Uniqueness and asymptotic stability of timeperiodic solution for the fractal Burgers equation
Abstract
The paper is concerned with the timeperiodic (Tperiodic) problem of the fractal Burgers equation with a Tperiodic force on the real line. Based on the Galerkin approximates and Fourier series (transform) methods, we first prove the existence of Tperiodic solution to a linearized version. Then, the existence and uniqueness of Tperiodic solution to the nonlinear equation are established by the contraction mapping argument. Furthermore, we show that the unique Tperiodic solution is asymptotically stable. This analysis, which is carried out in energy space $ H^{1}(0,T;H^{\frac{\alpha}{2}}(R))\cap L^{2}(0,T;\dot{H}^{\alpha})$ with $1<\alpha<\frac{3}{2}$, extends the Tperiodic viscid Burgers equation in \cite{5} to the Tperiodic fractional case.
 Publication:

arXiv eprints
 Pub Date:
 July 2021
 arXiv:
 arXiv:2107.00772
 Bibcode:
 2021arXiv210700772Z
 Keywords:

 Mathematics  Analysis of PDEs