Well-posedness of a class of stochastic partial differential equations with fully monotone coefficients perturbed by Lévy noise

Abstract

In this article, we consider the following class of stochastic partial differential equations (SPDEs):

$$\begin{aligned} \left\{ \! \begin{aligned} \text {d} \textbf{X}(t)&=\text {A}(t,\textbf{X}(t))\text {d} t+\text {B}(t,\textbf{X}(t))\text {d}\text {W}(t)+\!\!\int _{\text {Z}}\!\gamma (t,\textbf{X}(t-),z)\widetilde{\pi }(\text {d} t,\text {d} z),\; t\!\in \![0,T],\\ \textbf{X}(0)&=\varvec{x} \in \mathbb {H}, \end{aligned} \right. \end{aligned}$$

with fully locally monotone coefficients in a Gelfand triplet \(\mathbb {V}\subset \mathbb {H}\subset \mathbb {V}^*\), where the mappings

$$\begin{aligned} \text {A}:[0,T]\times \mathbb {V}\rightarrow \mathbb {V}^*,\quad \text {B}:[0,T]\times \mathbb {V}\rightarrow \text {L}_2(\mathbb {U},\mathbb {H}), \quad \gamma :[0,T]\times \mathbb {V}\times \text {Z}\rightarrow \mathbb {H}, \end{aligned}$$

are measurable, \(\text {L}_2(\mathbb {U},\mathbb {H})\) is the space of all Hilbert-Schmidt operators from \(\mathbb {U}\rightarrow \mathbb {H}\), \(\text {W}\) is a \(\mathbb {U}\)-cylindrical Wiener process and \(\widetilde{\pi }\) is a compensated time homogeneous Poisson random measure. This class of SPDEs covers various fluid dynamic models and also includes quasi-linear SPDEs, the convection-diffusion equation, the Cahn-Hilliard equation, and the two-dimensional liquid crystal model. Under certain generic assumptions of \(\text {A},\text {B}\) 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 probabilistic weak solution as well as pathwise uniqueness of solution. We use the classical Yamada-Watanabe theorem to obtain the existence of a unique probabilistic strong solution. Furthermore, we establish a result on the continuous dependence of the solutions on the initial data. Finally, we allow both diffusion coefficient \(\text {B}(t,\cdot )\) 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. Under some assumptions on the growth coefficient corresponding to the \(\mathbb {V}\)-norm, we establish the global solvability results also.