Nonlinear diffusive-dispersive limits for scalar multidimensional conservation laws

We consider a class of multidimensional conservation laws with vanishing nonlinear diffusion and dispersion terms. Under a condition on the relative size of the diffusion and dispersion coefficients, we show that the approximate solutions converge in a strong topology to the entropy solution of a scalar conservation law. Our proof is based on methodology developed in [S. Hwang, A.E. Tzavaras, Kinetic decomposition of approximate solutions to conservation laws: Applications to relaxation and diffusion-dispersion approximations, Comm. Partial Differential Equations 27 (2002) 1229-1254] which uses the averaging lemma.     

Vanishing viscosity with short wave-long wave interactions for multi-D scalar conservation laws

We consider a system coupling a multidimensional semilinear Schrödinger equation and a multidimensional nonlinear scalar conservation law with viscosity, which is motivated by a model of short wave-long wave interaction introduced by Benney (1977). We prove the global existence and uniqueness of the solution of the Cauchy problem for this system. We also prove the convergence of the whole sequence of solutions when the viscosity ε and the interaction parameter α approach zero so that α=o(ε1/2). We also indicate how to extend these results to more general systems which couple multidimensional semilinear systems of Schrödinger equations with multidimensional nonlinear systems of scalar conservation laws mildly coupled.     

Existence theory for nonclassical entropy solutions of scalar conservation laws

We establish a general existence theory for the Cauchy problem associated with a scalar conservation law in one-space dimension. The flux-function is assumed to be nonconvex and we consider nonclassical entropy solutions selected by a kinetic relation. To solve the Cauchy problem, we construct a sequence of approximate solutions using a wave-front tracking scheme. The main difficulty is deriving a uniform estimate on the total variation of the approximate solutions. This is achieved here by introducing a generalized total variation functional, which is decreasing in time and, additionally, reduces to the standard total variation functional when the solutions contain only classical shocks. This functional seems sufficiently robust to be useful for systems as well.Received: June 3, 2002; revised: November 12, 2002     

Existence theory for nonclassical entropy solutions of scalar conservation laws

We establish a general existence theory for the Cauchy problem associated with a scalar conservation law in one-space dimension. The flux-function is assumed to be nonconvex and we consider nonclassical entropy solutions selected by a kinetic relation. To solve the Cauchy problem, we construct a sequence of approximate solutions using a wave-front tracking scheme. The main difficulty is deriving a uniform estimate on the total variation of the approximate solutions. This is achieved here by introducing a generalized total variation functional, which is decreasing in time and, additionally, reduces to the standard total variation functional when the solutions contain only classical shocks. This functional seems sufficiently robust to be useful for systems as well.     

Small viscosity solution of linear scalar 1-D conservation laws with one discontinuity of the coefficient

While linear conservations laws have a classical well-defined solution for sufficiently regular coefficients, it is not the case when the coefficients are, for instance, discontinuous across a fixed hypersurface. In this case, another approach has to be proposed in order to answer the double concern of existence and uniqueness of a solution to the problem. We will focus mainly on showing such concerns can be solved by means of a small viscosity approach in 1-D scalar frameworks, in particular for expansive discontinuities of the coefficient. The obtained small viscosity solution is also the solution in the sense Bouchut and James or LeFloch for scalar equations. To cite this article: B. Fornet, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Viscosity methods for piecewise smooth solutions to scalar conservation laws

It is proved that for scalar conservation laws, if the flux function is strictly convex, and if the entropy solution is piecewise smooth with finitely many discontinuities (which includes initial central rarefaction waves, initial shocks, possible spontaneous formation of shocks in a future time and interactions of all these patterns), then the error of viscosity solution to the inviscid solution is bounded by in the -norm, which is an improvement of the upper bound. If neither central rarefaction waves nor spontaneous shocks occur, the error bound is improved to .

     

Existence and nonexistence of TV bounds for scalar conservation laws with discontinuous flux

For the scalar conservation laws with discontinuous flux, an infinite family of (A, B)‐interface entropies are introduced and each one of them is shown to form an L¹‐contraction semigroup (see [2]). One of the main unsettled questions concerning conservation law with discontinuous flux is boundedness of total variation of the solution. Away from the interface, boundedness of total variation of the solution has been proved in a recent paper [6]. In this paper, we discuss this particular issue in detail and produce a counterexample to show that the solution, in general, has unbounded total variation near the interface. In fact, this example illustrates that smallness of the BV norm of the initial data is immaterial. We hereby settle the question of determining for which of the aforementioned (A, B) pairs the solution will have bounded total variation in the case of strictly convex fluxes. © 2010 Wiley Periodicals, Inc.     

High-resolution numerical algorithm for one-dimensional scalar conservation laws with a constrained solution

The CABARET computational algorithm is generalized to one-dimensional scalar quasilinear hyperbolic partial differential equations with allowance for inequality constraints on the solution. This generalization can be used to analyze seepage of liquid radioactive wastes through the unsaturated zone.     

Existence of global strong solution for the compressible Navier–Stokes equations with degenerate viscosity coefficients in 1D

We consider Navier–Stokes equations for compressible viscous fluids in the one‐dimensional case. We prove the existence of global strong solution with large initial data for compressible Navier–Stokes equation with viscosity coefficients of the form with (it includes in particular the important physical case of the viscous shallow water system when ). The key ingredient of the proof relies to a new formulation of the compressible equations involving a new effective velocity v (see 13 , 14 , 16 , 17 ) such that the density verifies a parabolic equation. We estimate v in norm which enables us to control the norm of by using the maximum principle.     

On nonlinear artificial viscosity,discrete maximum principle and hyperbolic conservation laws

A finite element method for Burgers’ equation is studied. The method is analyzed using techniques from stabilized finite element methods and convergence to entropy solutions is proven under certain hypotheses on the artificial viscosity. In particular we assume that a discrete maximum principle holds. We then construct a nonlinear artificial viscosity that satisfies the assumptions required for convergence and that can be tuned to minimize artificial viscosity away from local extrema. The theoretical results are exemplified on a numerical example. AMS subject classification (2000)  65M20, 65M12, 35L65, 76M10     

Existence of global smooth solution to the initial boundary value problem for p-system with damping

This paper is concerned with the initial boundary value problem for the p$p$-system with damping. We prove the existence of the global smooth solution under the assumption that only the C⁰$C^0$-norm of the derivative of the initial data is sufficiently small, while the C⁰$C^0$-norm of the initial data is not necessarily small. The proof is based on several key a priori estimates, the maximum principle and the characteristic method.     

Potential comparison and asymptotics in scalar conservation laws without convexity

Two kinds of optimal convergence orders in L¹-norm to a self-similar solution are proved or conjectured for various evolutionary problems so far. The first convergence order is of the magnitude of the similarity solution itself and the second one is of order 1/t. Employing a potential comparison technique to scalar conservation laws we may easily see that these asymptotic convergence orders are related to space and time translation of potentials. We present the technique clearly in the simple setting of scalar conservation laws in one space dimension.     

Regularity of viscosity solutions of a degenerate parabolic equation

We study the Cauchy problem for the nonlinear degenerate parabolic equation of second order

and present regularity results for the viscosity solutions.

     

IBVPs for scalar conservation laws with time discontinuous fluxes

The initial boundary value problem for a class of scalar nonautonomous conservation laws in 1 space dimension is proved to be well posed and stable with respect to variations in the flux. Targeting applications to traffic, the regularity assumptions on the flow are extended to a merely dependence on time. These results ensure, for instance, the well‐posedness of a class of vehicular traffic models with time‐dependent speed limits. A traffic management problem is then shown to admit an optimal solution.     

Equilibrium schemes for scalar conservation laws with stiff sources

We consider a simple model case of stiff source terms in hyperbolic conservation laws, namely, the case of scalar conservation laws with a zeroth order source with low regularity. It is well known that a direct treatment of the source term by finite volume schemes gives unsatisfactory results for both the reduced CFL condition and refined meshes required because of the lack of accuracy on equilibrium states. The source term should be taken into account in the upwinding and discretized at the nodes of the grid. In order to solve numerically the problem, we introduce a so-called equilibrium schemes with the properties that (i) the maximum principle holds true; (ii) discrete entropy inequalities are satisfied; (iii) steady state solutions of the problem are maintained. One of the difficulties in studying the convergence is that there are no estimates for this problem. We therefore introduce a kinetic interpretation of upwinding taking into account the source terms. Based on the kinetic formulation we give a new convergence proof that only uses property (ii) in order to ensure desired compactness framework for a family of approximate solutions and that relies on minimal assumptions. The computational efficiency of our equilibrium schemes is demonstrated by numerical tests that show that, in comparison with an usual upwind scheme, the corresponding equilibrium version is far more accurate. Furthermore, numerical computations show that equilibrium schemes enable us to treat efficiently the sources with singularities and oscillating coefficients.

     

Global smooth solution for nonlinearly damped p-system with large initial data

In this paper, we prove the existence of global smooth solution for the Cauchy problem of nonlinearly damped p-system with large initial data. The analysis is based on several key a prioriestirmates. which are obtained by the tnaxirnum principle. Our results extend the corresponding results in [l0,11].     

Vanishing viscosity solutions for conservation laws with regulated flux

In this paper we introduce a concept of “regulated function” $v(t,x)$ of two variables, which reduces to the classical definition when v is independent of t. We then consider a scalar conservation law of the form $u_t+F{(v(t,x),u)}_x=0$, where F is smooth and v is a regulated function, possibly discontinuous w.r.t. both t and x. By adding a small viscosity, one obtains a well posed parabolic equation. As the viscous term goes to zero, the uniqueness of the vanishing viscosity limit is proved, relying on comparison estimates for solutions to the corresponding Hamilton–Jacobi equation.As an application, we obtain the existence and uniqueness of solutions for a class of $2\times 2$ triangular systems of conservation laws with hyperbolic degeneracy.     

Diffusion-dispersion limits for multidimensional scalar conservation laws with source terms

In this paper we consider conservation laws with diffusion and dispersion terms. We study the convergence for approximation applied to conservation laws with source terms. The proof is based on the Hwang and Tzavaras's new approach [Seok Hwang, Athanasios E. Tzavaras, Kinetic decomposition of approximate solutions to conservation laws: Application to relaxation and diffusion-dispersion approximations, Comm. Partial Differential Equations 27 (5-6) (2002) 1229-1254] and the kinetic formulation developed by Lions, Perthame, and Tadmor [P.-L. Lions, B. Perthame, E. Tadmor, A kinetic formulation of multidimensional scalar conservation laws and related equations, J. Amer. Math. Soc. 7 (1) (1994) 169-191].