# Khokhlov-Zabolotskaya-Kuznetsov Equation. Mathematical Analyze, Validation of the Approximation and Contol Method

Abstract : This work consists of two parts. In the first part we consider the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation $(u_t - u u_x -\beta u_{xx})_x -\gamma \Delta_y u =0$ in Sobolev spaces of functions periodic on $x$ and with mean value zero. The derivation of KZK from the nonlinear isentropic Navier Stokes equations and approximation their solutions (for viscous and non viscous cases), the results of the existence, uniqueness, stability and blow-up of solution of KZK equation are obtained, also a result of existence of a smooth solution of Navier-Stokes system in the half space with periodic in time mean value zero boundary conditions. In the second part we prove the local controllability of moments for two systems described by a nonlinear evolution equation in Banach space and by a nonlinear heat equation when the control is a multiplier on the right-hand side. For this two systems with integral overdetermination we obtain sufficient conditions on the size of the neighborhood from which we can take the function from the overdetermination condition so that the inverse problem is uniquely solvable. We also prove the controllability result for linearized KZK equation.
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https://tel.archives-ouvertes.fr/tel-00126487
Contributor : Anna Rozanova-Pierrat <>
Submitted on : Thursday, January 25, 2007 - 10:48:54 AM
Last modification on : Wednesday, December 9, 2020 - 3:08:55 PM
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• HAL Id : tel-00126487, version 1

### Citation

Anna Rozanova-Pierrat. Khokhlov-Zabolotskaya-Kuznetsov Equation. Mathematical Analyze, Validation of the Approximation and Contol Method. Mathematics [math]. Université Pierre et Marie Curie - Paris VI, 2006. English. ⟨tel-00126487⟩

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