| Contents |
| This is our current work in progress on proving a conjecture related to the viscous Burgers equation
\begin{equation*}
u_t+u\cdot u_x-\nu u_{xx}=f(t,x),
\end{equation*}
where the forcing $f(t,x)$ is a bounded and continuous with respect to time function.
The conjecture is that the attractor of the viscous Burgers equation on the line with periodic boundary conditions and nonautonomous forcing
is composed of an unique solution $u_\infty$ for any forcing bounded, continuous in time, and periodic in space. Moreover, the convergence
towards $u_\infty$ is exponential, and $||u_\infty||\approx O(1/c)$, where $c=\int_\Omega u_0$.
The conjecture is supported by previous works by H.R. Jauslin, H.O. Kreiss, J. Moser (1999), Y. Sinai (1991), and our recent computer
assisted proof of existence of globally attracting solutions of the viscous Burgers equation on the line with periodic boundary conditions and
nonautonomous forcing.
Methods we apply are general and, if successful, shall be applied to other similar partial differential equations including the
Navier-Stokes equations. |
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