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Register a new userOn the periodic Zakharov-Kuznetsov equation
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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-linear type for initial data in any Sobolev space on the torus, in sharp contrast with its semi-linear character in the $mathbb{R}^2$ and $mathbb{R}times mathbb{T}$ settings.
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We solve the group classification problem for the $2+1$ generalized quantum Zakharov-Kuznetsov equation. Particularly we consider the generalized equation $u_{t}+fleft( uright) u_{z}+u_{zzz}+u_{xxz}=0$, and the time-dependent Zakharov-Kuznetsov equation $u_{t}+delta left( tright) uu_{z}+lambda left( tright) u_{zzz}+varepsilon left( tright) u_{xxz}=0$% . Function $fleft( uright) $ and $delta left( tright) ,~lambda left( tright) $,~$varepsilon left( tright) $ are determine in order the equations to admit additional Lie symmetries. Finally, we apply the Lie invariants to find similarity solutions for the generalized quantum Zakharov-Kuznetsov equation.
We consider the cubic Hyperbolic Schrodinger equation eqref{eq:nls} on torus $T^2$. We prove that sharp $L^4$ Strichartz estimate, which implies that eqref{eq:nls} is analytic locally well-posed in in $H^s(T^2)$ with $s>1/2$, meanwhile, the ill-posedness in $H^s(T^2)$ for $s<1/2$ is also obtained. The main difficulty comes from estimating the number of representations of an integer as a difference of squares.
The aim of this paper is to investigate the Cauchy problem for the periodic fifth order KP-I equation [partial_t u - partial_x^5 u -partial_x^{-1}partial_y^2u + upartial_x u = 0,~(t,x,y)inmathbb{R}timesmathbb{T}^2] We prove global well-posedness for constant $x$ mean value initial data in the space $mathbb{E} = {uin L^2,~partial_x^2 u in L^2,~partial_x^{-1}partial_y u in L^2}$ which is the natural energy space associated with this equation.
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.
The periodic Benjamin-Ono equation is an autonomous Hamiltonian system with a Gibbs measure on $L^2({mathbb T})$. The paper shows that the Gibbs measures on bounded balls of $L^2$ satisfy some logarithmic Sobolev inequalities. The space of $n$-soliton solutions of the periodic Benjamin-Ono equation, as discovered by Case, is a Hamiltonian system with an invariant Gibbs measure. As $nrightarrowinfty$, these Gibbs measures exhibit a concentration of measure phenomenon. Case introduced soliton solutions that are parameterised by atomic measures in the complex plane. The limiting distributions of these measures gives the density of a compressible gas that satisfies the isentropic Euler equations.
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