## Well-posedness and Ill-posedness of the Nonlinear Beam Equation

Please use this identifier to cite or link to this item: http://hdl.handle.net/1928/23378

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Title

Well-posedness and Ill-posedness of the Nonlinear Beam Equation

Author(s)

Wang, Shuxin

Advisor(s)

Blair, Matthew

Committee Member(s)

Blair, Matthew

Pereyra, M. Cristina

Lorenz, Jens

Metcalfe, Jason L

Pereyra, M. Cristina

Lorenz, Jens

Metcalfe, Jason L

Department

University of New Mexico. Dept. of Mathematics and Statistics

Degree Level

Doctoral

Abstract

The dissertation consists of two parts, Well-posedness and ill-posedness for the nonlinear beam equation and Strichartz estimates of the beam equation on the domains.
In the first part, we will work to introduce the further studies of Strichartz estimates with initial data both in homogeneous Sobolev spaces $\dot{H}^s\times\dot{H}^{s-2}$ and in inhomogeneous Sobolev space ${H}^s\times{H}^{s-2}$. We take advantage of the Strichartz estimates to build well-posedness theorems of the nonlinear beam equations for rough data by the Picard iteration method. We will apply these methods on the nonlinear beam equation with ``energy critical, subcritical" and ``energy supercritical" cases. Since the beam equation does not satisfy finite speed propagation, we introduce the further result of the fractional chain rule to deal with the ``energy super critical" case. We obtain the global well-posedness with initial data in homogeneous Sobolev space $\dot{H}^s\times\dot{H}^{s-2}$ and local well-posedness with initial data in inhomogeneous Sobolev space ${H}^s\times{H}^{s-2}$. At the same time, we extend the range of order $s$. With the global existence for small data, we prove the scattering and asymptotic completeness result for the nonlinear beam equation. Last we prove the nonlinear beam equation is ill-posed in defocusing case $\omega=-1$ when $ 0<s<s_c=\frac{n}{2}-\frac{4}{\kappa-1}$ by small dispersion analysis of M. Christ, J. Colliander and T. Tao.
In the second part, we will study Strichartz estimates on Riemannian manifolds ($\Omega$, g) with boundary, for both the compact case and the case that is the exterior of a smooth, non-trapping obstacle in Euclidean space for the beam equation.

Date

May 2013

Subject(s)

Strichartz estimates,

Well-posedness

Ill-posedness

beam equation

Well-posedness

Ill-posedness

beam equation

##### Collections

- Mathematics [34]