Existence and extinction in finite time for Stratonovich gradient noise porous media equations

Abstract

We study existence and uniqueness of distributional solutions to the stochastic partial differential equation dX − ν∆X + ∆ψ(X) dt = N i=1 b i , ∇X • dβ i in ]0, T [ × O, with X(0) = x(ξ) in O and X = 0 on ]0, T [ × ∂O. Moreover, we prove extinction in finite time of the solutions in the special case of fast diffusion model and of self-organized criticality model.

Finite time extinction for the 1D stochastic porous medium equation with transport noise

“…This yields the claim for the noise term appearing in (48). We observe that all the other terms can be dealt with based on the estimates (15)- (18) from Lemma 4, which in turn concludes the proof of (33).…”

“…However, by the bounds (18) and (30), the convergence (31) and the fact that sign(a−b) = sign(a m −b m ) due to the monotonicity of r → r m we infer by an application of Lebesgue's dominated convergence theorem based on the regularity of the Wong-Zakai approximation u ε from Lemma 4 that the term on the right hand side of the latter bound vanishes as δ → 0.…”

“…Gess treats the case of stochastic sign fast diffusion equations in [13]. In a very recent work of Turra [18], finite time extinction in the fast diffusion regime is established for transport noise. The slow diffusion case with transport noise, however, is left open.…”

Stochastic evolution equations with singular drift and gradient noise via curvature and commutation conditions

“…[13,14]) or an approach via entropy solutions [20] or renormalized solutions [48]. We also refer to [21,24,53] for porous media type stochastic equations with gradient noise.…”

Finite time extinction for the 1D stochastic porous medium equation with transport noise

“…Gess treats the case of stochastic sign fast diffusion equations in [15]. In a very recent work of Turra [20], finite time extinction in the fast diffusion regime is established for transport noise. The slow diffusion case with transport noise, however, is left open.…”