Well-posedness of a class of stochastic partial differential equations with fully monotone coefficients perturbed by L\'evy noise

TL;DR

This paper establishes the well-posedness of a broad class of stochastic partial differential equations with fully monotone coefficients influenced by Lévy noise, covering many fluid dynamics models and quasilinear SPDEs.

Contribution

It proves existence and uniqueness of solutions for these SPDEs under general conditions, extending previous results to include coefficients depending on both solution norms and gradients.

Findings

Existence of probabilistic weak solutions.

Pathwise uniqueness of solutions.

Global solvability with gradient-dependent coefficients.

Abstract

In this article, we consider the following class of stochastic partial differential equations (SPDE): \begin{equation*} \left\{\begin{aligned}\mathrm{d} \mathbf{X}(t)&=\mathrm{A}(t,\mathbf{X}(t))\mathrm{d} t+\mathrm{B}(t,\mathbf{X}(t))\mathrm{d}\mathrm{W}(t)+\int_{\mathrm{Z}}\gamma(t,\mathbf{X}(t-),z)\widetilde{\pi}(\mathrm{d} t,\mathrm{d} z),\; t\in[0,T],\\ \mathbf{X}(0)&=\boldsymbol{x} \in \mathbb{H},\end{aligned} \right.\end{equation*} with fully locally monotone coefficients in a Gelfand triplet $\mathbb{V}\subset \mathbb{H}\subset\mathbb{V}^*$, where the mappings \begin{align*} \mathrm{A}:[0,T]\times \mathbb{V}\to\mathbb{V}^*,\quad \mathrm{B}:[0,T]\times \mathbb{V}\to\mathrm{L}_2(\mathbb{U},\mathbb{H}), \quad \gamma:[0,T]\times\mathbb{V}\times\mathrm{Z}\to\mathbb{H}, \end{align*} are measurable, $\mathrm{L}_2(\mathbb{U},\mathbb{H})$ is the space of all Hilbert-Schmidt operators from $\mathbb{U}\to\mathbb{H}$, $\mathrm{W}$ is a $\mathbb{U}$-cylindrical Wiener process and $\widetilde{\pi}$ is a compensated time homogeneous Poisson random measure. Such kind of SPDE cover a large class of quasilinear SPDE and a good number of fluid dynamic models. Under certain generic assumptions of $\mathrm{A},\mathrm{B}$ and $\gamma$, using the classical Faedo-Galekin technique, a compactness method and a version of Skorokhod's representation theorem, we prove the existence of a \emph{probabilistic weak solution} as well as \emph{pathwise uniqueness of solution}. We use the classical Yamada-Watanabe theorem to obtain the existence of a \emph{unique probabilistic strong solution}. Finally, we allow both diffusion coefficient $\mathrm{B}(t,\cdot)$ and jump noise coefficient $\gamma(t,\cdot,z)$ to depend on both $\mathbb{H}$-norm and $\mathbb{V}$-norm, which implies that both the coefficients could also depend on the gradient of solution. We establish the global solvability results.