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

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}$}γ(t,$\mathbf{X}$(t-),z)\widetildeπ($\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 $\mathbb{V}\subset \mathbb{H}\subset {\mathbb{V}}^{\ast }$, 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 γ:[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 $\tilde{\pi }$ 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.