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Register a new userOn the Cauchy problem for the periodic fifth-order KP-I equation
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Authors
Tristan Robert
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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.
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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.
Let $H$ be a norm of ${bf R}^N$ and $H_0$ the dual norm of $H$. Denote by $Delta_H$ the Finsler-Laplace operator defined by $Delta_Hu:=mbox{div},(H( abla u) abla_xi H( abla u))$. In this paper we prove that the Finsler-Laplace operator $Delta_H$ acts as a linear operator to $H_0$-radially symmetric smooth functions. Furthermore, we obtain an optimal sufficient condition for the existence of the solution to the Cauchy problem for the Finsler heat equation $$ partial_t u=Delta_H u,qquad xin{bf R}^N,quad t>0, $$ where $Nge 1$ and $partial_t:=partial/partial t$.
In this article, we address the Cauchy problem for the KP-I equation [partial_t u + partial_x^3 u -partial_x^{-1}partial_y^2u + upartial_x u = 0] for functions periodic in $y$. We prove global well-posedness of this problem for any data in the energy space $mathbb{E} = left{uin L^2left(mathbb{R}timesmathbb{T}right),~partial_x u in L^2left(mathbb{R}timesmathbb{T}right),~partial_x^{-1}partial_y u in L^2left(mathbb{R}timesmathbb{T}right)right}$. We then prove that the KdV line soliton, seen as a special solution of KP-I equation, is orbitally stable under this flow, as long as its speed is small enough.
This paper mainly investigates the Cauchy problem of the spatially weighted dissipative equation with initial data in the weighted Lebesgue space. A generalized Hankel Transform is introduced to derive the analytical solution and a special Youngs Inequality has been applied to prove the space-time estimates for this type of equation.
We are considering the asimptotic behavior as $ttoinfty$ of solutions of the Cauchy problem for parabolic second order equations with time periodic coefficients. The problem is reduced to considering degenerate time-homogeneous diffusion processes on the product of a unit circle and Euclidean space.
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