EXISTENCE AND UNIQUENESS OF THE ENTROPY SOLUTION TO A NONLINEAR HYPERBOLIC EQUATION

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Construction of Solutions and L^1-error Estimates of Viscous Methods for Scalar Conservation Laws with Boundary

This paper is concerned with an initial boundary value problem for strictly convex conservation laws whose weak entropy solution is in the piecewise smooth solution class consisting of finitely many discontinuities. By the structure of the weak entropy solution of the corresponding initial value problem and the boundary entropy condition developed by Bardos-Leroux Nedelec, we give a construction method to the weak entropy solution of the initial boundary value problem. Compared with the initial value problem, the weak entropy solution of the initial boundary value problem includes the following new interaction type： an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary. According to the structure and some global estimates of the weak entropy solution, we derive the global L^1-error estimate for viscous methods to this initial boundary value problem by using the matching travelling wave solutions method. If the inviscid solution includes the interaction that an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary, or the inviscid solution includes some shock wave which is tangent to the boundary, then the error of the viscosity solution to the inviscid solution is bounded by O（ε^1/2） in L^1-norm; otherwise, as in the initial value problem, the L^1-error bound is O（ε｜ In ε｜）.     

A kinetic decomposition for singular limits of non-local conservation laws

We consider a non-local regularization of nonlinear hyperbolic conservation laws in several space variables. The regularization is motivated by the theory of phase dynamics and is based on a convolution operator. We formulate the initial value problem and begin by deriving a priori estimates which are independent of the regularization parameter. Following Hwang and Tzavaras we establish a kinetic decomposition associated with the problem under consideration, and we conclude that the sequence of solutions generated by the non-local model converges to a weak solution of the corresponding hyperbolic problem. Depending on the scaling introduced in the non-local dispersive term, this weak limit is either a classical Kruzkov solution satisfying all entropy inequalities or, more interestingly, a nonclassical entropy solution in the sense defined by LeFloch, that is, a weak solution satisfying a single entropy inequality and containing undercompressive shock waves possibly selected by a kinetic relation. Finally, we illustrate our analytical conclusions with numerical experiments in one spatial variable.     

Global Existence and Asymptotic Limits of Weak Solutions of the Bipolar Hydrodynamic Model for Semiconductors

For the case of the adiabatic exponents being larger than , we establish the global existence of entropy weak solutions of the Cauchy problem to the bipolar hydrodynamic model for semiconductors. Using the theory of compensated compactness, we hence give finally a complete answer on the related existence problems with the -law pressure relation. A new kind of singular limit of the modified entropy weak solution is discussed. To some extent, the limit of this sort can provide some information about the uniform boundedness of the scaled solution sequences. The quasineutral-relaxation limit of the entropy weak solutions is also investigated.     

非凸单个守恒律初边值问题的整体弱熵解的构造

本文研究具有两段常数的初始值和常数边界值的非凸单个守恒律的初边值问题.在流函数具有一个拐点的条件下,由相应的初始值问题弱熵解的结构和Bardos-Leroux-Nedelec提出的边界熵条件,给出初边值问题整体弱熵解的一个构造方法,澄清弱熵解在边界附近的结构.与严格凸的单个守恒律初边值问题相比,非凸单个守恒律初边值问题的弱熵解中包括下列新的相互作用类型:一个接触或非接触激波碰到边界,边界弹回一个非接触激波.     

Convergence of finite volume monotone schemes for scalar conservation laws on bounded domains

Summary. This paper is devoted to the study of the finite volume methods used in the discretization of conservation laws defined on bounded domains. General assumptions are made on the data: the initial condition and the boundary condition are supposed to be measurable bounded functions. Using a generalized notion of solution to the continuous problem (namely the notion of entropy process solution, see [9]) and a uniqueness result on this solution, we prove that the numerical solution converges to the entropy weak solution of the continuous problem in for every . This also yields a new proof of the existence of an entropy weak solution. Received May 18, 2000 / Revised version received November 21, 2000 / Published online June 7, 2001     

Non-uniqueness of weak solutions for the fractal Burgers equation

The notion of Kruzhkov entropy solution was extended by the first author in 2007 to conservation laws with a fractional Laplacian diffusion term; this notion led to well-posedness for the Cauchy problem in the L^∞$L^{\infty }$-framework. In the present paper, we further motivate the introduction of entropy solutions, showing that in the case of fractional diffusion of order strictly less than one, uniqueness of a weak solution may fail.     

On a time-splitting method for a scalar conservation law with a multiplicative stochastic perturbation and numerical experiments

In this paper, we present a numerical scheme for a first-order hyperbolic equation of nonlinear type perturbed by a multiplicative noise. The problem is set in a bounded domain D of ${\mathbb{R}^{d}}$ and with homogeneous Dirichlet boundary condition. Using a time-splitting method, we are able to show the existence of an approximate solution. The result of convergence of such a sequence is based on the work of Bauzet–Vallet–Wittbold (J Funct Anal, 2013), where the authors used the concept of measure-valued solution and Kruzhkov’s entropy formulation to show the existence and uniqueness of the stochastic weak entropy solution. Then, we propose numerical experiments by applying this scheme to the stochastic Burgers’ equation in the one-dimensional case.     

On Majda' Model For Dynamic Combustion

Majda's model of dynamic combustion, consists of the system,

In this paper the Cauchy problem is considered. A weak entropy solution for this system is defined, existence, uniqueness and continuous dependence on initial data are proved, as well as finite propagation speed, for initial data in . The existence is proved via the "vanishing viscosity method". Furthermore it is proved that the solution to the Riemann problem converges as to the Z–N–D traveling wave solution. In the appendices, a second order numerical scheme for the model is described, and some numerical results are presented.     

A priori error estimates for approximate solutions to convex conservation laws

Summary. We introduce a new technique for proving a priori error estimates between the entropy weak solution of a scalar conservation law and a finite–difference approximation calculated with the scheme of Engquist-Osher, Lax-Friedrichs, or Godunov. This technique is a discrete counterpart of the duality technique introduced by Tadmor [SIAM J. Numer. Anal. 1991]. The error is related to the consistency error of cell averages of the entropy weak solution. This consistency error can be estimated by exploiting a regularity structure of the entropy weak solution. One ends up with optimal error estimates. Received December 21, 2001 / Revised version received February 18, 2002 / Published online June 17, 2002     

Construction of Approximate Entropy Measure-Valued Solutions for Hyperbolic Systems of Conservation Laws

Entropy solutions have been widely accepted as the suitable solution framework for systems of conservation laws in several space dimensions. However, recent results in De Lellis and Székelyhidi Jr (Ann Math 170(3):1417–1436, 2009) and Chiodaroli et al. (2013) have demonstrated that entropy solutions may not be unique. In this paper, we present numerical evidence that state-of-the-art numerical schemes need not converge to an entropy solution of systems of conservation laws as the mesh is refined. Combining these two facts, we argue that entropy solutions may not be suitable as a solution framework for systems of conservation laws, particularly in several space dimensions. We advocate entropy measure-valued solutions, first proposed by DiPerna, as the appropriate solution paradigm for systems of conservation laws. To this end, we present a detailed numerical procedure which constructs stable approximations to entropy measure-valued solutions, and provide sufficient conditions that guarantee that these approximations converge to an entropy measure-valued solution as the mesh is refined, thus providing a viable numerical framework for systems of conservation laws in several space dimensions. A large number of numerical experiments that illustrate the proposed paradigm are presented and are utilized to examine several interesting properties of the computed entropy measure-valued solutions.     

Stability of steady multi-wave configurations for the full Euler equations of compressible fluid flow

We are concerned with the stability of steady multi-wave configurations for the full Euler equations of compressible fluid flow. In this paper, we focus on the stability of steady four-wave configurations that are the solutions of the Riemann problem in the flow direction, consisting of two shocks, one vortex sheet, and one entropy wave, which is one of the core multi-wave configurations for the two-dimensional Euler equations. It is proved that such steady four-wave configurations in supersonic flow are stable in structure globally, even under the BV perturbation of the incoming flow in the flow direction. In order to achieve this, we first formulate the problem as the Cauchy problem (initial value problem) in the flow direction, and then develop a modified Glimm difference scheme and identify a Glimm-type functional to obtain the required BV estimates by tracing the interactions not only between the strong shocks and weak waves, but also between the strong vortex sheet/entropy wave and weak waves. The key feature of the Euler equations is that the reflection coefficient is always less than $1$, when a weak wave of different family interacts with the strong vortex sheet/entropy wave or the shock wave, which is crucial to guarantee that the Glimm functional is decreasing. Then these estimates are employed to establish the convergence of the approximate solutions to a global entropy solution, close to the background solution of steady four-wave configuration.     

Mathematical analysis of a scalar multidimensional conservation law with discontinuous flux

We deal in this paper with a scalar conservation law, set in a bounded multidimensional domain, and such that the convective term is discontinuous with respect to the space variable. First, we introduce a weak entropy formulation for the homogeneous Dirichlet problem associated with the first-order reaction-convection equation that we consider. Then, we establish an existence and uniqueness property for the weak entropy solution. The method of doubling variables and a pointwise reasoning along the curve of discontinuity are used to state uniqueness. Finally, the vanishing viscosity method allows us to prove the existence result. Another method to obtain the existence of a solution, which relies on the regularization of the flux, is also detailled, at least for a particular case.     

Global well-posedness for the diffusion equation of population genetics

In this paper, we study the forward diffusion equation of population genetics. We prove the global existence of smooth solutions if the initial value is smooth. We also show that if the initial value is singular on the boundary, in a weighted Sobolev space, the diffusion equation exists a unique weak solution which is a probability density function. Moreover, we investigate the asymptotic behavior of the weak solution by the entropy method.     

The unsaturated flow in porous media with dynamic capillary pressure

In this paper we consider a degenerate pseudoparabolic equation for the wetting saturation of an unsaturated two-phase flow in porous media with dynamic capillary pressure-saturation relationship where the relaxation parameter depends on the saturation. Following the approach given in [13] the existence of a weak solution is proved using Galerkin approximation and regularization techniques. A priori estimates needed for passing to the limit when the regularization parameter goes to zero are obtained by using appropriate test-functions, motivated by the fact that considered PDE allows a natural generalization of the classical Kullback entropy. Finally, a special care was given in obtaining an estimate of the mixed-derivative term by combining the information from the capillary pressure with the obtained a priori estimates on the saturation.     

Entropy-stable and entropy-dissipative approximations of a fourth-order quantum diffusion equation

Structure-preserving numerical schemes for a nonlinear parabolic fourth-order equation, modeling the electron transport in quantum semiconductors, with periodic boundary conditions are analyzed. First, a two-step backward differentiation formula (BDF) semi-discretization in time is investigated. The scheme preserves the nonnegativity of the solution, is entropy stable and dissipates a modified entropy functional. The existence of a weak semi-discrete solution and, in a particular case, its temporal second-order convergence to the continuous solution is proved. The proofs employ an algebraic relation which implies the G-stability of the two-step BDF. Second, an implicit Euler and $q$ -step BDF discrete variational derivative method are considered. This scheme, which exploits the variational structure of the equation, dissipates the discrete Fisher information (or energy). Numerical experiments show that the discrete (relative) entropies and Fisher information decay even exponentially fast to zero.     

Weak–strong uniqueness for heat conducting non-Newtonian incompressible fluids

In this work, we introduce a notion of dissipative weak solution for a system describing the evolution of a heat-conducting incompressible non-Newtonian fluid. This concept of solution is based on the balance of entropy instead of the balance of energy and has the advantage that it admits a weak–strong uniqueness principle, justifying the proposed formulation. We provide a proof of existence of solutions based on finite element approximations, thus obtaining the first convergence result of a numerical scheme for the full evolutionary system including temperature dependent coefficients and viscous dissipation terms. Then we proceed to prove the weak–strong uniqueness property of the system by means of a relative energy inequality.     

Conservation law with the flux function discontinuous in the space variable—II: Convex–concave type fluxes and generalized entropy solutions

We deal with a single conservation law with discontinuous convex–concave type fluxes which arise while considering sign changing flux coefficients. The main difficulty is that a weak solution may not exist as the Rankine–Hugoniot condition at the interface may not be satisfied for certain choice of the initial data. We develop the concept of generalized entropy solutions for such equations by replacing the Rankine–Hugoniot condition by a generalized Rankine–Hugoniot condition. The uniqueness of solutions is shown by proving that the generalized entropy solutions form a contractive semi-group in L¹$L^1$. Existence follows by showing that a Godunov type finite difference scheme converges to the generalized entropy solution. The scheme is based on solutions of the associated Riemann problem and is neither consistent nor conservative. The analysis developed here enables to treat the cases of fluxes having at most one extrema in the domain of definition completely. Numerical results reporting the performance of the scheme are presented.     

Mathematical and numerical study of a system of conservation laws

The system of equations (f (u))_t − (a(u)v + b(u))_x = 0 and u_t − (c(u)v + d(u))_x = 0, where the unknowns u and v are functions depending on , arises within the study of some physical model of the flow of miscible fluids in a porous medium. We give a definition for a weak entropy solution (u, v), inspired by the Liu condition for admissible shocks and by Krushkov entropy pairs. We then prove, in the case of a natural generalization of the Riemann problem, the existence of a weak entropy solution only depending on x/t. This property results from the proof of the existence, by passing to the limit on some approximations, of a function g such that u is the classical entropy solution of u_t − ((cg + d)(u))_x = 0 and simultaneously w = f (u) is the entropy solution of w_t − ((ag + b)(f⁽⁻¹⁾(w)))_x = 0. We then take v = g(u), and the proof that (u, v) is a weak entropy solution of the coupled problem follows from a linear combination of the weak entropy inequalities satisfied by u and f (u). We then show the existence of an entropy weak solution for a general class of data, thanks to the convergence proof of a coupled finite volume scheme. The principle of this scheme is to compute the Godunov numerical flux with some interface functions ensuring the symmetry of the finite volume scheme with respect to both conservation equations.     

On the regularity of weak solutions to Burgers’ equation with finite entropy production

Bounded weak solutions of Burgers’ equation \(\partial _tu+\partial _x(u^2/2)=0\) that are not entropy solutions need in general not be BV. Nevertheless it is known that solutions with finite entropy productions have a BV-like structure: a rectifiable jump set of dimension one can be identified, outside which u has vanishing mean oscillation at all points. But it is not known whether all points outside this jump set are Lebesgue points, as they would be for BV solutions. In the present article we show that the set of non-Lebesgue points of u has Hausdorff dimension at most one. In contrast with the aforementioned structure result, we need only one particular entropy production to be a finite Radon measure, namely \(\mu =\partial _t (u^2/2)+\partial _x(u^3/3)\). We prove Hölder regularity at points where \(\mu \) has finite \((1+\alpha )\)-dimensional upper density for some \(\alpha >0\). The proof is inspired by a result of De Lellis, Westdickenberg and the second author : if \(\mu _+\) has vanishing 1-dimensional upper density, then u is an entropy solution. We obtain a quantitative version of this statement: if \(\mu _+\) is small then u is close in \(L^1\) to an entropy solution.