On Nonlinear Stochastic Balance Laws

Abstract

We are concerned with multidimensional stochastic balance laws. We identify a class of nonlinear balance laws for which uniform spatial $BV$ bounds for vanishing viscosity approximations can be achieved. Moreover, we establish temporal equicontinuity in $L^1$ of the approximations, uniformly in the viscosity coefficient. Using these estimates, we supply a multidimensional existence theory of stochastic entropy solutions. In addition, we establish an error estimate for the stochastic viscosity method, as well as an explicit estimate for the continuous dependence of stochastic entropy solutions on the flux and random source functions. Various further generalizations of the results are discussed

Continuous dependence estimate for conservation laws with Lévy noise

“…The existence was proven using stochastic version of compensated compactness and it was valid for one spatial dimension. A number of authors have contributed since then, and we mention the works of Debussche and Vovelle [9], Chen et al [6], Bauzet et al [1] and Biswas and Majee [3]. We want to specifically mention the work [6] of Chen et al, where well-posedness of entropy solution is established in L p ∩ BV , via BV framework.…”

Convergence of monotone finite volume schemes for hyperbolic scalar conservation laws with multiplicative noise

“…Only few papers have been devoted to the theoretical study of scalar conservation laws with a multiplicative stochastic forcing, let us mention in chronological order the contributions of [4][5][6]8,10,13,15]. Concerning the study of numerical approximation of these stochastic problems, there is also, to our knowledge, few papers.…”

“…We will combine it below with corresponding integrals resulting from the stochastic integrals and show that the sum of these terms vanishes as η → | · |. 8 We come now to the estimate of the stochastic integrals. Using the same techniques as in [4] (p. 687), which use properties of the stochastic integral and the Itô formula, one shows that I 7 + J 9 can be written in the following way And, using again the same techniques as in [4] (pp.…”

Invariant measure of scalar first-order conservation laws with stochastic forcing

“…The first step was to have a satisfactory framework for existence and uniqueness of solutions. This has been accomplished by several authors and this point is now rather well understood (see [2,[9][10][11]14,20,21,26]). …”

“…Recall that η was defined in (6) and satisfies the decay condition (9). Consequently, we have the estimate…”