Structure of Entropy Solutions for Multi-Dimensional Scalar Conservation Laws

An entropy solution u of a multi-dimensional scalar conservation law is not necessarily in BV, even if the conservation law is genuinely nonlinear. We show that u nevertheless has the structure of a BV function in the sense that the shock location is codimension-one rectifiable. This result highlights the regularizing effect of genuine nonlinearity in a qualitative way; it is based on the locally finite rate of entropy dissipation. The proof relies on the geometric classification of blow-ups in the framework of the kinetic formulation.     

Entropy Solutions for Nonlinear Degenerate Problems

We consider a class of elliptic-hyperbolic degenerate equations g(u)-Db(u) +\divgf(u) = fg(u)-\Delta b(u) +\divg\phi (u) =f with Dirichlet homogeneous boundary conditions and a class of elliptic-parabolic-hyperbolic degenerate equations g(u)_t-Db(u) +\divgf(u) = fg(u)_t-\Delta b(u) +\divg\phi (u) =f with homogeneous Dirichlet conditions and initial conditions. Existence of entropy solutions for both problems is proved for nondecreasing continuous functions g and b vanishing at zero and for a continuous vectorial function J satisfying rather general conditions. Comparison and uniqueness of entropy solutions are proved for g and b continuous and nondecreasing and for J continuous.     

Boundary Layers in Weak Solutions of Hyperbolic Conservation Laws

. This paper is concerned with the initial‐boundary‐value problem for a nonlinear hyperbolic system of conservation laws. We study the boundary layers that may arise in approximations of entropy discontinuous solutions. We consider both the vanishing‐viscosity method and finite‐difference schemes (Lax‐Friedrichs‐type schemes and the Godunov scheme). We demonstrate that different regularization methods generate different boundary layers. Hence, the boundary condition can be formulated only if an approximation scheme is selected first. Assuming solely uniform bounds on the approximate solutions and so dealing with solutions, we derive several entropy inequalities satisfied by the boundary layer in each case under consideration. A Young measure is introduced to describe the boundary trace. When a uniform bound on the total variation is available, the boundary Young measure reduces to a Dirac mass. From the above analysis, we deduce several formulations for the boundary condition which apply whether the boundary is characteristic or not. Each formulation is based on a set of admissible boundary values, following the terminology of Dubois & LeFloch[15]. The local structure of these sets and the well‐posedness of the corresponding initial‐boundary‐value problem are investigated. The results are illustrated with convex and nonconvex conservation laws and examples from continuum mechanics. (Accepted July 2, 1998)     

Uniqueness of Weak Solutions to Systems of Conservation Laws

Consider a strictly hyperbolic system of conservation laws in one space dimension: Relying on the existence of the Standard Riemann Semigroup generated by , we establish the uniqueness of entropy-admissible weak solutions to the Cauchy problem, under a mild assumption on the variation of along space-like segments.     

Strong Traces for Solutions of Multidimensional Scalar Conservation Laws

In this paper we consider multidimensional scalar conservation laws without BV estimates defined in a subset Ω??⁺×? ^d . We show that, with a non-degeneracy hypothesis on the flux, we can define a strong notion of trace at the boundary of Ω reached by L ¹ convergence.     

Dissipative and Entropy Solutions to Non-Isotropic Degenerate Parabolic Balance Laws

We propose a new notion of weak solutions (dissipative solutions) for non-isotropic, degenerate, second-order, quasi-linear parabolic equations. This class of solutions is an extension of the notion of dissipative solutions for scalar conservation laws introduced by L. C. Evans. We analyze the relationship between the notions of dissipative and entropy weak solutions for non-isotropic, degenerate, second-order, quasi-linear parabolic equations. As an application we prove the strong convergence of a general relaxation-type approximation for such equations.     

Long-time Stability in Systems of Conservation Laws,Using Relative Entropy/Energy

We study the long-time stability of shock-free solutions of hyperbolic systems of conservation laws, under an arbitrarily large initial disturbance in L ²∩ L ^∞. We use the relative entropy method, a robust tool which allows us to consider rough and large disturbances. We display practical examples in several space dimensions, for scalar equations as well as isentropic gas dynamics. For full gas dynamics, we use a trick from Chen [1], in which the estimate is made in terms of the relative mechanical energy instead of the relative mathematical entropy.     

Global BV Entropy Solutions and Uniqueness for Hyperbolic Systems of Balance Laws

We consider the Cauchy problem for n×n strictly hyperbolic systems of nonresonant balance laws each characteristic field being genuinely nonlinear or linearly degenerate. Assuming that and are small enough, we prove the existence and uniqueness of global entropy solutions of bounded total variation as limits of special wave-front tracking approximations for which the source term is localized by means of Dirac masses. Moreover, we give a characterization of the resulting semigroup trajectories in terms of integral estimates.     

Zero-Dissipation Limit of Solutions with Shocks for Systems of Hyperbolic Conservation Laws

We consider the convergence of solutions of conservation laws with viscosity to solutions having shocks of hyperbolic conservation laws without viscosity as the viscosity tends to zero. Our analysis reveals a rich structure of nonlinear wave interactions due to the presence of shocks and initial layers. These interactions generate four different wave patterns: initial layers, shock layers, diffusion waves and coupling waves. We study the propagation and interactions of the four wave patterns by a detailed pointwise analysis. (Accepted February 19, 1998)     

Strong Traces for Solutions to Scalar Conservation Laws with General Flux

In this paper we consider bounded weak solutions u of scalar conservation laws, not necessarily of class BV, defined in a subset . We define a strong notion of trace at the boundary of reached by L ¹ convergence for a large class of functionals of u, G(u). The functionals G depend on the flux function of the conservation law and on the boundary of . The result holds for a general flux function and a general subset.     

Structure of Solutions of Multidimensional Conservation Laws with Discontinuous Flux and Applications to Uniqueness

We investigate the structure of solutions of conservation laws with discontinuous flux under quite general assumption on the flux. We show that any entropy solution admits traces on the discontinuity set of the coefficients and we use this to prove the validity of a generalized Kato inequality for any pair of solutions. Applications to uniqueness of solutions are then given.     

Asymptotic Stability of Riemann Solutions for a Class of Multidimensional Systems of Conservation Laws with Viscosity

We prove the asymptotic stability of two-state nonplanar Riemann solutions for a class of multidimensional hyperbolic systems of conservation laws when the initial data are perturbed and viscosity is added. The class considered here is those systems whose flux functions in different directions share a common complete system of Riemann invariants, the level surfaces of which are hyperplanes. In particular, we obtain the uniqueness of the self-similar L^∞ entropy solution of the two-state nonplanar Riemann problem. The asymptotic stability to which the main result refers is in the sense of the convergence as t→∞ in L_loc¹ of the space of directions ξ = x/t. That is, the solution u(t, x) of the perturbed problem satisfies u(t, tξ)→R(ξ) as t→∞, in L_loc¹(ℝⁿ), where R(ξ) is the self-similar entropy solution of the corresponding two-state nonplanar Riemann problem.     

Relative Entropy and the Stability of Shocks and Contact Discontinuities for Systems of Conservation Laws with non-BV Perturbations

We develop a theory based on relative entropy to show the uniqueness and L ² stability (up to a translation) of extremal entropic Rankine?CHugoniot discontinuities for systems of conservation laws (typically 1-shocks, n-shocks, 1-contact discontinuities and n-contact discontinuities of large amplitude) among bounded entropic weak solutions having an additional trace property. The existence of a convex entropy is needed. No BV estimate is needed on the weak solutions considered. The theory holds without smallness conditions. The assumptions are quite general. For instance, strict hyperbolicity is not needed globally. For fluid mechanics, the theory handles solutions with vacuums.     

Homogenization of Non-linear Scalar Conservation Laws

We study the limit as ε → 0 of the entropy solutions of the equation . We prove that the sequence u ^ε two-scale converges toward a function u(t, x, y), and u is the unique solution of a limit evolution problem. The remarkable point is that the limit problem is not a scalar conservation law, but rather a kinetic equation in which the macroscopic and microscopic variables are mixed. We also prove a strong convergence result in .     

Stability and Asymptotic Behavior of Periodic Traveling Wave Solutions of Viscous Conservation Laws in Several Dimensions

Under natural spectral stability assumptions motivated by previous investigations of the associated spectral stability problem, we determine sharp L ^p estimates on the linearized solution operator about a multidimensional planar periodic wave of a system of conservation laws with viscosity, yielding linearized L ¹ ∩ L ^p → L ^p stability for all p \geqq 2{p \geqq 2} and dimensions d \geqq 1{d \geqq 1} and nonlinear L ¹ ∩ H ^s → L ^p ∩ H ^s stability and L ²-asymptotic behavior for p\geqq 2{p\geqq 2} and d\geqq 3{d\geqq 3} . The behavior can in general be rather complicated, involving both convective (that is, wave-like) and diffusive effects.     

Solutions for Linear Conservation Laws with Velocity Fields in $$L^{\rm \infty}$$

A Space-Time Integrated Least Squares (STILS) method is derived for solving the linear conservation law with a velocity field in . An existence and uniqueness result is given for the solution of this equation. A maximum principle is established and finally a comparison with a renormalized solution is presented.     

On Nonlinear Stochastic Balance Laws

We are concerned with multidimensional stochastic balance laws. We identify a class of nonlinear balance laws for which uniform spatial BV bound for vanishing viscosity approximations can be achieved. Moreover, we establish temporal equicontinuity in L ¹ of the approximations, uniformly in the viscosity coefficient. Using these estimates, we supply a multidimensional existence theory of stochastic entropy solutions. In addition, we establish an error estimate for the stochastic viscosity method, as well as an explicit estimate for the continuous dependence of stochastic entropy solutions on the flux and random source functions. Various further generalizations of the results are discussed.