Related papers: Bounded Mean Oscillation and the Uniqueness of Act…
We establish existence, uniqueness and optimal regularity results for very weak solutions to certain nonlinear elliptic boundary value problems. We introduce structural asymptotic assumptions of Uhlenbeck type on the nonlinearity, which are…
This paper investigates the local regularity of solutions to stationary Fokker-Planck equations on an open set $U \subset \mathbb{R}^d$ with $d \geq 2$. A central objective is to relax the classical assumptions on the coefficients by…
We give an example of a well posed, finite energy, 2D incompressible active scalar equation with the same scaling as the surface quasi-geostrophic equation and prove that it can produce finite time singularities. In spite of its simplicity,…
We study the quantitative stability for the classical Brezis-Nirenberg problem associated with the critical Sobolev embedding $H^1_0(\Omega) \hookrightarrow L^{\frac{2n}{n-2}}(\Omega)$ in a smooth bounded domain $\Omega \subset…
We establish anisotropic uncertainty principles (UPs) for general metaplectic operators acting on $L^2(\mathbb{R}^d)$, including degenerate cases associated with symplectic matrices whose $B$-block has nontrivial kernel. In this setting,…
We establish a new local well-posedness result in the space of finite Borel measures for mild solutions of the parabolic-elliptic Patlak-Keller-Segel (PKS) model of chemotactic aggregation in two dimensions. Our result only requires that…
The Neumann problem of linear elasticity is singular with a kernel formed by the rigid motions of the body. There are several tricks that are commonly used to obtain a non-singular linear system. However, they often cause reduced accuracy…
This paper studies limit measures of stationary measures of stochastic ordinary differential equations on the Euclidean space and tries to determine which invariant measures of an unperturbed system will survive. Under the assumption for…
We consider a noninteracting unbounded spin system with conservation of the mean spin. We derive a uniform logarithmic Sobolev inequality (LSI) provided the single-site potential is a bounded perturbation of a strictly convex function. The…
In this paper, we begin with extended Majda--Biello system (BSAB equations): $$ \left\{\begin{array}{l} 0=A_t-DA_3+\mu A_1+\Gamma_S B^S_1+\Gamma_A B_1^A+\left(AB^S\right)_x \\ 0=B^S_t-B_3^S+\Gamma_SA_1+\lambda B_1^S+\sigma B^A_1+AA_1 \\…
In the present paper, we consider the Chern-Simons-Schr\"{o}dinger system \begin{equation} \left\{ \begin{aligned} &-\varepsilon^{2}\Delta u+V(x)u+(A_{0}+A_{1}^{2}+A_{2}^{2})u=|u|^{p-2}u,\,\,\,\,x\in \mathbb{R}^2,\\ &\partial_1 A_0 = A_2…
We are interested in the long-time behaviour of the kinetic Vicsek equation, rigorously derived as the mean-field limit~\cite{bolley2012meanfield} of a coupled system of~$N$ stochastic differential equations describing particles moving at…
In this article we investigate the existence, uniqueness and exponential decay of asymptotically almost periodic solutions of the parabolic-elliptic Keller-Segel system on a real hyperbolic manifold. We prove the existence and uniqueness of…
In our previous work [SIAM J. Sci. Comput. 43(3) (2021) B784-B810], an accurate hyper-singular boundary integral equation method for dynamic poroelasticity in two dimensions has been developed. This work is devoted to studying the more…
We consider a stochastic nonlinear defocusing Schr\"{o}dinger equation with zero-order linear damping, where the stochastic forcing term is given by a combination of a linear multiplicative noise in the Stratonovich form and a nonlinear…
We prove existence and uniqueness of strong (pointwise) solutions to a linear nonlocal strongly coupled hyperbolic system of equations posed on all of Euclidean space. The system of equations comes from a linearization of a nonlocal model…
Motivated by an equation arising in magnetohydrodynamics, we address the well-posedness theroy for the non-diffusive magneto-geostrophic equation. Namely, an active scalar equation in which the divergence-free drift velocity is one…
This paper considers the strong error analysis of the Euler and fast Euler methods for nonlinear overdamped generalized Langevin equations driven by the fractional noise. The main difficulty lies in handling the interaction between the…
This paper is concerned with a chemotaxis aggregation model for cells, more precisely with a parabolic-elliptic semilinear Patlak-Keller-Segel system in a ball of $\mathbb{R}^N$ for $N\geq 2$. For $N=2$, this system is well known for its…
In this paper, we develop a new reduced basis (RB) method, named as Single Eigenvalue Acceleration Method (SEAM), for second-order parabolic equations with homogeneous Dirichlet boundary conditions. The high-fidelity numerical method adopts…