Abstract
We prove the existence and uniqueness of entropy solutions for nonlinear diffusion equations with nonlinear conservative gradient noise. As particular applications our results include stochastic porous media equations, as well as the one-dimensional stochastic mean curvature flow in graph form.
Highlights
In this work we consider stochastic partial differential equations of the type ∞du = ∆Φ(u) + ∇ · G(x, u) dt + ∇ · σk(x, u) ◦ dβk(t) on (0, T ) × Td k=1 (1.1)u(0, x) = ξ(x), where Td is the d-dimensional torus, βk are independent R-valued Brownian motions, Φ : R → R is a monotone function and the coefficients G : Td × R → Rd, σk : Td × R → Rd are regular enough
The main results of this work are the existence and uniqueness of entropy solutions to (1.1) (Theorem 2.7 below) and the stability of (1.1) with respect to Φ (Theorem 4.1 below)
Stochastic partial differential equations of the type (1.1) arise as limits of interacting particle systems driven by common noise, with notable relation to the theory of mean field games [35, 36, 37], in the graph formulation of the stochastic mean curvature/curve shortening flow [30, 47, 9, 11] and as simplified approximating models of fluctuations
Summary
In the most recent contribution [14] the path-by-path well-posedness of kinetic solutions to (1.1), with Φ(u) = u|u|m−1 for m ∈ (0, ∞) (fast and slow diffusion), was proved for the first time for non-negative initial data, while for sign-changing data the uniqueness was restricted to the case m > 2 As it is well-known from the theory of rough paths, such path-by-path methods require stronger regularity assumptions on the diffusion coefficients than what would be expected based on probabilistic methods. The key aims of the current work are to obtain well-posedness without sign restrictions on the initial data that covers the full spectrum of m for the slow diffusion (m > 1), to relax the regularity assumptions on the diffusion coefficients σk, and to treat a general class of diffusion nonlinearities Φ These aims are achieved by developing a probabilistic entropy approach to (1.1) leading to the relaxed regularity assumption (cf Assumption 2.3 below for details) σk(x, u) ∈ Cb3(Td × R) ∀k ∈ N.
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