ﻻ يوجد ملخص باللغة العربية
In this paper, we consider a 1D periodic transport equation with nonlocal flux and fractional dissipation $$ u_{t}-(Hu)_{x}u_{x}+kappaLambda^{alpha}u=0,quad (t,x)in R^{+}times S, $$ where $kappageq0$, $0<alphaleq1$ and $S=[-pi,pi]$. We first establish the local-in-time well-posedness for this transport equation in $H^{3}(S)$. In the case of $kappa=0$, we deduce that the solution, starting from the smooth and odd initial data, will develop into singularity in finite time. If adding a weak dissipation term $kappaLambda^{alpha}u$, we also prove that the finite time blowup would occur.
In dimension n isolated singularities -- at a finite point or at infinity -- for solutions of finite total mass to the n-Liouville equation are of logarithmic type. As a consequence, we simplify the classification argument in arXiv:1609.03608 and est
We consider the Cauchy problem associated with the Zakharov-Kuznetsov equation, posed on $mathbb{T}^2$. We prove the local well-posedness for given data in $H^s(mathbb{T}^2)$ whenever $s>5/3$. More importantly, we prove that this equation is of quasi
We prove the discontinuity for the weak $ L^2(T) $-topology of the flow-map associated with the periodic Benjamin-Ono equation. This ensures that this equation is ill-posed in $ H^s(T) $ as soon as $ s<0 $ and thus completes exactly the well-posedness result obtained by the author.
We prove that, for some irrational torus, the flow map of the periodic fifth-order KP-I equation is not locally uniformly continuous on the energy space, even on the hyperplanes of fixed x-mean value.
We study the existence and stability of the standing waves for the periodic cubic nonlinear Schrodinger equation with a point defect determined by a periodic Dirac distribution at the origin. This equation admits a smooth curve of positive periodic s