Hölder estimates for parabolic and transport equations with Besov coefficients
Résumé
For the parabolic and the transport equations whose coefficient of the first term, denoted by $b$, can be in a negative Besov space,
we provide a control of the solution in the H\"older space $C^\gamma$, $\gamma \in (0,1)$, regardless of $b$ and matching
with the source functions regularity. We define some kind of solutions
which do not require $b$ to be Lipschitz continuous.
\begin{trivlist}
\item For a ``very weak" solution, called in the paper \textit{mild vanishing viscous}, in $C^\gamma$, there is no regularity constraint on $b$.
\item If $b$ lies in a $\tilde \gamma$-H\"older space, $\tilde \gamma>1-\gamma$, we establish that there is a weak solution in a $\gamma$-H\"older space.
\item If $b$ is supposed to be divergence free, we obtain the same result for $b$ having a negative regularity in space, precisely in $L^\infty(B_{\infty,\infty}^{-\beta})$ for $\beta<\gamma$.
\item Finally, if $\tilde \gamma > \frac{1}{1+\gamma}$
with a \textit{vanishing viscous} condition, then the selected solution is unique.
In this case, there is somehow a \textit{regularisation} by turbulence (corresponding to the Reynolds number going to $+ \infty$); the \textit{vanishing viscosity} overwhelms the potential blowing up of the rough coefficients.
\end{trivlist}
Importantly, as a by-product of our analysis, we are able to give a meaning of a product of distributions.
For $b$ lying in a $C^\gamma$, we obtain the same condition as for the usual Bony's para-product; but in a weaker solution framework, the product is defined beyond the para-product condition and even with no constraint at all in the \textit{mild vanishing viscous} context.
We also obtain that the time averaging of the distributions product is $\gamma$-H\"older continuous.
These new results happens because, in the considered product, one of the distribution is the gradient of a solution of a Partial Differential Equation.
Thanks to our analysis, we also get a H\"older control of a solution of the inviscid Burgers' equation. Under some regularity and vanishing viscous constraint, the solution is a weak solution and is unique in a certain sense.
The vanishing viscous procedure allows to avoid the well-known critical time
of the solution built by characteristics.
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