偏微分方程分析

We prove the so-called second case of the fractional Korn inequality for uniform domains. We obtain this result as an application of a novel fractional Korn-type inequality formulated in terms of truncated seminorms, which turns out to be…

This paper investigates the global controllability properties of the Cahn--Hilliard equation posed on the $d$-dimensional flat torus $\mathbb{T}^d$. We first establish small-time global approximate controllability of the system by means of…

In this paper we extend some results presented in \cite{julio} to the case of the $p$-Laplacian operator. More precisely, we consider a model that couples a local $p$-Laplacian operator with a nonlocal $p$-Laplacian operator through source…

The sharpness of various Hardy-type inequalities is well-understood in the reversible Finsler setting; while infinite reversibility implies the failure of these functional inequalities, cf. Krist\'aly, Huang, and Zhao [Trans. Am. Math.…

We extend a classical model of continuous opinion formation to explicitly include an age-structured population. We begin by considering a stochastic differential equation model which incorporates ageing dynamics and birth/death processes,…

We formulate new admissibility criteria for initial value problems motivated by the least action principle. These are applied to a two-dimensional Riemann initial value problem for the isentropic compressible Euler fluid flow. It is shown…

We study the energy-critical half-wave maps equation: \[ \partial_t \mathbf{u} = \mathbf{u} \times |D| \mathbf{u} \] for $\mathbf{u} : [0, T) \times \mathbb{R} \to \mathbb{S}^2$. Our main result establishes the global existence and…

Some qualitative properties of radially symmetric solutions to the non-homogeneous heat equation with critical density and weighted source $$ |x|^{-2}\partial_tu=\Delta u+|x|^{\sigma}u^p, \quad (x,t)\in\mathbb{R}^N\times(0,T), $$ are…

The analysis of non-local regularisations of scalar conservation laws is an active research program. Applications of such equations are found in the modelling of physical phenomena such as traffic flow. In this paper, we propose a novel…

In this article we prove the Pohozaev identity for the semilinear Dirichlet problem of the form $-\Delta u + a(-\Delta)^s u = f(u)$ in $\Omega$, and $u=0$ in $\Omega^c$, where $a$ is a non-negative constant and $\Omega$ is a bounded $C^2$…

We consider a boundary value problem of a stationary advection equation in a bounded domain with Lipschitz boundary. It is known to be well-posed in $L^p$-based function spaces for $1 < p < \infty$ under the separation condition of the…

We consider the Cauchy problem to the 3D fractional Schr\"odinger equation with quadratic interaction of $u\bar u$ type. We prove the global existence of solutions and scattering properties for small initial data. For the proof, one novelty…

In present paper, we study the limit behavior of normalized ground states for the following mass critical Kirchhoff equation $$ \left\{\begin{array}{ll} -(a+b\int_{\Omega}|\nabla u|^2\mathrm{d}x)\Delta u+V(x)u=\mu…

We study the existence of solutions to the fractional semilinear heat equation with a singular inhomogeneous term. For this aim, we establish decay estimates of the fractional heat semigroup in several uniformly local Zygumnd spaces.…

We propose an alternative proof of the classical result of Type-I blowup with log correction for the semilinear heat equation. Compared with previous proofs, we use a novel idea of enforcing stable normalizations for perturbations around…

The nonlinear Hartree equation (NLH) in the Heisenberg picture admits steady states of the form $\gamma_f=f(-\Delta)$ representing quantum states of infinitely many particles. In this article, we consider the time evolution of perturbations…

In this paper we unveil novel monotonicity conditions applicable for Mean Field Games through the exploration of finite dimensional $canonical\ transformations$. Our findings contribute to establishing new global well-posedness results for…

We investigate the global-in-time existence and uniqueness of weak solutions for a family of equations introduced by Moffatt to model magnetic relaxation. These equations are topology-preserving and admit all stationary solutions to the…

We study the incoming boundary value problem for the stationary linearized Boltzmann equation in bounded convex domains. The geometry of the domain has a dramatic effect on the space of solutions. We prove the existence of solutions in…

In this paper, we consider the Strichartz estimates for orthonormal systems in the context of randomization. Frank, Lewin, Lieb, and Seiringer first proved the orthonormal Strichartz estimates. After that, many authors have studied this…