arXiv subjects

Jiahong Wu

Publications and source records attributed to Jiahong Wu.

112 records · Page 7Linked to original sources

arXiv2008

A new solution representation for the BBM equation in a quarter plane and the eventual periodicity

The initial- and boundary-value problem for the Benjamin-Bona-Mahony (BBM) equation is studied in this paper. The goal is to understand the periodic behavior (termed as eventual periodicity) of its solutions corresponding to periodic boundary condition or periodic forcing. To this aim, we derive a new formula representing solutions of this initial- and boundary-value problem by inverting the operator $\partial_t +α\partial_x -γ\partial_{xxt}$ defined in the space-time quarter plane. The eventual periodicity of the linearized BBM equation with periodic boundary data and forcing term is established by combining this new representation formula and the method of stationary phase.

math.AP↗

arXiv2007

Hölder continuity of solutions of supercritical dissipative hydrodynamic transport equations

We examine the regularity of weak solutions of quasi-geostrophic (QG) type equations with supercritical ($α<1/2$) dissipation $(-Δ)^α$. This study is motivated by a recent work of Caffarelli and Vasseur, in which they study the global regularity issue for the critical ($α= 1/2$) QG equation \cite{CV}. Their approach successively increases the regularity levels of Leray-Hopf weak solutions: from $L^2$ to $L^\infty$, from $L^\infty$ to Hölder ($C^δ$, $δ>0$), and from Hölder to classical solutions. In the supercritical case, Leray-Hopf weak solutions can still be shown to be $L^\infty$, but it does not appear that their approach can be easily extended to establish the Hölder continuity of $L^\infty$ solutions. In order for their approach to work, we require the velocity to be in the Hölder space $C^{1-2α}$. Higher regularity starting from $C^δ$ with $δ>1-2α$ can be established through Besov space techniques and will be presented elsewhere \cite{CW6}.

math.AP↗

arXiv2007

Regularity of Hölder continuous solutions of the supercritical quasi-geostrophic equation

We present a regularity result for weak solutions of the 2D quasi-geostrophic equation with supercritical ($α< 1/2$) dissipation $(-Δ)^α$ : If a Leray-Hopf weak solution is Hölder continuous $θ\in C^δ({\mathbb R}^2)$ with $δ>1-2α$ on the time interval $[t_0, t]$, then it is actually a classical solution on $(t_0,t]$.

math.AP↗

arXiv2000

On Solutions of Three Quasi-Geostrophic Models

We consider the 2D quasi-geostrophic model and its two different regularizations. Global regularity results are established for the regularized models with subcritical or critical indices. The proof of Onsager's conjecture concerning weak solutions of the 3D Euler equations and the notion of dissipative solutions of Duchon and Robert are extended to weak solutions of the quasi-geostrophic equation.

math.AP↗