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A study of Long time behaviours of solutions of Zakharov-Kuznetsov Equations

Abstract : In this thesis, we study different aspects of behaviors in long time of the generalized Zakharov-Kuznetsov equations: \begin{align}\tag{gZK}\label{gZK_resume_en} \partial_t u + \partial_1 \left( \Delta u + u ^p \right)=0, \quad (t,\bx)\in \mathbb{R}_t\times \mathbb{R}^d_\bx, \quad d\in \left\{ 2, 3 \right\}, \quad p\geq 2. \end{align} The first chapter sums up the state-of-the-art knowledges on those equations. The second chapter is dedicated to polynomial growth of Sobolev norms $H^s(\mathbb{R}^2)$ for the equation \eqref{gZK_resume_en} in dimension $d=2$ and a power of nonlinearity $p=2$. In the third chapter, we prove existence and uniqueness of multi-solitons associated with different equation of Zakharov-Kuznetsov equations ($(d,p)=(2,2)$, $(2,3)$ et $(3,2)$). In the last chapter, we introduce some ideas of construction of a $2$-solitons with strong interaction for the \eqref{gZK_resume_en} equation in dimension $d=2$ and a non-linearity $f(u)=\vert u \vert^{p-1}u$, with $2
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Submitted on : Monday, August 24, 2020 - 11:23:29 AM
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Frédéric Valet. A study of Long time behaviours of solutions of Zakharov-Kuznetsov Equations. Analysis of PDEs [math.AP]. Université de Strasbourg, 2020. English. ⟨tel-02886810⟩

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