| Contents |
| We prove the existence of globally attracting solutions of the viscous
Burgers equation with periodic boundary conditions on the line for some
particular choices of viscosity and non-autonomous forcing
\begin{equation*}
u_t+u\cdot u_x-\nu u_{xx}=f(t,x).
\end{equation*}
The attracting solution is periodic if the forcing is periodic.
The convergence towards attracting solution is exponential.
The proof is computer assisted.
The method is general and can be applied to other similar partial differential equations
including the Navier-Stokes equations.
The technique we use is not restricted to some particular type of equation nor to the dimension one,
as we are not using any maximum principles, nor unconstructive functional analysis techniques.
We need some kind of 'energy' decay as a global
property of our dissipative PDEs and then if the system exhibits an attracting
orbit, then we should in principle be able to prove it independent of the dimensionality
of of the system. At the present state our technique strongly relies on the existence of good coordinates,
the Fourier modes in the considered example. We hope that the further development of the rigorous numerics
for dissipative PDEs based on other function bases, e.g. for example the finite elements, should
allow to treat also different domains and boundary conditions in near future. |
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