Abstract. TBA

Abstract. We consider nonstationary Stokes equations in nondivergence form with variable viscosity coefficients and Navier slip boundary conditions with slip coefficient $\alpha$ in a domain $\Omega$. On the one hand, if $\alpha$ is sufficiently smooth, then we obtain a priori local regularity estimates for solutions near a curved portion of the boundary of the domain. On the other hand, if $\alpha$ depends on the curvature of the boundary of the domain, then we obtain local boundary estimates of Hessians of solutions where the right-hand side does not contain the pressure. Our results are new even if the viscosity coefficients are constant.

Abstract. We consider an asymptotic behavior of solutions to the Vlasov-Riesz system of order $\alpha$ in $\mathbb{R}^3$ which is a kinetic model induced by Riesz interactions. We prove small data scattering when $1/2<\alpha<1$ and modified scattering when $1<\alpha<1+\delta$ for some $\delta>0$. Moreover, we show the existence of (modified) wave operators for such a regime. To the best of our knowledge, this is the first result on the existence of modified scattering with polynomial correction in kinetic models.

Abstract. In this paper, we investigate a Stokes-Magneto system with fractional diffusions. We first deal with the non-resistive case in $\mathbb{T}^d$ and establish the local and global well-posedness with initial magnetic field $\boldsymbol{b}_0\in H^s(\mathbb{T}^d)$. We also show the existence of a unique mild solution of the resisitive case with initial data $\boldsymbol{b}_0$ in the critical $L^p(\mathbb{R}^d)$ space. Moreover, we show that the $L^p$-norm of $\boldsymbol{b}(t)$ converges to zero as $t\rightarrow\infty$ if the initial data is sufficiently small.

Abstract. We obtain weighted mixed norm Sobolev estimates in the whole space for nonstationary Stokes equations in divergence and nondivergence form with variable viscosity coefficients that are merely measurable in time variable and have small mean oscillation in spatial variables in small cylinders. As an application, we prove interior mixed norm derivative estimates for solutions to both equations. We also discuss boundary mixed norm Hessian estimates for solutions to equations in nondivergence form under the Lions boundary conditions.

Abstract. We consider a Stokes-Magneto system in $\mathbb{R}^d$ ($d\geq 2)$ with fractional diffusions $\Lambda^{2\alpha}\boldsymbol{u}$ and $\Lambda^{2\beta}\boldsymbol{b}$ for the velocity $\boldsymbol{u}$ and the magnetic field $\boldsymbol{b}$, respectively. Here $\alpha,\beta$ are positive constants and $\Lambda^s = (-\Delta)^{s/2}$ is the fractional Laplacian of order $s$. We establish global existence of weak solutions of the Stokes-Magneto system for any initial data in $L_{2}$ when $\alpha$, $\beta$ satisfy $1/2<\alpha<(d+1)/2$, $\beta >0$, and $\min(\alpha+\beta,2\alpha+\beta-1)>d/2$. It is also shown that weak solutions are unique if $\beta \geq 1$ and $\min (\alpha+\beta,2\alpha+\beta-1)\geq d/2+1$, in addition.

Abstract. We consider the Dirichlet problem for second-order linear elliptic equations in divergence form

in bounded Lipschitz domain $\Omega$ in $\mathbb{R}^2$, where $A:\mathbb{R}^2\rightarrow \mathbb{R}^{2^2}$, $\mathbf{b}: \Omega\rightarrow \mathbb{R}^2$, and $\lambda \geq 0$ are given. If $2<p<\infty$ and $A$ has a small mean oscillation in small balls, $\Omega$ has small Lipschitz constant, and $\mathrm{div } A, \mathbf{b}\in L^2(\Omega;\mathbb{R}^2)$, then we prove existence and uniqueness of weak solutions in $W_0^{1,p}(\Omega)$ of the problem. Similar result also holds for the dual problem.

Abstract. We consider the following Dirichlet problems for elliptic equations with singular drift $\mathbf{b}$:

where $\Omega$ is a bounded Lipschitz domain in $\mathbb{R}^n$, $n\geq 2$. Assuming that $\mathbf{b}\in L^{n,\infty}(\Omega)^n$ has non-negative weak divergence in $\Omega$, we establish existence and uniqueness of weak solution in $W^{1,2+\varepsilon}_0(\Omega)$ of the problem (b) when $A$ is bounded and uniformly elliptic. As an application, we prove unique solvability of weak solution $u$ in $W^{1,2-}_0(\Omega)$ for the problem (a) for every $f\in W^{-1,2-}(\Omega)$.

Abstract. We consider the Dirichlet and Neumann problems for second-order linear elliptic equations:

in a bounded Lipschitz domain $\Omega$ in $\mathbb{R}^n$ ($n\geq 3$), where $\mathbf{b}:\Omega\rightarrow \mathbb{R}^n$ is a given vector field. Under the assumption that $\mathbf{b}\in L^n(\Omega)^n$, we first establish existence and uniqueness of solutions in $L^p_{\alpha}(\Omega)$ for the Dirichlet and Neumann problems. Here $L^p_{\alpha}(\Omega)$ denotes the Sobolev space (or Bessel potential space) with the pair $(\alpha,p)$ satisfying certain conditions. These results extend the classical works of Jerison-Kenig (1995) and Fabes-Mendez-Mitrea (1998) for the Poisson equation. We also prove existence and uniqueness of solutions of the Dirichlet problem with boundary data in $L^2(\partial\Omega)$. Our results for the Dirichlet problems hold even for the case $n=2$.