# 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} }

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…

## 4 Citations

### Renormalized solutions for stochastic $p$-Laplace equations with $L^1$-initial data: The multiplicative case

- Mathematics
- 2021

We consider a p-Laplace evolution problem with multiplicative noise on a bounded domain D ⊂ R with homogeneous Dirichlet boundary conditions for 1 < p < ∞. The random initial data is merely…

### On temporal regularity for strong solutions to stochastic $p$-Laplace systems

- Mathematics
- 2021

where S(ξ) := (κ+ |ξ|) ξ, ξ ∈ R , p ∈ [2,∞) and κ ≥ 0. In applications, the time and space regularity of solutions to (1.1) are of great importance. When designing numerical algorithms, the…

### An averaged space-time discretization of the stochastic p-Laplace system

- Computer Science, MathematicsArXiv
- 2022

A sampling algorithm is provided to construct the necessary random input in an e-cient way and two new space-time discretizations based on the approximation of time-averaged values are proposed.

### Stability and moment estimates for the stochastic singular $\Phi$-Laplace equation

- Mathematics
- 2021

Abstract. We study stability, long-time behavior and moment estimates for stochastic evolution equations with additive Wiener noise and with singular drift given by a divergence type quasilinear…

## 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

- MathematicsProceedings 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

- MathematicsThe 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

- Mathematics
- 2013

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

- Mathematics
- 2014

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…