 2, 3,
(2.25)
where φ is a null Lagrangian, namely φ(F) = tr(EF) + tr(ΘF∗ ) + α det F + β,
(2.26)
with F∗ = (det F)F−1 , E and Θ arbitrary constant 3 × 3 matrices, and α and β arbitrary scalars. In that case, η = 12 |v|2 + σ (F) + φ(F) becomes a (standard) entropy for (2.15). Furthermore, σ (F) and σ (F) + φ(F) satisfy (2.20) for the same μ, because null Lagrangians are affine in the direction of the involution cone.
3. Stability of classical solutions We consider the system of conservation laws (2.1), equipped with the involution (2.3) and the contingent entropy–entropy flux pair (η, Q), where η is uniformly convex (2.10) in the direction of the involution cone C, defined by (2.9). A bounded measurable Rn -valued function V, defined on Rm × [0, T], is a local weak solution of (2.1) if it satisfies the equation m ∂α Gα (V) = 0 (3.1) ∂t V + α=1
× [0, T], in the sense of distributions. As (3.1) is in divergence form, V can be normalized on [7, §4.3], so that the function t → V(·, t) is continuous on [0, T] in L∞ (Rm ) weak∗ . It is in that sense that one may assign initial conditions V(x, 0) = V0 (x), with V0 ∈ L∞ (Rm ). We restrict our consideration to weak solutions satisfying the involution Rm
m
Mα ∂α V = 0,
(3.2)
α=1
in the sense of distributions. This will be the case when the initial data satisfy the involution: Mα ∂α V0 = 0, on Rm , in the sense of distributions. The contingent entropy–entropy flux pair provides a vehicle for disqualifying (at least some of) the spurious solutions. The weak solution V, satisfying the involution (3.2), shall be deemed admissible if the entropy inequality ∂t η(V) +
m
∂α Qα (V) ≤ 0
(3.3)
α=1
holds, in the sense of distributions, on Rm × [0, T]. Because the left-hand side of (3.3) is a measure, the function t → η(U(·, t)) is continuous in L∞ weak∗ on a subset F of [0, T] whose complement is at most countable (see [2, §4.5]). As is customary, we complement the admissibility condition (3.3) with the requirement 0 ∈ F , in which case η(V(·, t)) → η(V0 (·)), in L∞ weak∗ .
as t ↓ 0,
(3.4)
......................................................
(2.23)
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Σ(F)F = FΣ(F)
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Theorem 3.2. Let U be a classical C1 solution and V an admissible weak solution of the Cauchy problem for (2.1), defined on Rm × [0, T] and taking values in some open, bounded convex subset O of Rn , with initial data U0 ∈ C1 (Rm ) and V0 ∈ L∞ (Rm ) that satisfy the involution. Assume that V(·, t) − U(·, t) lies in H0 , for t ∈ [0, T], and
V(·, t) − U(·, t) 0 ≤ α,
t ∈ [0, T],
(3.5)
for some constant α. Furthermore, let the asymptotic oscillation of both U and V, as well as the local oscillation of V be sufficiently small, namely for some ρ > 0 and a small, depending solely on bounds of derivatives of η on O, |U(x, t) − U(y, t)| < aμ, |V(x, t) − V(y, t)| < aμ,
|x| > ρ, |y| > ρ, t ∈ [0, T]
(3.6)
and |V(x, t) − V(y, τ )| < aμ,
lim sup
y ∈ Rm , τ ∈ [0, T],
(3.7)
x→y, t→τ
where μ is the constant in (2.10). Under the above assumptions,
V(·, t) − U(·, t) 0 ≤ ceλt V0 (·) − U0 (·) 0 ,
t ∈ [0, T],
(3.8)
where c depends on O, while λ depends on O, on T, and on the maximum of |∇U| over Rm × [0, T]. In particular, U is the unique solution of the Cauchy problem for (2.1), with initial data U0 , within the class of admissible weak solutions with sufficiently small local oscillation (3.7). The following proposition will play a pivotal role in the proof of the above theorem, as well as in §4 of this paper. Accordingly, its proof, which is found in [2, §5.4], will be reproduced here, adapted to the current assumptions and notations. Lemma 3.3. Let P be a bounded measurable symmetric n × n matrix-valued function on Rm , such that X P(x)X ≥ 12 μ|X|2 ,
X ∈ C, x ∈ Rm ,
(3.9)
where C is the involution cone (2.9). Assume further that there is a finite covering of Rm by the union of open sets Ω0 , Ω1 , . . . , ΩK with the property that for J = 0, 1, . . . , K, |P(y) − P(x)| < 14 μ,
x, y ∈ ΩJ .
Then, there is b, depending solely on the covering, such that 1 Z(x) P(x)Z(x) dx ≥ μ Z 20 − b Z 2−1 , m 8 R
(3.10)
(3.11)
holds for any Z ∈ H0 that satisfies the involution m α=1
in the sense of distributions on Rm .
Mα ∂α Z = 0,
(3.12)
......................................................
Notation 3.1. For = −1, 0, 1, . . . , H will denote the Sobolev space [W ,2 (Rm )]n . The norm in H will be denoted by · .
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In particular, any classical C1 solution U of (2.1) on Rm × [0, T], satisfying the involution (2.3), is admissible, as (2.5) holds, by virtue of (2.4). In what follows, it will be shown that classical solutions to the Cauchy problem for (2.1) are L2 -stable, and thereby unique, not just in the class of classical solutions, but even within a broader class of admissible weak solutions. Variants of this, with sketched proofs, are found in [3] and in successive editions of [2]. Here, we present a more general version, with detailed proof.
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Proof. Fix U ∈ Rn and consider the linear differential operator (3.13)
α=1
We construct Φ ∈ H1 such that
LΦ + Φ = Z.
(3.14)
We solve this equation by use of Fourier transform. Recalling (2.11), ˆ ). ˆ ) = Z(ξ {i|ξ |Λ(|ξ |−1 ξ ; U) + I}Φ(ξ
(3.15)
The above linear system is solvable, because all the eigenvalues of Λ(ν; U) are real. Furthermore, ˆ ) lies in the kernel of N(|ξ |−1 ξ ). This, in turn, implies that Φ(ξ ˆ ) also by virtue of (3.12) and (2.8), Z(ξ −1 −1 lies in the kernel of N(|ξ | ξ ), on which all eigenvalues of Λ(|ξ | ξ ; U) are non-zero. Therefore, ˆ )|2 , ˆ )|2 ≤ b2 (1 + |ξ |2 )−1 |Z(ξ |Φ(ξ 1 for some constant b1 , whence
ξ ∈ Rm ,
Φ 0 ≤ b1 Z −1 .
(3.16) (3.17)
Next, we fix a partition of unity ψ0 , ψ1 , . . . , ψK subordinate to the covering Ω0 , Ω1 , . . . , ΩK , i.e. for J = 0, 1, . . . , K, ψJ ∈ C∞ (Rm ), sptψJ ⊂ ΩJ and K
ψJ2 (x) = 1,
x ∈ Rm .
(3.18)
J=0
We also fix yJ ∈ ΩJ , J = 0, 1, . . . , K, and write Rm
Z(x) P(x)Z(x) dx =
=
K m J=0 R
K m J=0 R
+
ψJ2 (x)Z(x) P(x)Z(x) dx ψJ2 (x)Z(x) P(yJ )Z(x) dx
K m J=0 R
ψJ2 (x)Z(x) [P(x) − P(yJ )]Z(x) dx.
(3.19)
By virtue of (3.10), K J=0
Rm
1 ψJ2 (x)Z(x) [P(x) − P(yJ )]Z(x) dx ≥ − μ Z 20 . 4
(3.20)
For each J = 0, 1, . . . , K, we split ψJ Z into ψJ Z = SJ + TJ ,
(3.21)
SJ = L(ψJ Φ)
(3.22)
where and
TJ = ψJ I −
m
∂α ψJ DGα (U) Φ.
(3.23)
α=1
Note that
N(|ξ |−1 ξ )SˆJ (ξ ) = i|ξ |N(|ξ |−1 ξ )Λ(|ξ |−1 ξ ; U)(ψ
J Φ) = 0,
(3.24)
so both the real and the imaginary parts of SˆJ (ξ ) are in C, for any ξ ∈ Rm and J = 0, 1, . . . , K. Thus, applying Parseval’s relation and using (3.9) results in 1 SJ (x) P(yJ )SJ (x) dx = |SJ (x)|2 dx. (3.25) SˆJ (ξ )∗ P(yJ )SˆJ (ξ ) dξ ≥ μ 2 Rm Rm Rm
......................................................
DGα (U)∂α .
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L=
m
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Furthermore, from (3.23) and (3.17) we infer, for J = 0, 1, . . . , K, |TJ (x)|2 dx ≤ b2 Z 2−1 .
8
Rm
≥
7 8
≥
7 μ 16
Rm
SJ (x) P(yJ )SJ (x) dx − 8
Rm
Rm
TJ (X) P(yJ )TJ (x) dx
|SJ (x)|2 dx − b3 Z 2−1 .
Again by (3.21) and (3.26), Rm
|SJ (x)|2 dx ≥
7 8
(3.27)
Rm
ψJ2 (x)|Z(x)|2 dx − b4 Z 2−1 .
Combining (3.18)–(3.20), (3.27) and (3.28), we arrive at (3.11). The proof is complete.
(3.28)
Proof of theorem 3.2. On Rn × Rn , we define the functions h(V, U) = η(V) − η(U) − Dη(U)[V − U],
(3.29)
Yα (V, U) = Qα (V) − Qα (U) − Dη(U)[Gα (V) − Gα (U)] − Ξ (U) Mα [V − U]
(3.30)
Γα (V, U) = Dη(U)[Gα (V) − Gα (U) − DGα (U)[V − U]],
(3.31)
and
all of quadratic order in V − U, by virtue of (2.4). We evaluate the above functions along the classical solution U and the weak solution V. Recalling that U satisfies (2.1), (2.3) and (2.5), whereas V satisfies (3.1)–(3.3), we deduce ∂t h(V, U) +
m
∂α Yα (V, U) = −∂t Dη(U)[V − U]
α=1
−
m
∂α Dη(U)[Gα (V) − Gα (U)]
α=1
−
m
∂α Ξ (U) Mα [V − U],
(3.32)
α−1
in the sense of distributions. On account of (2.1), (2.6), (2.7) and (2.3), ∂t Dη(U) = ∂t U A(U) = −
m
∂α U DGα (U) A(U)
α=1
=−
m
∂α U A(U)DGα (U) −
α=1
m
∂α Ξ (U) Mα .
(3.33)
α=1
Therefore, (3.32) and (3.31) yield ∂t h(V, U) +
m α=1
∂α Yα (V, U) ≤ −
m
∂α U Γα (V, U).
(3.34)
α=1
We fix any t ∈ F and integrate (3.34) over Rm × [0, t] (see [7]). Recalling (3.5), we obtain t h(V(x, t), U(x, t)) dx ≤ h(V0 (x), U0 (x)) dx − ∂α U Γα (V, U) dx dτ . (3.35) Rm
Rm
0 Rm
From (3.29), h(V, U) = (V − U) H(V, U)(V − U),
(3.36)
......................................................
We now return to (3.19). From (3.21), (3.25) and (3.26), it follows that ψJ2 (x)Z(x) P(yJ )Z(x) dx
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(3.26)
Rm
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where H(V, U) =
1
(3.37)
At the same time, by (2.1) and (3.1),
V(·, t) − U(·, t) −1 ≤ V0 (·) − U0 (·) −1 + ≤ V0 (·) − U0 (·) 0 +
t 0
∂τ [V(·, τ ) − U(·, τ )] −1 dτ
t m 0 α=1
≤ V0 (·) − U0 (·) 0 + c0
t
0
Gα (V(·, τ )) − Gα (U(·, τ )) 0 dτ
V(·, τ ) − U(·, τ ) 0 dτ ,
(3.39)
where c0 depends solely on O. Hence
V(·, t) − U(·, t) 2−1 ≤ 2 V0 (·) − U0 (·) 20 + 2c20 T
t 0
V(·, τ ) − U(·, τ ) 20 dτ .
Setting ω(t) = V(·, t) − U(·, t) 20 and combining (3.35), (3.38), (3.40) and (3.31) yields t ω(t) ≤ c2 ω(0) + 2λ ω(τ ) dτ ,
(3.40)
(3.41)
0
for any t ∈ F . However, as t → ω(t) is lower semicontinuous, (3.41) holds for all t ∈ [0, T]. Then, (3.8) follows by Gronwall’s lemma. The proof is complete.
4. Existence and regularity of solutions Here, we establish the existence of locally defined classical solutions to the Cauchy problem for systems of conservation laws equipped with involutions and contingent entropies, convex in the direction of the involution cone. We follow closely the treatment of this problem in [2, §5.4], but the assumptions are here slightly weaker, whereas the conclusion is improved. In fact, we demonstrate that the solutions are as regular as the solutions to systems endowed with convex entropies. Theorem 4.1. Assume that the system (2.1) is equipped with the involution (2.3) and the contingent entropy η, that is uniformly convex (2.10) in the direction of the involution cone C, defined by (2.9). Given U0 ∈ H with > m/2 + 1, satisfying the involution, there exists a unique C1 classical solution U to the Cauchy problem for (2.1), with initial data U0 , defined on a time interval [0, T∞ ), with 0 < T∞ ≤ ∞, and U(·, t) ∈
Ck ([0, T∞ ); H−k ).
(4.1)
k=0
The interval [0, T∞ ) is maximal in that if T∞ < ∞ then lim sup ∇U(·, t) L∞ = ∞. t↑T∞
(4.2)
......................................................
for any X ∈ C. In particular, X H(V, U)X ≥ On account of (3.6), there is ρ > 0 such that, for each fixed t ∈ [0, T], the oscillation of H(V(·, t), U(·, t)) over the set Ω0 = {x ∈ Rm : |x| > ρ} is less than 14 μ. Furthermore, when (3.7) holds with a sufficiently small, there exists δ > 0 such that, for each t ∈ [0, T], the oscillation of H(V(·, t), U(·, t)) over any ball of radius δ in Rm is less than 14 μ. Therefore, we may cover Rm by the union Ω0 ∪ Ω1 ∪ · · · ∪ ΩK , where, for J = 1, . . . , K, ΩJ is a ball of radius δ, so that, for each fixed t ∈ [0, T], H(V(·, t), U(·, t)) satisfies the assumptions for P in lemma 3.3. Consequently, 1 h(V(x, t), U(x, t)) dx ≥ μ V(·, t) − U(·, t) 20 − b V(·, t) − U(·, t) 2−1 . (3.38) m 8 R 1 2 2 μ|X| ,
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0
9 sA(sU + (1 − s)V) ds.
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Proof. As in [2], the solution U will be constructed by the vanishing viscosity method, namely as the ε ↓ 0 limit of solutions Uε to the parabolic system ∂α Gα (U) = εU,
(4.3)
α=1
under the same initial condition Uε (x, 0) = U0 (x), x ∈ Rm . This method is effective because (4.3) inherits the involution (2.3) from (2.1). As shown in [2, lemma 5.1.2], for any ω > 0 and U0 ∈ H with U0 < ω, the Cauchy problem for (4.3) admits a solution Uε on [0, Tε ) such that Uε (·, t) ∈ C0 ([0, Tε ); H ) ∩ L2 ([0, Tε ); H+1 )
(4.4)
and
Uε (·, t) < ω,
t ∈ [0, Tε ),
(4.5)
and [0, Tε ) is maximal in that if Tε < ∞ then Uε (·, t) → ω, as t ↑ Tε . The aim is to show that if
U0 < cω, with c sufficiently small, then, for all ε > 0, Tε > T > 0, in which case one may derive the solution of (2.1) on [0, T] by passing to the limit ε ↓ 0. We thus fix ε > 0, sufficiently small, and simplify the notation by dropping the subscript ε and denoting the solution of (4.3) simply by U. Once ω has been fixed, the range of U will be confined in a ball B of Rn . In what follows, c will denote a generic constant that may depend on bounds of G(U), η(U), Ξ (U), and their derivatives, on B. In order to secure sufficient regularity for our calculations, we assume temporarily that U0 ∈ H+2 , and therefore U(·, t) ∈ H+2 . We fix any multi-index r = (r1 , . . . , rm ), of order |r| ≤ , and apply ∂ r to (4.3). Upon setting Ur = ∂ r U, ∂t Ur +
m
∂ r ∂α Gα (U) = εUr .
(4.6)
α=1
We multiply (4.6) by ∂t Ur and integrate over Rm × [0, t], 0 < t < Tε , which yields t ε |∂t Ur |2 dx dτ + |∇Ur (x, t)|2 dx 2 Rm 0 Rm t m ε |∇U0r (x)|2 dx − ∂t Ur ∂ r ∂α Gα (U) dx dτ . = 2 Rm 0 Rm
(4.7)
α=1
By standard Sobolev space estimates,
∂ r ∂α Gα (U) 0 ≤ c U ≤ cω,
(4.8)
for any r of order |r| ≤ − 1, and τ ∈ [0, t]. Hence, applying the Cauchy–Schwarz inequality on the right-hand side of (4.7) and summing over all r of order 0 ≤ |r| ≤ − 1, we deduce t
∂t U 2−1 dτ ≤ (ε + ct)ω2 . (4.9) 0
In turn, (4.9) implies
U(·, t) − U0 (·) 2−1 ≤ t(ε + ct)ω2
(4.10)
max |U(x, t) − U0 (x)| ≤ c(ε + t)1/2 t1/2 ω.
(4.11)
and x∈Rm
Next, we rewrite (4.6) as ∂t Ur +
m α=1
DGα (U)∂α Ur =
m α=1
{DGα (U)∂α Ur − ∂ r [DGα (U)∂α U]} + εUr ,
(4.12)
......................................................
m
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∂t U +
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multiply by Ur A(U) and integrate over Rm × [0, t]. We note the identities
and m
Ur A(U)Ur =
∂α [Ur A(U)∂α Ur ] −
α=1
m
∂α Ur A(U)∂α Ur −
α=1
m
Ur ∂α A(U)∂α Ur .
(4.14)
α=1
Furthermore, recalling (2.3) and (2.7), m
2Ur A(U)DGα (U)∂α Ur =
α=1
m
2Ur [A(U)DGα (U) + DΞ (U) Mα ]∂α Ur
α=1
=
m
∂α {Ur [A(U)DGα (U) + DΞ (U) Mα ]Ur }
α=1
−
m
Ur ∂α [A(U)DGα (U) + DΞ (U) Mα ]Ur .
(4.15)
α=1
Therefore, 1 2
Rm
Ur (x, t)A(U(x, t))Ur (x, t) dx + ε
+ε
m Rm α=1
1 2
=
−
+
m Rm α=1
∂α Ur (x, t)A(U(x, t))∂α Ur (x, t) dx
Ur (x, t)∂α A(U(x, t))∂α Ur (x, t) dx
U0r (x)A(U0 (x))U0r (x) dx +
Rm
1 2
t
m
0 Rm α=1
t
0 Rm
1 2
t 0 Rm
Ur ∂α A(U)Ur dx dτ
Ur ∂α [A(U)DGα (U) + DΞ (U) Mα ]Ur dx dτ
Ur A(U)
m
{DGα (U)∂α Ur − ∂ r [DGα (U)∂α U]} dx dτ .
(4.16)
α=1
We fix r of order |r| = . By standard Sobolev space estimates,
DGα (U)∂α Ur − ∂ r [DGα (U)∂α U] 0 ≤ c ∇U L∞ U .
(4.17)
From (4.11) and since |U0 (x)| → 0 as |x| → ∞, we deduce that when ρ is sufficiently large and T is sufficiently small, the oscillation of A(U(·, t)) over the set Ω0 = {x ∈ Rm : |x| > ρ} is less than 12 μ, for any t ∈ [0, T]. Furthermore, as ∇U L∞ is bounded by cω, the oscillation of A(U(·, t)) over any ball of sufficiently small radius δ is likewise less than 12 μ. These, together with (2.10) imply that 1 2 A(U(·, t)) satisfies the assumptions for P(·) in lemma 3.3. Therefore, 1 2
Rm
1 Ur (x, t)A(U(x, t))Ur (x, t) dx ≥ μ Ur (·, t) 20 − b U(·, t) 2−1 , 8
(4.18)
and
m Rm
α=1
1 ∂α Ur (x, t)A(U(x, t))∂α Ur (x, t) dx ≥ μ ∇Ur (·, t) 20 − 2b U(·)t 2 . 4
(4.19)
......................................................
(4.13)
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t ∈ [0, T].
(4.20)
Thus, if c U0 2 < ω2 and T is sufficiently small, then (4.20) yields U(·, t) < ω, for t ∈ [0, T] and any ε 1. We have now laid the preparation for constructing the solution to the Cauchy problem for (2.1), under initial data U0 ∈ H , on the time interval [0, T]. For that purpose, we fix a sequence {U0ν } in H+2 and a sequence {εν }, such that U0ν → U0 in H and εν → 0, as ν → ∞. We let Uν denote the solution of (4.3), with ε = εν and initial data U0ν . As shown earlier, {Uν } is contained in a bounded set of L∞ ([0, T]; H ) and {∂t Uν } is contained in a bounded set of L2 ([0, T]; H−1 ). Thus, {Uν } is uniformly equicontinuous on Rm × [0, T]. Hence, we may extract a subsequence, again denoted by {Uν }, that converges to a continuous function U, uniformly on compact subsets of Rm × [0, T]. Clearly, U is a distributional solution of (2.1) on Rm × [0, T], with initial value U0 . Furthermore, U(·, t) ∈ L∞ ([0, T]; H ), and this together with (2.1) implies U(·, t) ∈
W k,∞ ([0, T]; H−k ).
(4.21)
k=0
In fact, t → U(·, t) is weakly continuous in H on [0, T]. We may extend the domain of this solution beyond [0, T], by restarting at t = T, with initial data U(·, T), and repeating the above process. On account of (4.20), this may proceed for as long as ∇U L∞ stays bounded. Therefore, the maximal interval [0, T∞ ) of existence will be finite when ∇U(·, t) blows up as t ↑ T∞ , i.e. (4.2) holds. It remains to prove that the solution U is endowed with regularity (4.1). As (2.1) is invariant under time translations and reflections, it will suffice to show that t → U(·, t) is right-continuous in H at t = 0. We begin with a sequence {U0ν } in H+2 , such that U0ν → U0 in H , as ν → ∞, and consider the solution Uν of the Cauchy problem for (2.1), with initial data U0ν , defined on a time interval [0, T], independent of ν. By virtue of (4.11) and the uniform boundedness of ∇Uν , for T small, both the asymptotic and local oscillations of A(U(·, t)) and A(Uν (·, t)) are uniformly small. We may thus apply theorem 3.2 to deduce that Uν (·, t) → U(·, t) strongly in H0 , for t ∈ [0, T]. On the other hand, Uν (·, t) → U(·, t) weakly in H . It follows that Uν (·, t) → U(·, t),
strongly in H−1 , t ∈ [0, T].
(4.22)
We now fix r of order |r| ≤ , and apply ∂ r to (2.1) thus arriving at (4.12), with ε = 0. Since Uν (·, t) ∈ H+2 , there is sufficient regularity to justify (4.13) and (4.15), so that Uν satisfies (4.16), with ε = 0. In particular, Uνr (x, t)A(Uν (x, t))Uνr (x, t) dx = U0νr (x)A(U0ν (x))U0νr (x) dx + O(t). (4.23) Rm
Rm
Next, we write the identity [Uνr (x, t) − Ur (x, t)] A(U(x, t))[Uνr (x, t) − Ur (x, t)] dx Rm
=
Uνr (x, t)A(Uν (x, t))Uνr (x, t) dx −
Rm
−
Rm
Rm
Ur (x, t)A(U(x, t))Ur (x, t) dx
Uνr (x, t)[A(Uν (x, t)) − A(U(x, t))]Uνr (x, t) dx
−2
Rm
[Uνr (x, t) − Ur (x, t)]A(U(x, t))Ur (x, t) dx
(4.24)
......................................................
U(·, t) 2 ≤ c U0 (·) 2 + ct[ε + t + ∇U L∞ ]ω2 ,
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Next, we sum (4.16) over all r of order |r| = , use (4.18) and (4.19) to estimate the left-hand side, use (4.17) and Schwarz’s inequality to estimate the right-hand side, and combine the result with (4.10) to conclude that for ε 1,
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1 ≥ μ Uνr (·, t) − Ur (·, t) 20 − 2b Uν (·, t) − U(·, t) 2−1 . 4
(4.25)
As ν → ∞, the last term on the right-hand side of (4.25) tends to zero, by virtue of (4.22). We thus conclude Ur (x, t)A(U(x, t))Ur (x, t) dx ≤ lim inf Uνr (x, t)A(Uν (x, t))Uνr (x, t) dx. (4.26) ν→∞
Rm
Combining (4.23) with (4.26) yields Ur (x, t)A(U(x, t))Ur (x, t) dx ≤ Rm
Rm
Rm
U0r (x)A(U0 (x))U0r (x) dx + O(t).
(4.27)
Finally, we write the identity [Ur (x, t) − U0r (x)] A(U0 (x))[Ur (x, t) − U0r (x)] dx Rm
=
Ur (x, t)A(U(x, t))Ur (x, t) dx −
Rm
−
Rm
Rm
U0r (x)A(U0 (x))U0r (x) dx
Ur (x, t)[A(U(x, t)) − A(U0 (x))]Ur (x, t) dx
−2
Rm
[Ur (x, t) − U0r (x)] A(U0 (x))U0r (x) dx
(4.28)
and let t ↓ 0. On the right-hand side, the third term tends to zero, by the Lebesgue dominated convergence theorem, as U(x, t) → U0 (x); and the last term also tends to zero, as Ur (·, t) → U0r (·) weakly in H0 ; furthermore, the limit superior of the difference of the first two terms is non-positive, because of (4.27). For the left-hand side, lemma 3.3 yields [Ur (x, t) − U0r (x)] A(U0 (x))[Ur (x, t) − U0r (x)] dx Rm
1 ≥ μ Ur (·, t) − U0r (·) 20 − 2b U(·, t) − U0 (·) 2−1 . 4
(4.29)
Recalling (4.10), we conclude that Ur (·, t) − U0r (·) 0 → 0, as t ↓ 0, i.e. U(·, t) → U0 (·), strongly in H . The proof is complete.
References 1. Benzoni-Gavage S, Serre D. 2007 Multi-dimensional hyperbolic partial differential equations. Oxford, UK: Oxford University Press. 2. Dafermos CM. 2010 Hyperbolic conservation laws in continuum physics, 3rd edn. Heidelberg, Germany: Springer. 3. Dafermos CM. 1986 Quasilinear hyperbolic systems with involutions. Arch. Ration. Mech. Anal. 94, 373–389. (doi:10.1007/BF00280911) 4. Boillat G. 1988 Involutions des systèmes conservatifs. C.R. Acad. Sci. Paris, Série I 307, 891–894. 5. Serre D. 2004 Hyperbolicity of the non-linear models of Maxwell’s equations. Arch. Ration. Mech. Anal. 172, 309–331. (doi:10.1007/s00205-003-0303-4) 6. Ball JM. 1977 Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63, 337–403. (doi:10.1007/BF00279992) 7. Chen G-Q, Frid H. 1999 Divergence measure fields and conservation laws. Arch. Ration. Mech. Anal. 147, 89–118. (doi:10.1007/s002050050146)
......................................................
Rm
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and let ν → ∞. On the right-hand side, the third term tends to zero, by the Lebesgue dominated convergence theorem, as Uν → U; and the last term also tends to zero, as Uνr (·, t) → Ur (·, t), weakly in H0 . On the left-hand side, by lemma 3.3, [Uνr (x, t) − Ur (x, t)] A(U(x, t))[Uνr (x, t) − Ur (x, t)] dx
