Related papers: The stochastic $p$-Laplace equation on $\mathbb{R}…
This paper is concerned with the existence of invariant measure for 3D stochastic primitive equations driven by linear multiplicative noise under non-periodic boundary conditions. The common method is to apply Sobolev imbedding theorem to…
We investigate an example of noise-induced stabilization in the plane that was also considered in (Gawedzki, Herzog, Wehr 2010) and (Birrell, Herzog, Wehr 2011). We show that despite the deterministic system not being globally stable, the…
In this paper we investigate a nonlinear stochastic partial differential equation (spde in short) perturbed by a space-correlated Gaussian noise in arbitrary dimension $d\geq1$, with a non-Lipschitz coefficient noisy term. The equation…
The existence of random attractors for singular stochastic partial differential equations (SPDE) perturbed by general additive noise is proven. The drift is assumed only to satisfy the standard assumptions of the variational approach to…
We consider a nonlinear stochastic partial differential equation (SPDE) that takes the form of the Camassa--Holm equation perturbed by a convective, position-dependent, noise term. We establish the first global-in-time existence result for…
We study the Evolutionary p-Laplace Equation in the singular case 1 < p < 2. We prove that a weak solution has a time derivative in Sobolev's sense and that the time derivative is locally summable to some power > 1.
Ergodicity for local and nonlocal stochastic singular $p$-Laplace equations is proven, without restriction on the spatial dimension and for all $p\in[1,2)$. This generalizes previous results from [Gess, T\"{o}lle; J. Math. Pures Appl.,…
We establish the unique ergodicity of a fully discrete scheme for monotone SPDEs with polynomial growth drift and bounded diffusion coefficients driven by multiplicative white noise. The main ingredient of our method depends on the…
Given $1<p<N$ and two measurable functions $V\left( r\right) \geq 0$ and $K\left( r\right) >0$, $r>0$, we define the weighted spaces \[ W=\left\{ u\in D^{1,p}(\mathbb{R}^{N}):\int_{\mathbb{R}^{N}}V\left( \left| x\right| \right) \left|…
We consider the Navier-Stokes equations in $\mathbb R^d$ ($d=2,3$) with a stochastic forcing term which is white noise in time and coloured in space; the spatial covariance of the noise is not too regular, so It\^o calculus cannot be…
In this paper, we are concerned with the local and global existence for the stochastic Prandtl equation in two and three dimensions, which governs the velocity field inside the boundary layer that appears in the inviscid limit of the…
In this paper, we establish the well-posedness and optimal trajectory regularity for the solution of stochastic evolution equations with generalized Lipschitz-type coefficients driven by general multiplicative noises. To ensure the…
We study a class of stochastic evolution equations with a dissipative forcing nonlinearity and additive noise. The noise is assumed to satisfy rather general assumptions about the form of the covariance function; our framework covers…
We study a stochastic boundary value problem on $(0,1)^d$ of elliptic type in dimension $d\ge 4$, driven by a coloured noise. An approximation scheme based on a suitable discretization of the Laplacian on a lattice of $(0,1)^d$ is…
In this paper, we establish the existence and uniqueness of invariant measures for a class of semilinear stochastic partial differential equations driven by multiplicative noise on a bounded domain. The main results can be applied to SPDEs…
The purpose of this work is twofold. First, we construct probabilistically strong solutions to the three-dimensional Euler equations perturbed by additive noise that are $\mathbb{P}$-almost surely continuous in time, H\"older in space, and…
A fundamental open problem in fluid dynamics is whether solutions to $2$D Euler equations with $(L^1_x\cap L^p_x)$-valued vorticity are unique, for some $p\in [1,\infty)$. A related question, more probabilistic in flavour, is whether one…
We study the existence and uniqueness of solutions of 2D Stochastic Navier-Stokes equation with space irregular jump noise for initial data in certain Sobolev spaces of negative order. Comparing with the Galerkin approximation method, the…
We study stochastic evolution equations driven by Gaussian noise. The key features of the model are that the operators in the deterministic and stochastic parts can have the same order and the noise can be time-only, space-only, or…
Martingale solutions of stochastic Navier-Stokes equations in 2D and 3D possibly unbounded domains, driven by the L\'evy noise consisting of the compensated time homogeneous Poisson random measure and the Wiener process are considered.…