We review the theory of the (extended) divergence-measure fields providing an up-to-date account of its basic results established by Chen and Frid (1999, 2002), as well as the more recent important ...contributions by Silhavý (2008, 2009). We include a discussion on some pairings that are important in connection with the definition of normal trace for divergence-measure fields. We also review its application to the uniqueness of Riemann solutions to the Euler equations in gas dynamics, as given by Chen and Frid (2002). While reviewing the theory, we simplify a number of proofs allowing an almost self-contained exposition.
We establish the well-posedness of the Neumann problem for stochastic conservation laws with multiplicative noise. As a crucial point in order to prove the uniqueness of the kinetic solution to the ...referred problem we establish a new strong trace theorem for stochastic conservation laws, which extends to the stochastic context the pioneering theorem by Vasseur. Existence of kinetic solutions is proved through the vanishing viscosity method and the detailed analysis of the corresponding stochastic parabolic problem is also made here for the first time, as far as the authors know.
We prove the well-posedness and the asymptotic decay to the mean value of Besicovitch almost periodic solutions to nonlinear anisotropic degenerate parabolic-hyperbolic equations.
We consider a mixed type boundary value problem for a class of degenerate parabolic–hyperbolic equations. Namely, we consider a Cartesian product domain and split its boundary into two parts. In one ...of them we impose a Dirichlet boundary condition; in the other, we impose a Neumann condition. We apply a normal trace formula for
L
2
-divergence-measure fields to prove a new strong trace property in the part of the boundary where the Neumann condition is imposed. We prove the existence and uniqueness of the entropy solution.
We study the well-posedness of the initial value problem and the long-time behavior of almost periodic solutions to stochastic scalar conservation laws in any space dimension, under the assumption of ...Lipschitz continuity of the flux functions and a non-degeneracy condition. We show the existence and uniqueness of an invariant measure in a separable subspace of the space of Besicovitch almost periodic functions.
We establish the well-posedness of an initial-boundary value problem of mixed type for a stochastic nonlinear parabolic-hyperbolic equation on a space domain O=O′×O″ where a Neumann boundary ...condition is imposed on ∂O′×O″, the hyperbolic boundary, and a Dirichlet condition is imposed on O′×∂O″, the parabolic boundary. Among other points to be highlighted in our analysis of this problem we mention the new strong trace theorem for the special class of stochastic nonlinear parabolic-hyperbolic equations studied here, which is decisive for the uniqueness of the kinetic solution, and the new averaging lemma for the referred class of equations which is a vital part of the proof of the strong trace property. We also provide a detailed analysis of the approximate nondegenerate problems, which is also made here for the first time, as far as the authors know, whose solutions we prove to converge to the solution of our initial-boundary value problem.
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