# Well-posedness of renormalized solutions for a stochastic p -Laplace equation with L^1 -initial data

@article{Sapountzoglou2021WellposednessOR,
title={Well-posedness of renormalized solutions for a stochastic p -Laplace equation with L^1 -initial data},
author={Niklas Sapountzoglou and Aleksandra Zimmermann},
journal={Discrete \& Continuous Dynamical Systems - A},
year={2021}
}
• Published 29 August 2019
• Mathematics
• Discrete & Continuous Dynamical Systems - A
We consider a $p$-Laplace evolution problem with stochastic forcing on a bounded domain $D\subset\mathbb{R}^d$ with homogeneous Dirichlet boundary conditions for $1<p<\infty$. The additive noise term is given by a stochastic integral in the sense of Ito. The technical difficulties arise from the merely integrable random initial data $u_0$ under consideration. Due to the poor regularity of the initial data, estimates in $W^{1,p}_0(D)$ are available with respect to truncations of the solution…
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## References

SHOWING 1-10 OF 39 REFERENCES

### On the Boltzmann Equation with Stochastic Kinetic Transport: Global Existence of Renormalized Martingale Solutions

• Mathematics
• 2016
This article studies the Cauchy problem for the Boltzmann equation with stochastic kinetic transport. Under a cut-off assumption on the collision kernel and a coloring hypothesis for the noise

### On a nonlinear parabolic problem arising in some models related to turbulent flows

• Mathematics
• 1994
This paper studies the Cauchy–Dirichlet problem associated with the equation \[ b(u)_t - {\operatorname{div}}\left( {| {\nabla u - K(b(u)){\bf e}} |^{p - 2} (\nabla u - K(b(u)){\bf e})} \right) +

### Renormalized Solutions for Stochastic Transport Equations and the Regularization by Bilinear Multiplicative Noise

• Mathematics
• 2010
A linear stochastic transport equation with non-regular coefficients is considered. Under the same assumption of the deterministic theory, all weak L ∞-solutions are renormalized. But then, if the

### Renormalised solutions of nonlinear parabolic problems with L1 data: existence and uniqueness

• Mathematics
Proceedings of the Royal Society of Edinburgh: Section A Mathematics
• 1997
In this paper we prove the existence and uniqueness of a renormalised solution of the nonlinear problem where the data f and u0 belong to L1(Ω × (0, T)) and L1 (Ω), and where the function a:(0, T) ×

### On the Cauchy problem for Boltzmann equations: global existence and weak stability

• Mathematics
• 1989
We study the large-data Cauchy problem for Boltzmann equations with general collision kernels. We prove that sequences of solutions which satisfy only the physically natural a priori bounds converge

### Well-posedness and regularity for quasilinear degenerate parabolic-hyperbolic SPDE

• Mathematics
The Annals of Probability
• 2018
We study quasilinear degenerate parabolic-hyperbolic stochastic partial differential equations with general multiplicative noise within the framework of kinetic solutions. Our results are twofold:

### On ergodicity of some Markov processes

• Mathematics
• 2010
We formulate a criterion for the existence and uniqueness of an invariant measure for a Markov process taking values in a Polish phase space. In addition, weak- * ergodicity, that is, the weak

### Stochastic Evolution Equations

In the paper, the concept of the Lévy process with values in a real separable Hilbert space is introduced and some of its properties, in particular the Lévy-Khinchin decomposition, is described.

### Lp-solutions of the stochastic transport equation

• Mathematics
• 2011
Abstract. We consider the stochastic transport linear equation and we prove existence and uniqueness of weak Lp-solutions. Moreover, we obtain a representation of the general solution and a

### A theory of regularity structures

We introduce a new notion of “regularity structure” that provides an algebraic framework allowing to describe functions and/or distributions via a kind of “jet” or local Taylor expansion around each