A particular case of initial data for the two-dimensional Euler equations is studied numerically. The results show that the Godunov method does not always converge to the physical solution, at least not on feasible grids. Moreover, they suggest that entropy solutions (in the weak entropy inequality sense) are not well posed.
@cite_26 proposes the EEF condition for scalar conservation laws ( @math ), proves that it is implied by the VV condition under some circumstances and notes that there is a large set of convex entropies. Apparently independently, @cite_5 obtained analogous results for systems. @cite_32 contains the first use of the term entropy condition'' for the EEF condition. Various forms of the EEF condition had been known and in use for special systems such as the Euler equations for a long time (e.g. by the name of Clausius-Duhem inequality ), especially as shock relations; however, the above references seem to be the first to define the general notion of strictly convex EEF pairs, to propose the EEF condition as a mathematical tool for arbitrary systems of conservation laws and to formulate it in the weak form rather than the special case .
@cite_5 @cite_26 @cite_32
1992812527 205360716
In this paper we construct a theory of generalized solutions in the large of Cauchy's problem for the equations in the class of bounded measurable functions. We define the generalized solution and prove existence, uniqueness and stability theorems for this solution. To prove the existence theorem we apply the "vanishing viscosity method"; in this connection, we first study Cauchy's problem for the corresponding parabolic equation, and we derive a priori estimates of the modulus of continuity in of the solution of this problem which do not depend on small viscosity.Bibliography: 22 items. Publisher Summary This chapter provides an overview of shock waves and entropy. It describes systems of the first order partial differential equations in conservation form: ∂ t U + ∂ X F = 0, F = F(u). In many cases, all smooth solutions of the first order partial differential equations in conservation form satisfy an additional conservation law where U is a convex function of u. The chapter discusses that for all weak solutions of ∂ t u j +∂ x f j = 0, j=1,…, m, f j =f j (u 1 ,…, u m ), which are limits of solutions of modifications ∂ t u j +∂ x f j = 0, j=1,…, m, f j =f j (u 1 ,…, u m ) , by the introduction of various kinds of dissipation, satisfy the entropy inequality, that is, ∂ t U + ∂ x F≦ 0. The chapter also explains that for weak solutions, which contain discontinuities of moderate strength, ∂ t U + ∂ x F≦ 0 is equivalent to the usual shock condition involving the number of characteristics impinging on the shock. The chapter also describes all possible entropy conditions of ∂ t U + ∂ x F≦ 0 that can be associated to a given hyperbolic system of two conservation laws.