$\mathbf{\nabla^2 u + f (u) = 0}$上下解的比较定理及其在涡量水波中的应用

拟线性方程 $\Delta_p u = V|u|^{p-2}u$ 及一般拟线性方程类的下解增长性

Growth of subsolutions of $Δ_p u = V|u|^{p-2}u$ and of a general class of quasilinear equations

本文在完备黎曼流形上研究一般拟线性方程下解正部的最小增长积分估计，不依赖额外几何假设，并由此得到缓慢增长解的 Liouville 型定理。

Local boundedness for $p$-Laplacian with degenerate coefficients

We study local boundedness for subsolutions of nonlinear nonuniformly elliptic equations whose prototype is given by $\nabla \cdot (λ|\nabla u|^{p-2}\nabla u)=0$, where the variable coefficient $0\leqλ$ and its inverse $λ^{-1}$ are allowed to be unbounded. Assuming certain integrability conditions on $λ$ and $λ^{-1}$ depending on $p$ and the dimension, we show local boundedness. Moreover, we provide counterexamples to regularity showing that the integrability conditions are optimal for every $p>1$.

Vorticity convergence from Boltzmann to 2D incompressible Euler equations below Yudovich class

It is challenging to perform a multiscale analysis of mesoscopic systems exhibiting singularities at the macroscopic scale. In this paper, we study the hydrodynamic limit of the Boltzmann equations $$\mathrm{St} \partial_t F + v\cdot \nabla_x F = \frac{1}{\mathrm{Kn}} Q(F ,F ) $$ toward the singular solutions of 2D incompressible Euler equations whose vorticity is unbounded $$\partial_t u + u \cdot \nabla_x u + \nabla_x p = 0,\text{div }u =0.$$ We obtain a microscopic description of the singularity through the so-called kinetic vorticity and understand its behavior in the vicinity of the macro…

H-principle for the 2D incompressible porous media equation with viscosity jump

In this work we extend the results in [6,32] on the 2D IPM system with constant viscosity (Atwood number $A_μ=0$) to the case of viscosity jump ($|A_μ|<1$). We prove a h-principle whereby (infinitely many) weak solutions in $C_tL_{w^*}^{\infty}$ are recovered via convex integration whenever a subsolution is provided. As a first example, non-trivial weak solutions with compact support in time are obtained. Secondly, we construct mixing solutions to the unstable Muskat problem with initial flat interface. As a byproduct, we check that the connection, established by Székelyhidi for $A_μ=0$, betwe…

Sobolev spaces and operators vorticity and the gradient of the divergence

In a bounded domain $G$ with smooth border studied boundary value and spectral problems for operators of the rotor (vortex) and the gradient of the divergence $+λ\,I$ in the Sobolev spaces. For $λ\neq 0$ these operators are reducible ( by B. Veinberg and V. Grushin method) to elliptical matrices and the boundary value problems satisfy the conditions of V. Solonnikov's ellipticity. Useful properties of solutions of these spectral problems follow from the theory and estimates. The $\nabla \text{div}$ and $ \text{rot}$ operators have self-adjoint extensions $\mathcal{N}_d$ and $\mathcal{S}$ in or…

Vortex motion for the lake equations

The lake equations $$\left\{\begin{aligned} \nabla \cdot \big( b \, \mathbf{u}\big) &= 0 & & \text{on}\ \mathbb{R}\times D,\\ \partial_t\mathbf{u} + (\mathbf{u}\cdot \nabla)\mathbf{u} &= -\nabla h & & \text{on}\ \mathbb{R}\times D ,\\ \mathbf{u} \cdot \boldsymbolν &= 0 & & \text{on}\ \mathbb{R}\times\partial D . \end{aligned}\right.$$ model the vertically averaged horizontal velocity in an inviscid incompressible flow of a fluid in a basin whose variable depth $b : D \to [0, + \infty)$ is small in comparison with the size of its two-dimensional projection $D \subset \mathbb{R}^2$. When the dep…