We study stochastic Cahn-Hilliard equations in bounded smooth domains with a double-well potential, transport-type noise, and natural Neumann boundary conditions in dimensions $d\le 4$. By employing stochastic maximal regularity techniques and deriving suitable energy estimates, we prove local and global well-posedness. The initial data considered here are allowed to belong to the critical trace space $B^{d/q-1}_{q,p}$, which is locally invariant under the natural scaling of the Cahn-Hilliard equation. In particular, for arbitrary $\varepsilon>1/3$ one can find $q$ sufficiently large such that uniqueness and global existence of a probabilistically strong solution hold for every initial datum $u_0\in H^{\varepsilon, q}(\mathscr{O})$. If $u_0\in H^1(\mathscr{O})$, then these global solutions have $L_t^2H_x^3 \cap C_tH_x^1$-regularity on finite time intervals.
This paper investigates the uniform-in-time strong convergence rates of a fully discrete approximation for the stochastic Cahn--Hilliard equation driven by multiplicative noise in spatial dimensions $d\in\{1,2,3\}$. The proposed scheme combines a spectral Galerkin method in space with a backward Euler scheme in time. T...
Jia-Qi He, Nan Deng, Shuhan Zhang et al.· 1 citation
We investigate the dynamical behavior of a class of semilinear stochastic wave equations on a bounded smooth domain $\Ocal\subset\R^d$ driven by additive trace-class noise, where the elastic response is governed by a \emph{mixed local--nonlocal} operator $\Acal=-\theta\Delta+\beta(-\Delta)^s$ with $s\in(0,1)$. A fundam...
Francisco Delgado-Vences, Jose Julian Pavon-Español· 0 citations
We consider the stochastic heat equation $\partial_t u(t,x) = \Delta u(t,x) + \dot{W}_\alpha(t,x)$ on a bounded Lipschitz domain with zero Dirichlet boundary condition and zero initial condition, where $\dot{W}_\alpha$ is a Gaussian noise that is white in time and whose spatial covariance is the kernel of $(-\Delta)^{-...
We establish interior maximal $L^{q_c}$-regularity for bounded strong solutions of $u_t-\Delta u+|Du|^\gamma=f$ in $\mathbb{T}^d\times(0,T)$, where $d\geq 2$, $\gamma>2$, and $q_c=(d+2)(\gamma-1)/\gamma$. The estimates are uniform for uniformly bounded families of solutions whose source terms range over a bounded, unif...
We prove that strong well-posedness is generic for McKean--Vlasov stochastic differential equations with jumps in a complete coefficient space that allows genuinely discontinuous state dependence. For every $T>0$, there exists a dense residual subset $\mathfrak R_T$, independent of the initial state, such that every co...
We prove the optimal global regularity of admissible solutions to a transformed very fast diffusion equation in the range $-1<p<0$, posed on smooth bounded domains with zero Dirichlet boundary data and initial data comparable to the distance function. More precisely, we establish existence and uniqueness and show that...
Tian-Ling Jin, Xu-Shan Tu, Jingang Xiong et al.· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
