Related papers: Sharp pointwise and uniform estimates for $\bar\pa…

We prove the following estimate \[ \|{e^{it\partial_x^2}f}\|_{L_{(t,x)\in \mathbb{T}^2}^6}\leq C (\log N)^{{1/6}} \|f\|_{L^2_x(\mathbb{T})}, \] assuming $\mbox{supp} (\hat f)\subset [-N,N]$ for $N>1$. The bound $(\log N)^{{1/6}}$ is sharp…

We prove a sharp Bernstein-type inequality for complex polynomials which are positive and satisfy a polynomial growth condition on the positive real axis. This leads to an improved upper estimate in the recent work of Culiuc and Treil on…

We establish a strategy for finding sharp upper and lower numerical bounds of the Poincar\'e constant on a class of planar domains with piecewise self-similar boundary. The approach consists of four main components: W1) tight inner-outer…

We extend the slow blow up solutions of Krieger, Schlag, and Tataru to semilinear wave equations on a curved background. In particular, for a class of manifolds $(M,g)$ we show the existence of a family of blow-up solutions with finite…

We establish a new BKM-type blow-up criterion for solutions of the incompressible Euler equations that belong to Sobolev or H\" older spaces. Our criterion involves the $L^2$ norm in time of the $L^\infty$ norm of the first order tangential…

We introduce a high-order numerical scheme for fractional ordinary differential equations with the Caputo derivative. The method is developed by dividing the domain into a number of subintervals, and applying the quadratic interpolation on…

The article provides upper bounds for the blow-up time of a system of fractional differential equations in the Caputo sense. Furthermore, concrete examples of blow-up time estimation are given using a numerical algorithm of the…

In this paper, based on the local comparison principle in [12], we study the local behavior of the difference of two spacelike graphs in a neighborhood of a second contact point. Then we apply it to the constant mean curvature equation in…

Let $G \subset \mathbb{C}^2$ be a smoothly bounded pseudoconvex domain and assume that the Bergman kernel of $G$ is algebraic of degree $d$. We show that the boundary $\partial G $ is of finite type and the type $r$ satisfies $r\leq 2d$.…

We study families of strongly elliptic, second order differential operators with singular coefficients on domains with conical points. We obtain uniform estimates on their inverses and on the regularity of the solutions to the associated…

In this paper, we study the strong and weak convergence rates for multi-scale one-dimensional stochastic Burgers equation. Based on the techniques of Galerkin approximation, Kolmogorov equation and Poisson equation, we obtain the slow…

We prove local well-posedness and finite-time blow-up for a restricted fourth-order Prandtl equation posed on the half-line with clamped boundary conditions. The equation arises from a two-dimensional fourth-order Prandtl system via an…

We establish resolvent estimates in spaces of bounded solenoidal functions for the Stokes operator in a bounded domain $\Omega$ in $R^d$ under the assumptions that $\Omega$ is $C^1$ for $d\ge 3$ and Lipschitz for $d=2$. As a corollary, it…

Let V $\subset$ C n be an equidimensional algebraic set and g be an n-variate polynomial with rational coefficients. Computing the critical points of the map that evaluates g at the points of V is a cornerstone of several algorithms in real…

We show that if a bounded pseudoconvex domain satisfies the solvability of the bounded $\bar{\partial}$ problem, then the ideal of bounded holomorphic functions vanishing at a point in the domain is finitely generated. We also prove a…

We prove fine higher regularity results of Calder\'on-Zygmund-type for equations involving nonlocal operators modelled on the fractional $p$-Laplacian with possibly discontinuous coefficients of VMO-type. We accomplish this by establishing…

We obtain new two-sided norm estimates for the family of Bergman-type projections arising from the standard weights $(1-|z|^2)^{\alpha}$ where $\alpha>-1$. As $\alpha\to -1$, the lower bound is sharp in the sense that it asymptotically…

Pointwise error analysis of the linear finite element approximation for $-\Delta u + u = f$ in $\Omega$, $\partial_n u = \tau$ on $\partial\Omega$, where $\Omega$ is a bounded smooth domain in $\mathbb R^N$, is presented. We establish…

We study dynamical properties of blowup solutions to the focusing $L^2$-supercritical nonlinear fractional Schr\"odinger equation \[ i\partial_t u -(-\Delta)^s u = -|u|^\alpha u, \quad u(0) = u_0, \quad \text{on } [0,\infty) \times…

The linearly constrained matrix rank minimization problem is widely applicable in many fields such as control, signal processing and system identification. The tightest convex relaxation of this problem is the linearly constrained nuclear…