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.

Appendix A: Some useful results

In this section, we state some useful results for the tightness of the laws as well as the well-known Skorokhod’s representation theorem. Let $(\mathbb {Y},\mathcal {B}(\mathbb {Y}))$ be a separable and complete metric space.

Lemma A.1

(Prokhorov Theorem, [9, Sect. 5]). A sequence of probability measures $\{\mu _m\}_{m\in \mathbb {N}}$ on $(\mathbb {Y},\mathcal {B}(\mathbb {Y}))$ is tight if and only if it is relatively compact, that is, there exists a subsequence $\{\mu _{m_k}\}_{k\in \mathbb {N}}$ which converges weakly to a probability measure $\mu$.

Lemma A.2

[62, Theorem 5]. Let $1\le p<\infty$. Let $\mathbb {V},\mathbb {H}$ and $\mathbb {Y}$ be Banach spaces satisfying the embedding $\mathbb {V}\subset \mathbb {H}\subset \mathbb {Y}$. Suppose that the embedding $\mathbb {V}\subset \mathbb {H}$ is compact. If $\Gamma$ is a bounded subset of $\text {L}^p(0,T;\mathbb {V})$ satisfying

$$\begin{aligned} \lim _{\delta \rightarrow 0^+} \sup _{f\in \Gamma }\int _0^{T-\delta }\Vert f(t+\delta )-f(t)\Vert _{\mathbb {Y}}^p\text {d}t=0, \end{aligned}$$

then $\Gamma$ is a relatively compact subset of $\text {L}^p(0,T;\mathbb {H})$.

Based on the previous lemma, the following result has been established in the work [57] which prove the tightness of the laws in $\text {L}^p(0,T;\mathbb {H})$.

Lemma A.3

[57, Lemma 5.2]. Let $1\le p<\infty$. Let $\mathbb {V},\mathbb {H}$ and $\mathbb {Y}$ be Banach spaces satisfying the embedding $\mathbb {V}\subset \mathbb {H}\subset \mathbb {Y}$. Suppose that the embedding $\mathbb {V}\subset \mathbb {H}$ is compact. Let $\{\textbf{Y}_m\}_{m\in \mathbb {N}}$ be a sequence of the stochastic processes. If

$$\begin{aligned} \lim _{N\rightarrow \infty }\sup _{m\in \mathbb {N}}\mathbb {P}\bigg (\int _0^T\Vert \textbf{Y}_m(t)\Vert _\mathbb {V}^p\text {d}t>N\bigg )=0, \end{aligned}$$

(A.1)

and for any $\epsilon >0$,

$$\begin{aligned} \lim _{\delta \rightarrow 0^+}\sup _{m\in \mathbb {N}}\mathbb {P}\bigg (\int _0^{T-\delta }\Vert \textbf{Y}_m(t+\delta )-\textbf{Y}_m(t)\Vert _{\mathbb {Y}}^p\text {d}t>\epsilon \bigg )=0. \end{aligned}$$

(A.2)

Then, $\{\textbf{Y}_m\}_{m\in \mathbb {N}}$ is tight in $\text {L}^p(0,T;\mathbb {H})$.

Let us now recall the following version of the Skorokhod’s representation theorem.

Theorem A.4

[50, Theorem A.1], [13, Theorem C.1]. Let $(\Omega ,\mathscr {F},\mathbb {P})$ be a probability space and $\mathbb {Y}^1,\mathbb {Y}^2$ be two complete separable metric spaces. Let $\chi _m:\Omega \rightarrow \mathbb {Y}^1\times \mathbb {Y}^2,\; m\in \mathbb {N}$, be a sequence of weakly convergent random variables with laws $\{\rho _m\}_{m\in \mathbb {N}}$. For $j=1,2$ let $\Pi _j:\mathbb {Y}^1\times \mathbb {Y}^2\rightarrow \mathbb {Y}^j$ denote the natural projection, that is,

$$\begin{aligned} \Pi _j(\chi ^1,\chi ^2)=\chi ^j,\ \text { for all }\ (\chi ^1,\chi ^2)\in \mathbb {Y}^1\times \mathbb {Y}^2. \end{aligned}$$

Finally, we assume that the random variables $\Pi _1(\chi _m)=\chi _m^1$ on $\mathbb {Y}^1$ have the same law, independent of m.

Then, there exists a family of $\mathbb {Y}^1\times \mathbb {Y}^2$-valued random variables $\{\widehat{\chi }_m\}_{m\in \mathbb {N}}$ on the probability space $(\widehat{\Omega },\widehat{\mathscr {F}},\widehat{\mathbb {P}}):=\big ([0,1)\times [0,1), \mathcal {B}([0,1)\times [0,1)), \text { Lebesgue measure}\big )$ and a random variable $\widehat{\chi }_\infty$ on $(\widehat{\Omega },\widehat{\mathscr {F}},\widehat{\mathbb {P}})$ such that the following statements hold:

• (1) The law of $\{\widehat{\chi }_m\}$ is $\{\rho _m\}$, for every $m\in \mathbb {N}$,

• (2) $\widehat{\chi }_m\rightarrow \widehat{\chi }_\infty$ in $\mathbb {Y}^1\times \mathbb {Y}^2,\;\widehat{\mathbb {P}}$-a.s.,

• (3) $\widehat{\chi }_m^1=\widehat{\chi }_\infty ^1$ everywhere on $\widehat{\Omega }$, for every $m\in \mathbb {N}$.