HYPERBOLIC BALANCE LAWS WITH RELAXATION

. This expository paper surveys the progress in a research program aiming at establishing the existence and long time behavior of BV solutions to the Cauchy problem for hyperbolic systems of balance laws modeling relaxation phenomena.


Introduction. A system of conservation laws in one space dimension has the form
where U is the state vector and F is the flux. The system is strictly hyperbolic when the eigenvalues of the Jacobian matrix of the flux are real and distinct. Peter Lax has made major contributions in that area of partial differential equations. In fact, it is fair to state that his seminal paper [15], which coined the term "hyperbolic system of conservation laws" set the directions for the active development of this field over the past fifty years.
The root of the difficulties encountered in the analysis of hyperbolic conservation laws with nonlinear flux lies in their distinctive feature of wave breaking: Even when the initial values of the state vector are smooth, solutions to the Cauchy problem (1.1), (1.2) develop spontaneously singularities that propagate on as shock waves. The state of the art is that when the total variation of U 0 is sufficiently small there exists a unique admissible weak solution of class BV on the upper half-plane (−∞, ∞) × [0, ∞), which may be constructed by any one of the following three methods: (a) the random choice algorithm of Glimm [14], improved by several authors, most notably Liu [16]; (b) front tracking, developed by the Italian school headed by Bressan [4]; and (c) the vanishing viscosity approach of Bianchini and Bressan [2]. The key estimate is T V U (·, t) ≤ cT V U 0 (·), 0 ≤ t < ∞. (1. 3) The restriction that T V U 0 must be small is essential, as there are cases of systems in which specially constructed initial data with large variation generate solutions with variation that explodes in finite time. There are indications that this pathology may be present even in the classical Euler equations of gas dynamics.
Turning from systems (1.1) of conservation laws to systems ∂ t U (x, t) + ∂ x F (U (x, t)) + P (U (x, t)) = 0 (1.4) of balance laws, with source P , it is a routine matter to construct local solutions to the Cauchy problem by employing any one of the aforementioned three methods, in conjunction with an operator splitting scheme. However, it is not generally possible to extend local to global solutions, because in the presence of the source the estimate (1.3) must be revised into (1.5) where µ is typically positive. Consequently, as time increases, T V U (·, t) may eventually exceed the range of applicability of the method. The present paper will survey a research program by the author with objective to establish the existence and long time behavior of BV solutions to the Cauchy problem for hyperbolic systems of balance laws (1.4) modeling relaxation phenomena. In such systems, the effect of the source is dissipative and this raises the expectation for the existence of solutions in the large.
In a parallel direction, there is voluminous literature on the existence of classical solutions in the large to the Cauchy problem for systems in the above class, under initial data that are smooth and small. For the state of the art and references to earlier seminal work, the reader may consult [3]. The theories of classical and BV weak solutions may share goals, but technically are quite distinct, as the former relies on Sobolev space estimates, while the latter hinges on L 1 and BV type bounds.
The challenge to the analysis stems from the fact that in the systems of interest the source incurs only partial damping. This can be seen by noting that systems (1.4) modeling relaxation phenomena typically result from the coupling of conservation laws with balance laws and thereby assume the form It is clear that the source term cannot affect directly the damping of the V component of the state vector. This difficulty may be overcome by exploiting the synergy between source and flux encoded in the so-called Kawashima condition.
The existence and long time behavior of BV solutions to the Cauchy problems for systems (1.4) in the form (1.6) is established in Sections 2,3 and 4, under the assumption that the V component carries zero "mass". The complications arising when the V component is allowed to carry nonzero "mass" are discussed in Section 5. Section 6 is a survey of similar results for certain systems arising in Physics, modeling continuous media with memory, in which the flux depends not only on the present value but also on the past history of the state vector.
For systems with parametrized source, a central problem that is still open, at least in the BV setting, is the behavior of solutions as the relaxation parameter µ tends to zero. An answer, in the context of a special system and very special initial data, is found in [18].
2. The Cauchy problem. We consider systems of strictly hyperbolic balance laws (1.4) under the following assumptions, typical for systems modeling relaxation phenomena: The state vector U takes values in R n . The flux F is a given smooth function from R n to R n . For any U ∈ R n , the Jacobian matrix DF (U ) possesses real eigenvalues λ 1 (U ) < · · · < λ n (U ) and thereby linearly independent sets of left (row) eigenvectors L 1 (U ), . . . , L n (U ) and right (column) eigenvectors R 1 (U ), . . . , R n (U ), normalized by The source P , also a given smooth function from R n to R n , vanishes at the origin, P (0) = 0, so that U ≡ 0 is an equilibrium solution of (1.4). Typically, the null space of DP (0) has dimension k, 1 ≤ k < n, in which case one may assume, without loss of generality, that (1.4) is of the form with V taking values in R k , W taking values in R , = n − k, and X, Y of quadratic order at the origin. In fact, as noted in the Introduction, these systems often appear in the form (1.6).
The flux and the source are coupled by the Kawashima condition which guarantees that the linearized system does not admit solutions in the form u(x − λ i (0)t)R i (0), manifesting undamped propagating fronts.
The system (1.4) is endowed with an entropy-entropy flux pair (η, q), with η normalized by η(0) = 0, Dη(0) = 0. Furthermore, the Hessian D 2 η(0) is positive definite, so η is convex on some neighborhood of the origin. Consequently, admissible solutions of (1.4) with values in a small neighborhood of the origin must satisfy the entropy inequality The source is dissipative in that the entropy production DηP satisfies with a > 0, for U in some neighborhood of the origin. Since the null space of DP (0) is nontrivial, the entropy production is merely positive semidefinite and the damping exerted unilaterally by the source is only partial. To achieve full damping will require the synergy between source and flux expressed by the Kawashima condition (2.3).
We assign initial data U 0 with bounded variation, that decay as |x| → ∞ sufficiently fast to render the integral finite. The analysis is simplified substantially when the system (1.4) is in the form (1.6) and the V 0 component of U 0 carries zero "mass": (2.10) (2.14) As already noted in the Introduction, the existence of a local solution to the Cauchy problem (1.4), (1.2) follows from a tedious, but fairly routine, application of the standard algorithms for treating hyperbolic systems of conservation laws, in conjunction with operator splitting, so as to account for the effects of the source. This analysis is found, for instance, in [13]. Thus the challenge for proving Theorem 2.1 lies in establishing the estimate (2.12), which will allow for extending the local solution into a global one. The derivation of this estimate will be the task of the following two sections, 3 and 4. In Section 5 we shall consider the complications arising when the system (1.4) is in its most general form (2.2) and/or the restriction (2.10) on the initial data is removed.
3. L 1 estimates. Under the conditions of Theorem 2.1, assuming that U = (V, W ) is a solution to the Cauchy problem for (1.6), with initial values U 0 = (V 0 , W 0 ), we establish here the assertions (2.11) and (2.13).
We will employ the "potential" function Clearly, Ψ is Lipschitz, with derivatives The role of the assumption (2.10) is to secure that the initial value Ψ 0 (x) of Ψ satisfies Ψ 0 (±∞) = 0. Then, integrating by parts, so that, by virtue of (2.9), We will be operating under the ansatz that, for some ω > 0, which will be verified later. The first step is to show By virtue of (2.7), integration of the inequality (2.6) over (−∞, ∞) × [0, ∞) readily yields that |W | 2 dxdt is bounded by cσ 2 . However, showing that this will also be the case for the complementary component V will require considerable effort, as it rests on the synergy between the partially dissipative source and the flux, encoded in the Kawashima condition. The following argument has been adapted from [17].
We introduce the notations (3.9) The k × k matrix K and the × matrix M are symmetric and positive definite. Furthermore, since D 2 η(U )DF (U ) is symmetric and η V W (0, 0) = 0, is an eigenvector of DF (0) with DP (0)R = 0, in contradiction to the Kawashima condition. It may then be shown (see [17 ]) that there exists a k × k matrix Ω such that ΩK is skew-symmetric and ΩKB is positive on the kernel of M E. We now define the following functions: where κ and γ are positive constants to be fixed below.
A lengthy but straightforward calculation, using (1.6), (3.2) and (2.6), yields We perform a (finite) Taylor expansion of Π(V, W, Ψ) about the origin. Using the symmetry relations (3.10) and recalling that |U | < ρ and |Ψ| < ω, we obtain The crucial observation is that the second term on the right-hand side of (3.16) is positive on the kernel of M E and the first term is positive on the complementary space, whence, for κ sufficiently small, Λ is positive definite. We thus fix ρ and ω sufficiently small and γ sufficiently large so that both Θ and Π become positive definite at the origin. Then, integrating (3.14) over (−∞, ∞) × [0, ∞) and using (2.9) and (3.4), we arrive at (3.6). The next step is to show that, always under the ansatz (3.5), for ω as fixed above, On account of (2.6) and (2.7), Integrating the above inequality over (−∞, ∞) × [0, t] and using (3.6), we arrive at (3.19). Now fix any > 0. By virtue of (3.6), there exists τ > 0 such that whence (3.20) follows.
We have now laid the preparation for establishing (2.11) and (2.13), while also verifying, in the process, the ansatz (3.5).
We identify λ > 0 such that holds for all U in a neighborhood of the origin.

Redistribution of damping and BV bounds.
To complete the proof of Theorem 2.1, we need to verify (2.12) and (2.14). For guidance, we turn to the linearized system (2.4). We assemble the left eigenvectors L 1 (0), . . . , L n (0) and the right eigenvectors R 1 (0), . . . , R n (0) as rows and columns of n × n matrices L(0) and R(0). In terms of the new state vector V = L(0)R, (2.4) becomes with 2) It is easily seen that if A is column-diagonally dominant, namely then the L 1 norm, and thereby also the total variation, of the solution V to the Cauchy problem for (4.1) are time-decreasing. In fact, it is known [1,5,13] that when (4.3) holds, the Cauchy problem for the nonlinear system (1.4), with initial data of sufficiently small total variation, possesses a unique admissible BV solution U on (−∞, ∞) × [0, ∞) and for some ν > 0. Unfortunately, the condition (4.3) is far too restrictive, as it presumes that the damping is widely distributed among the equations of the system, and is thus incompatible with systems in the form (1.6). The plan here is to devise a transformation of the state vector that redistributes the damping more equitably among the equations of the system. The first step is to show that (2.7) together with (2.3) imply is positive semidefinite. Multiplying, from the left, (2.5) by R i yields which shows that R i D 2 η is collinear to L i , and in particular We now multiply (4.7), from the left by R i (0) and from the right by R i (0). Using that D 2 θ(0) is positive semidefinite, together with (4.8) at U = 0, we conclude (4.10) Let us now assume that U is an admissible BV solution of the Cauchy problem (1.4), (1.2), on the upper half-plane, taking values in a ball of small radius ρ, centered at the origin. We introduce the functions U (y, t))dy, (4.12) where N is a n × n matrix to be specified below. We note that Φ is Lipschitz with t). (4.13) We replace U by the new state vector and rewrite (1.4) as a system forÛ , namely The motivation for switching from the relatively simple (1.4) to the cumbersome (4.15), which is not even a closed system, is that if one presumes that Φ and Z are somehow known, and regards (4.15) as an inhomogeneous system of balance laws forÛ , then, in the place of (4.2), one gets the matrixÂ with entrieŝ (4.18) where ∆ = L(0)N R(0). This presents the opportunity of makingÂ diagonally dominant by properly selecting the matrix N . In particular, the choice N = R(0)∆L(0), with ∆ ii = 0, for i = 1, . . . , n and renders a diagonalÂ, withÂ ii = A ii , for i = 1, . . . , n, andÂ ij = 0, for i = j. In that case, when (4.5) holds,Â is diagonally dominant. The principal task is to solve the Cauchy problem for the inhomogeneous system (4.15) of balance laws, estimating, in the process, the variation of the solutionÛ . IfF andP did not depend on Φ and Z, the diagonal dominance ofÂ would induce exponential decay in the variation ofÛ , as in (4.4). The difficulty stems from the fact that Φ and Z are not known in advance, as they depend on the solution. However, the saving grace is that the variation of these terms is already under control. Indeed, on account of (4.11), (4.12) and (2.11), we have In particular, (4.14) and (4.20) allow us to relate the variations of U andÛ : The construction ofÛ is attained by combining the random choice method of Glimm with an operator splitting algorithm, and it is lengthy and technical. It can be found in [6,7,12 ]. The conclusion is summed up in the following estimate: When ρ is sufficiently small, (4.21) implies the desired estimate (2.12). The assertion (2.14) follows easily from (2.12) and (2.13).
5. Nonzero mass. The proof of Theorem 2.1, and in particular the analysis in Section 3, rely heavily on the assumption that the system is in the form (1.6) and the mass carried by the V component is zero. The last condition is induced by (2.10). The decay (2.13) of the L 1 norm is a direct consequence of the above. The situation is substantially more complicated for systems (1.4) in the more general form (2.2), or even for systems in the form (1.6) when the assumption (2.10) is relaxed. We shall illustrate this here in the context of an example.
We consider the Cauchy problem for the simple system   where p (u) < 0, and α is a constant, positive, negative or zero. In the place of Theorem 2.1, we here have

3)
with σ and δ sufficiently small. Then there exists an admissible BV solution (u, v) where c 0 , c 1 , ν and b are positive constants, independent of the initial data, and θ is the solution to the heat equation, with M some constant depending on (u 0 , v 0 ).
Sketching the proof of the above theorem, we discuss, for simplicity, only the special case α = 0, so that (5.1) is still in the form (1.6). However, we no longer impose (2.10), assuming instead The details of the proof, together with the treatment of the general case α = 0, are found in [11].
The objective is to demonstrate that, for any 0 < t < ∞, where (û,v) is the solution to the system with the same initial values (u 0 , v 0 ) as (u, v). Thusû satisfies the porous media equation By means of lengthy but straightforward analysis, involving elementary "energy" estimates, it is possible to establish bounds for the solutionû of (5.10) and thereby forv, including the following: where θ is defined by (5.6). Thus proving (5.8) will establish the assertion (5.4). The asserted decay (5.5) in the variation will then easily follow.
With an eye to verifying (5.8), we set w = u −û and z = v −v, noting that (w, z) with zero initial data. In (5.16),p stands for the "relative pressure" defined bŷ The admissibility of solutions to (5.16) is encoded in the "relative entropy" inequality A priori bounds on (w, z) will be derived by combining (5.18) with the balance law (5.20) where Φ is the "potential function" Terms with the "good sign" appearing in (5.18) and (5.20) includeψ(w,û), z 2 , Φ 2 and −p(w,û)w. It is easy to see that, when σ is sufficiently small, the terms of indefinite sign may be balanced, with the help of (5.11), (5.12) and (5.13), against the terms with the good sign, yielding bounds To get the next round of estimates, we multiply (5.18), first by t and then by x 2 , which yields With the help of (5.22) and (5.23), together with (5.11), (5.12) and (5.13), one may balance the terms of indefinite sign in (5.24) and (5.25) against the terms tψ, x 2ψ , tz 2 and x 2 z 2 , with the good sign, to obtain the estimate (5.26) Finally, we apply Schwarz's inequality to (5.26) to get The fact that at the L 1 level solutions to systems (5.1) behave asymptotically as heat kernels is not unexpected, as similar behavior of classical solutions is familiar. The counterpart of Theorem 5.1 for general systems (1.4) is still unknown.
6. Viscous relaxation. The methodology expounded in the previous sections also applies to systems of conservation laws modeling viscous relaxation of the Boltzmann type, in which the flux depends not only on the present value but also on the past history of the state vector: We shall illustrate this by means of two examples. First we consider the system   which governs longitudinal oscillations of viscoelastic bars and shearing motions of viscoelastic slabs.
To avoid technicalities, we assume that the relaxation kernel a is in the form with α i > 0 and 0 < λ 1 < · · · < λ n . The functions f (u) and g(u) = f (u) − a(0)u, which encode the instantaneous and the relaxed elastic response of the medium must be increasing, f (u) > 0, g (u) > 0. Consequently, the system (6.2) is of "hyperbolic" type.
The memory has a dissipative effect. In particular, admissible solutions of (6.2) satisfy the entropy inequality Notice that, as in earlier sections, the entropy production is positive semidefinite but not positive definite. For simplicity, we assign initial conditions where u 0 is a function of bounded variation that decays as |x| → ∞ sufficiently fast to render the integral finite. T V u(·, t) + T V v(·, t) ≤ c 1 σ + c 2 δ, 0 ≤ t < ∞. (6.13) Furthermore, T V u(·, t) + T V v(·, t) → 0, as t → ∞. (6.14) The proof to the above theorem (under slightly different assumptions) is found in [8,10]. It follows the same pattern as the proof of Theorem 2.1 but requires new estimates.