BV Solutions for a Class of Viscous Hyperbolic Systems

The paper is concerned with the Cauchy problem for a nonlinear, strictly hyperbolic system with small viscosity: ut +A(u)ux = ε uxx, u(0, x) = ū(x). (∗) We assume that the integral curves of the eigenvectors ri of the matrix A are straight lines. On the other hand, we do not require the system (∗) to be in conservation form, nor do we make any assumption on genuine linearity or linear degeneracy of the characteristic fields. In this setting we prove that, for some small constant η0 > 0 the following holds. For every initial data ū ∈ L with Tot.Var.{ū} < η0, the solution u of (∗) is well defined for all t > 0. The total variation of u(t, ·) satisfies a uniform bound, independent of t, ε. Moreover, as ε → 0+, the solutions u(t, ·) converge to a unique limit u(t, ·). The map (t, ū) 7→ Stū . = u(t, ·) is a Lipschitz continuous semigroup on a closed domain D ⊂ L of functions with small total variation. This semigroup is generated by a particular Riemann Solver, which we explicitly determine. The above results can also be applied to strictly hyperbolic systems on a Riemann manifold. Although these equations cannot be written in conservation form, we show that the Riemann structure uniquely determines a Lipschitz semigroup of “entropic” solutions, within a class of (possibly discontinuous) functions with small total variation. The semigroup trajectories can be obtained as the unique limits of solutions to a particular parabolic system, as the viscosity coefficient approaches zero. The proofs rely on some new a priori estimates on the total variation of solutions for a parabolic system whose components drift with strictly different speeds. 0 1 Introduction Consider a strictly hyperbolic n× n system of conservation laws in one space dimension: ut + f(u)x = 0. (1.1) For initial data with small total variation, the global existence of weak solutions was proved in [8]. Moreover, the uniqueness and stability of entropy admissible BV solutions was recently established in a series of papers [3,4,5,6]. A long standing open question is whether these discontinuous solutions can be obtained as vanishing viscosity limits. More precisely, given a smooth initial data ū : IR 7→ IR with small total variation, consider the parabolic Cauchy problem u(0, x) = ū(x). (1.2) ut +A(u)ux = ε uxx. (1.3) Here A(u) . = Df(u) is the Jacobian matrix of f and ε > 0. It is then natural to expect that, as ε→ 0, the solution u of (1.2)-(1.3) converges to the unique entropy weak solution u of (1.1)-(1.2). Unfortunately, no general theorem in this direction is yet known. Some of the main results available in the literature are listed below. 1) In the case of a scalar conservation law, the entropic solutions of (1.1) determine a semigroup which is contractive w.r.t. the L-distance. In this case, a general convergence theorem for vanishing viscosity approximations was proved in the classical work of Kruzhkov [12]. 2) For various 2 × 2 systems, if a uniform L∞–bound on all functions u is available, one can consider a weak limit u ⇀ u. By a compensated compactness argument introduced by DiPerna [7], it then follows that u is actually a weak solution of the nonlinear system (1.1). For a comprehensive discussion of the compensated compactness method and its applications to conservation laws, see [18]. 3) For n × n Temple class systems, a proof of the convergence of the viscous solutions u to a solution of (1.1) can be found in [17,18]. 4) Assume that all characteristic fields of the system (1.1) are linearly degenerate. Then every solution with small total variation which is initially smooth remains smooth for all positive times [2]. Clearly such solution can be obtained as limit of vanishing viscosity approximations. By a density argument it follows that every weak solution of (1.1) with sufficiently small total variation is a limit of viscous approximations. 5) For a general n× n strictly hyperbolic system, let u be a piecewise smooth entropic solution of (1.1) with jumps along a finite number of smooth curves in the t-x plane. Thanks to this 1 additional regularity assumptions on u, it was proved in [10] that there exists a family of viscous solutions u converging to u in Lloc as ε→ 0. From our point of view, the major difficulty toward a general proof of the convergence u → u lies in deriving an a priori estimate on the total variation of the solution of (1.2)-(1.3), uniformly valid as ε → 0. To fix the ideas, assume ū ∈ C∞ c with Tot.Var.(ū) sufficiently small. Performing the rescalings t 7→ t/ε, x 7→ x/ε, the Cauchy problem becomes ut +A(u)ux = uxx, (1.4) u(0, x) = ū(εx). (1.5) Observe that, as ε → 0, the initial data u(0, ·) has constant total variation, all of its derivatives approach zero, but its L-norm approaches infinity. We thus need estimates on the total variation of a solution u(t, ·) of (1.4) which are independent of the L-norm of the initial data. To illustrate the heart of the matter, let us denote by λ1(u) < · · · < λn(u) the eigenvalues of the n × n Jacobian matrix A(u) . = Df(u), and call l, . . . , l, r1, . . . , rn, its left and right eigenvectors, normalized so that ∣∣ri(u)∣∣ ≡ 1, l(u) · rj(u) = { 1 if i = j, 0 if i 6= j. (1.6) The directional derivative of a function φ = φ(u) in the direction of the eigenvector ri is written ri • φ(u) . = lim h→0 φ ( u+ hri(u) ) − φ(u) h . Moreover, by ux . = l(u) · ux we denote the i-th component of the gradient ux w.r.t. the basis of right eigenvectors {r1, . . . , rn}. Recalling (1.6), this implies ux = ∑