偏微分方程分析

We prove global well-posedness for the cubic nonlinear Schr\"odinger equation for periodic initial data in the mass-critical dimension $d=2$ for initial data of arbitrary size in the defocusing case and data below the ground state threshold…

This work is concerned with the generation of decay estimates in the velocity variable for solutions of the space-inhomogeneous Boltzmann equation without cutoff on a bounded spatial domain for hard and moderately soft potentials. We work…

In this article, we consider $(p,q)$-extension operators, $1 < q \le p < \infty$, on Sobolev spaces. Based on composition operators on Sobolev spaces, we construct the extension operators in outward cuspidal domains with estimates of their…

We analyze a quasi-incompressible Cahn--Hilliard--Navier--Stokes system (qCHNS) for two-phase flows with unmatched densities. The order parameter is the volume fraction difference of the two fluids, while mass-averaged velocity is adopted.…

In this paper, we fully resolve the question of whether the Regularity problem for the parabolic PDE $-\partial_tu + \mbox{div}(A\nabla u)=0$ on a Lipschitz cylinder $\mathcal O\times\mathbb R$ is solvable for some $p\in (1,\infty)$ under…

Using the recent analysis of the output of the low-energy resolvent of Schr\"odinger operators on asymptotically conic manifolds (including Euclidean space) when the potential is short-range, we produce detailed asymptotic expansions for…

In this paper, we prove that flat free boundaries of solutions to inhomogeneous one-phase Stefan problem are $C^{1,\alpha}$. The method consists of employing a hodograph transform and deriving the regularity via a linearization technique,…

We consider the three-dimensional steady Navier-Stokes system in the exterior of an infinite cylinder under the action of an external force. We construct solutions in the class of vertically uniform flows which vanish at horizontal…

We study a mathematical model of in situ leaching of rare metals, in which the joint filtration of two liquids is governed by the microscopic model $\mathbb{A}^{\varepsilon}$. A key difficulty is the unknown (free) boundary $\Gamma(r)$…

We consider a power-law thin-film equation for strongly shear-thinning fluids. Weak solutions to this equation have been constructed more than twenty years ago by Ansini and Giacomelli. Here, we pass over to analyzing strong solutions with…

In this paper weighted Dirichlet-type inequalities for the decreasing rearrangement in cylinders are proved. A weighted isoperimetric inequality is also obtained.

We study the efficiency of the first Dirichlet eigenfunction $u$ on bounded convex domains $\Omega \subset \mathbb{R}^N$, defined as the ratio between the mean value of $u$ on $\Omega$ and its maximum value. By exploiting improved…

We consider the characteristic problem for the ultrahyperbolic equation in the Euclidean space. The value of a solution is prescribed on the characteristic hyperplane. A well-posed set-up of the problem is discussed. We obtain a certain…

This paper is devoted to investigating the interior $C^{1, \alpha}$ regularity of viscosity solutions to the nonlocal double phase equations $$ \int_{\mathbb{R}^d}…

For $L^2$-normalized joint eigenfunctions in a quantum integrable system, [GT20] gave polynomial improvements over the standard H\"omander bounds for typical points. In this paper, we improve their result by establishing a sharp bound of…

The large time behavior of solutions to the following generalized Burgers-Fisher-KPP equation $$ \partial_tu=u_{xx}+k(u^n)_x+u^p-u^q, \quad (x,t)\in\mathbb{R}\times(0,\infty), $$ with $n\geq2$, $p>q\geq1$ and $k\in\mathbb{R}$, is considered…

We consider the compressible Euler system for ideal gas flow in the absence of any forces except the internal thermodynamic pressure. In this setting, and in dimensions higher 1, it is known that wave-focusing can drive Euler solutions to…

In this paper, we establish Brezin-Li-Yau type lower bounds for averaged sums of Dirichlet eigenvalues of the Laplacian and poly-Laplacian on bounded domains in Euclidean spaces. By deriving expansions of two binary polynomials which may be…

We prove lossless Strichartz estimates at the critical exponent $q_c = \frac{2(n+1)}{n-1}$ and the endpoint exponent pair $\left(2,\frac{2(n-1)}{n-3}\right)$ for the Schr\"{o}dinger equation on rectangular tori of dimension $n-1$ with…

We consider steady water waves in a two-dimensional channel bounded below by a flat, rigid bottom and above by a free surface. Surface tension is neglected, and the flow is rotational with constant vorticity $a$. We analyze an analytic…