Existence of solutions to Burgers equations in domains that can be transformed into rectangles
This work is concerned with Burgers equation $\partial _{t}u+u\partial_x u-\partial _x^2u=f$ (with Dirichlet boundary conditions) in the non rectangular domain $\Omega =\{(t,x)\in R^2;\ 0<t<T,\; \varphi_1(t)<x<\varphi _2(t)\}$ (where $\varphi _1(t)<\varphi _2(t)$ for all $t\in [...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Texas State University
2016-06-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2016/157/abstr.html |
| Summary: | This work is concerned with Burgers equation
$\partial _{t}u+u\partial_x u-\partial _x^2u=f$
(with Dirichlet boundary conditions) in the
non rectangular domain
$\Omega =\{(t,x)\in R^2;\ 0<t<T,\; \varphi_1(t)<x<\varphi _2(t)\}$
(where $\varphi _1(t)<\varphi _2(t)$ for all $t\in [ 0;T]$).
This domain will be transformed into a rectangle by a regular change of
variables. The right-hand side lies in the Lebesgue space $L^2(\Omega )$,
and the initial condition is in the usual Sobolev space $H_0^{1}$.
Our goal is to establish the existence, uniqueness and the optimal regularity
of the solution in the anisotropic Sobolev space. |
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| ISSN: | 1072-6691 |