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Entropy Solution (entropy + solution)

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Selected Abstracts

Interaction of elementary waves for scalar conservation laws on a bounded domain

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 7 2003
Hongxia Liu
Abstract This paper is concerned with the interaction of elementary waves on a bounded domain for scalar conservation laws. The structure and large time asymptotic behaviours of weak entropy solution in the sense of Bardos et al. (Comm. Partial Differential Equations 1979; 4: 1017) are clarified to the initial,boundary problem for scalar conservation laws ut+,(u)x=0 on (0,1) × (0,,), with the initial data u(x,0)=u0(x):=um and the boundary data u(0,t)=u -,u(1,t)=u+, where u±,um are constants, which are not equivalent, and ,,C2 satisfies ,,,>0, ,(0)=f,(0)=0. We also give some global estimates on derivatives of the weak entropy solution. These estimates play important roles in studying the rate of convergence for various approximation methods to scalar conservation laws. Copyright © 2003 John Wiley & Sons, Ltd. [source]

Vanishing viscosity limit of the Navier-Stokes equations to the euler equations for compressible fluid flow

COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 11 2010
Gui-Qiang G. Chen
We establish the vanishing viscosity limit of the Navier-Stokes equations to the isentropic Euler equations for one-dimensional compressible fluid flow. For the Navier-Stokes equations, there exist no natural invariant regions for the equations with the real physical viscosity term so that the uniform sup-norm of solutions with respect to the physical viscosity coefficient may not be directly controllable. Furthermore, convex entropy-entropy flux pairs may not produce signed entropy dissipation measures. To overcome these difficulties, we first develop uniform energy-type estimates with respect to the viscosity coefficient for solutions of the Navier-Stokes equations and establish the existence of measure-valued solutions of the isentropic Euler equations generated by the Navier-Stokes equations. Based on the uniform energy-type estimates and the features of the isentropic Euler equations, we establish that the entropy dissipation measures of the solutions of the Navier-Stokes equations for weak entropy-entropy flux pairs, generated by compactly supported C2 test functions, are confined in a compact set in H,1, which leads to the existence of measure-valued solutions that are confined by the Tartar-Murat commutator relation. A careful characterization of the unbounded support of the measure-valued solution confined by the commutator relation yields the reduction of the measurevalued solution to a Dirac mass, which leads to the convergence of solutions of the Navier-Stokes equations to a finite-energy entropy solution of the isentropic Euler equations with finite-energy initial data, relative to the different end-states at infinity. © 2010 Wiley Periodicals, Inc. [source]

A quantitative compactness estimate for scalar conservation laws

COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 7 2005
Camillo de Lellis
In the case of a scalar conservation law with convex flux in space dimension one, P. D. Lax proved [Comm. Pure and Appl. Math.7 (1954)] that the semigroup defining the entropy solution is compact in L for each positive time. The present note gives an estimate of the ,-entropy in L of the set of entropy solutions at time t > 0 whose initial data run through a bounded set in L1. © 2005 Wiley Periodicals, Inc. [source]

Some existence results for conservation laws with source-term

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 13 2002
João-Paulo Dias
We prove some new results concerning the existence and asymptotic behaviour of entropy solutions of hyperbolic conservation laws containing non-smooth source-terms. Copyright © 2002 John Wiley & Sons, Ltd. [source]

Gauss-Green theorem for weakly differentiable vector fields, sets of finite perimeter, and balance laws

COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 2 2009
Gui-Qiang Chen
We analyze a class of weakly differentiable vector fields F : ,n , ,n with the property that F , L, and div F is a (signed) Radon measure. These fields are called bounded divergence-measure fields. The primary focus of our investigation is to introduce a suitable notion of the normal trace of any divergence-measure field F over the boundary of an arbitrary set of finite perimeter that ensures the validity of the Gauss-Green theorem. To achieve this, we first establish a fundamental approximation theorem which states that, given a (signed) Radon measure , that is absolutely continuous with respect to ,N , 1 on ,N, any set of finite perimeter can be approximated by a family of sets with smooth boundary essentially from the measure-theoretic interior of the set with respect to the measure ||,||, the total variation measure. We employ this approximation theorem to derive the normal trace of F on the boundary of any set of finite perimeter E as the limit of the normal traces of F on the boundaries of the approximate sets with smooth boundary so that the Gauss-Green theorem for F holds on E. With these results, we analyze the Cauchy flux that is bounded by a nonnegative Radon measure over any oriented surface (i.e., an (N , 1)-dimensional surface that is a part of the boundary of a set of finite perimeter) and thereby develop a general mathematical formulation of the physical principle of the balance law through the Cauchy flux. Finally, we apply this framework to the derivation of systems of balance laws with measure-valued source terms from the formulation of the balance law. This framework also allows the recovery of Cauchy entropy flux through the Lax entropy inequality for entropy solutions of hyperbolic conservation laws. © 2008 Wiley Periodicals, Inc. [source]