Unique Conservative Solutions to a Variational Wave Equation

Relying on the analysis of characteristics, we prove the uniqueness of conservative solutions to the variational wave equation $u_{tt}-c(u) (c(u)u_x)_x=0$. Given a solution $u(t,x)$, even if the wave speed $c(u)$ is only H\"older continuous in the $t$-$x$ plane, one can still define forward and backward characteristics in a unique way. Using a new set of independent variables $X,Y$, constant along characteristics, we prove that $t,x,u$, together with other variables, satisfy a semilinear system with smooth coefficients. From the uniqueness of the solution to this semilinear system, one obtains the uniqueness of conservative solutions to the Cauchy problem for the wave equation with general initial data $u(0,\cdot)\in H^1(\mathbb{R})$, $u_t(0,\cdot)\in L^2(\mathbb{R})$

Global solutions of quasi-linear Hamiltonian mKdV equation

We study the initial value problem of quasi-linear Hamiltonian mKdV equations. Our goal is to prove the global-in-time existence of a solution given sufficiently smooth, localized, and small initial data. To achieve this, we utilize the bootstrap argument, Sobolev energy estimates, and the dispersive estimate. This proof relies on the space-time resonance method, as well as a bilinear estimate developed by Germain, Pusateri, and Rousset.Comment: 25 page

Generic regularity of conservative solutions to Camassa-Holm type equations

This paper mainly proves the generic properties of the Camassa-Holm equation and the two-component Camassa-Holm equation by Thom's transversality Lemma. We reveal their differences in generic regularity and singular behavior

Initial-boundary value problems for Poiseuille flow of nematic liquid crystal via full Ericksen-Leslie model

In this paper, we study the initial-boundary value problem for the Poiseuille flow of hyperbolic-parabolic Ericksen-Leslie model of nematic liquid crystals in one space dimension. Due to the quasilinearity, the solution of this model in general forms cusp singularity. We prove the global existence of H\"older continuous solution, which may include cusp singularity, for initial-boundary value problems with different types of boundary conditions.Comment: 36 pages, 3 figure