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Register a new userGlobal well-posedness and attractors for the hyperbolic Cahn-Hilliard-Oono equation in the whole space
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We prove the global well-posedness of the so-called hyperbolic relaxation of the Cahn-Hilliard-Oono equation in the whole space R^3 with the non-linearity of the sub-quintic growth rate. Moreover, the dissipativity and the existence of a smooth global attractor in the naturally defined energy space is also verified. The result is crucially based on the Strichartz estimates for the linear Scroedinger equation in R^3.
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In this paper, we continue the study of the hyperbolic relaxation of the Cahn-Hilliard-Oono equation with the sub-quintic non-linearity in the whole space $R^3$ started in our previous paper and verify that under the natural assumptions on the non-linearity and the external force, the fractal dimension of the associated global attractor in the natural energy space is finite.
In this paper, we consider the almost sure well-posedness of the Cauchy problem to the Cahn-Hilliard-Navier-Stokes equation with a randomization initial data on a torus $mathbb{T}^3$. First, we prove the local existence and uniqueness of solution. Furthermore, we prove the global existence and uniqueness of solution and give the relative probability estimate under the condition of small initial data.
In this paper we consider the hyperbolic-elliptic Ishimori initial-value problem. We prove that such system is locally well-posed for small data in $H^{s}$ level space, for $s> 3/2$. The new ingredient is that we develop the methods of Ionescu and Kenig cite{IK} and cite{IK2} to approach the problem in a perturbative way.
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 article is devoted to review the known results on global well-posedness for the Cauchy problem to the Kirchhoff equation and Kirchhoff systems with small data. Similar results will be obtained for the initial-boundary value problems in exterior domains with compact boundary. Also, the known results on large data problems will be reviewed together with open problems.
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