Degenerate parabolic stochastic partial differential equations: Quasilinear case

Abstract

In this paper, we study the Cauchy problem for a quasilinear degenerate parabolic stochastic partial differential equation driven by a cylindrical Wiener process. In particular, we adapt the notion of kinetic formulation and kinetic solution and develop a well-posedness theory that includes also an L 1 -contraction property. In comparison to the previous works of the authors concerning stochastic hyperbolic conservation laws [J. Funct. Anal. 259 (2010) 1014-1042] and semilinear degenerate parabolic SPDEs [Stochastic Process. Appl. 123 (2013) 4294-4336], the present result contains two new ingredients that provide simpler and more effective method of the proof: a generalized Itô formula that permits a rigorous derivation of the kinetic formulation even in the case of weak solutions of certain nondegenerate approximations and a direct proof of strong convergence of these approximations to the desired kinetic solution of the degenerate problem.

Existence Theory for Stochastic Power Law Fluids

“…As a consequence of the convergence properties of the convolution a passage to the limit in (6.75) implies the claim (see [14] for more details).…”

Invariant Measures for Nonlinear Conservation Laws Driven by Stochastic Forcing

“…The well-posedness for nonlinear conservation laws driven by multiplicative noises is quite well understood from several different perspectives -the strong entropy stochastic solutions of Feng-Nualart [49] and of Chen-Ding-Karlsen [9], the viscosity solution methods of Bauzet-Vallet-Wittbold [3], and the kinetic approach of Debussche-Hofmanovà-Vovelle [33,34], as we have mentioned above. Nevertheless, the problem of long-time behavior of solutions is wide open, since there is no effective way to control u(t) L 1 .…”

“…Such a kinetic function has been popularized by [81] and has been used, inter alia, in [12,[33][34][35], and even as far back as [61]. The usefulness of the kinetic function can be seen in the kinetic formulation of scalar conservation laws, in which the kinetic variable takes the place of the solution in the nonlinear coefficients so that a degree of linearity is restored for analysis.…”

Quasilinear parabolic stochastic partial differential equations: Existence, uniqueness

“…Let 1 > a 1 > a 2 > · · · > a n · · · > 0 be a fixed sequence of decreasing positive numbers such that Suppose that u 1 , u 2 are two solutions to equation (2.4). We may apply the generalized Itô formula [DHV,Proposition A.1] to deduce Φ n (u 1 (t) − u 2 (t)) = (3.39)…”