Abstract
Abstract In this article, we deal with the stochastic perturbation of degenerate parabolic partial differential equations (PDEs). The particular emphasis is on analyzing the effects of a multiplicative Lévy noise on such problems and on establishing a well-posedness theory by developing a suitable weak entropy solution framework. The proof of the existence of a solution is based on the vanishing viscosity technique. The uniqueness of the solution is settled by interpreting Kruzhkov’s doubling technique in the presence of a noise.
Highlights
In this article we deal with stochastic perturbation of degenerate parabolic partial differential equations (PDEs)
The study of stochastic degenerate parabolic-hyperbolic equations has so far been limited to mainly equations with Brownian noise
Hyperbolic conservation laws with Brownian noise are the examples of such problems that have attracted the attention of many
Summary
And in the sequel, we denote by Nω2(0, T, L2(Rd)) the space of predictable L2(Rd)-valued processes u such that E ΠT |u|2 dt dx < +∞. 0≤t≤T (2) Given a nonnegative test function ψ ∈ Cc1,2([0, ∞) × Rd) and a convex entropy flux triple (β, ζ, ν), the following inequality holds: β(u(t, x))∂tψ(t, x) + ν(u(t, x))∆ψ(t, x) − ∇ψ(t, x) · ζ(u(t, x)) dx dt η(x, u(t, x); z)β′(u(t, x) + θ η(x, u(t, x); z))ψ(t, x) dθ N (dz, dt) dx (1 − θ)η2(x, u(t, x); z)β′′(u(t, x) + θ η(x, u(t, x); z))ψ(t, x) dθ m(dz) dx dt. Any entropy solution u(t, ·) of (1.1) satisfies the initial condition in the following sense: for every non negative test function ψ ∈ Cc2(Rd) such that supp (ψ) = K lim E 1 h→0 h h 0 u(t, x) − u0(x) ψ(x) dx dt.
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