We prove that, unlike in several space dimensions, there is no critical (nonlinear) diffusion coefficient for which solutions to the one dimensional quasilinear Smoluchowski-Poisson equation with small mass exist globally while finite time blowup could occur for solutions with large mass.
We prove a constructive stable ODE-type blowup result for open sets of solutions to a family of quasilinear wave equations in three spatial dimensions featuring a Riccati-type derivative-quadratic semilinear term. The singularity is more severe than a shock in that the solution itself blows up like the log of the distance to the blowup-time. We assume that the quasilinear terms satisfy certain structural assumptions, which in particular ensure that the "elliptic part" of the wave operator vanishes precisely at the singular points. The initial data are compactly supported and can be small or la…
Global existence, blowup phenomena, and asymptotic behavior for quasilinear Schrödinger equations
China People’s Police University（中国人民警察大学）；Hebei Key Laboratory of Information Support Technology for Smart Policing, China People’s Police University（中国人民警察大学智慧警务信息支撑技术河北省重点实验室）；Tianjin University（天津大学）；Xinjiang University of Technology（新疆理工学院）
For the first order 1D $n\times n$ quasilinear strictly hyperbolic system $\partial_tu+F(u)\partial_xu=0$ with $u(x, 0)=\varepsilon u_0(x)$, where $\varepsilon>0$ is small, $u_0(x)\not\equiv 0$ and $u_0(x)\in C_0^2(\mathbb R)$, when at least one eigenvalue of $F(u)$ is genuinely nonlinear, it is well-known that on the finite blowup time $T_{\varepsilon}$, the derivatives $\partial_{t,x}u$ blow up while the solution $u$ keeps to be small. For the 1D scalar equation or $2\times 2$ strictly hyperbolic system (corresponding to $n=1, 2$), if the smooth solution $u$ blows up in finite time, then the…
In three spatial dimensions, we study the Cauchy problem for the model wave equation $- \partial_t^2 Ψ+ (1 + Ψ)^P ΔΨ= 0$ for $P \in \lbrace 1,2 \rbrace$. We exhibit a stable form of finite-time Tricomi-type degeneracy formation that has not previously been studied for quasilinear wave equations. Specifically, using only energy methods and ODE-type techniques, we exhibit an open (in an appropriate Sobolev topology) set of data such that $Ψ$ is initially near $0$ while $1 + Ψ$ vanishes in finite time. In fact, generic data profiles, when appropriately rescaled, lead to the vanishing of $1 + Ψ$ i…
We construct examples of finite time singularity from smooth data for linear uniformly parabolic systems in the plane. We obtain similar examples for quasilinear systems with coefficients that depend only on the solution.
We study the validity of a Smoluchowski-Kramers approximation for a class of wave equations in a bounded domain of $\mathbb{R}^n$ subject to a state-dependent damping and perturbed by a multiplicative noise. We prove that in the small mass limit the solution converges to the solution of a stochastic quasilinear parabolic equation where a noise-induced extra drift is created.
Blowup of smooth solutions to 2D quasilinear wave equations with higher-order null conditions and short pulse initial data
School of Mathematical Sciences and IMS, Nanjing Normal University（南京师范大学数学科学学院和集成数学科学研究所）；Institute of Mathematical Sciences, The Chinese University of Hong Kong（香港中文大学数学科学研究所）
Unboundedness of solutions to a supercritical quasilinear parabolic-parabolic Keller-Segel system in dimension 1 and to a critical one in higher dimensions
Alinhac solved a long-standing open problem in 2001 and established that quasilinear wave equations in two space dimensions with quadratic null nonlinearities admit global-in-time solutions, provided that the initial data are compactly supported and sufficiently small in Sobolev norm. In this work, Alinhac obtained an upper bound with polynomial growth in time for the top-order energy of the solutions. A natural question then arises whether the time-growth is a true phenomena, despite the possible conservation of basic energy. Analogous problems are also of central importance for Schrödinger e…
We prove a stable shock formation result for a large class of systems of quasilinear wave equations in two spatial dimensions. We give a precise description of the dynamics all the way up to the singularity. Our main theorem applies to systems of two wave equations featuring two distinct wave speeds and various quasilinear and semilinear nonlinearities, while the solutions under study are (non-symmetric) perturbations of simple outgoing plane symmetric waves. The two waves are allowed to interact all the way up to the singularity. Our approach is robust and could be used to prove shock formati…
In this paper, we deal with the Cauchy problem of the quasilinear Schödinger equation \begin{equation*} \left\{ \begin{array}{lll} iu_t=Δu+2uh'(|u|^2)Δh(|u|^2)+(W(x)\ast|u|^2)u,\ x\in \mathbb{R}^N,\ t>0\\ u(x,0)=u_0(x),\quad x\in \mathbb{R}^N. \end{array}\right. \end{equation*} Here $h(s)$ and $W(x)$ are some real valued functions. Our focus is to investigate how the interplay between the potential $W(x)$ and the quasilinear presence $h(s)$ affects the blowup in finite time and global existence of the solution. In a special, we can obtain the watershed condition on $W(x)$ in the following sens…
Two global stability theorems for small-data $3D$ compressible Euler solutions: Unique maximal globally hyperbolic developments with gradient-blowup & smooth global existence in the entire spacetime
Georgia Institute of Technology（佐治亚理工学院）；Vanderbilt University（范德堡大学）
