Regularisation by turbulence for transport equations in Hölder space
Résumé
We consider a transport equation whose the coefficient b can be in a negative Besov space, we provide controls in γ-Hölder spaces, γ ∈ (0, 1), of some kind of vanishing viscosity solutions based on a parabolic approximation. The regularity in γ-Hölder space of the solution exactly matches the one of the source functions, but the regularisation occurs on the coefficients irregularity. Indeed, we define some kind of solutions of the transport equation which do not require b to be Lipschitz continuous. If b lies in a α-Hölder space, α > 1−γ, then we establish that there is a weak solution in a γ-Hölder space. If b is supposed to be divergence free, then we obtain the same result for b having a negative regularity in space, precisely in L ∞ (B −β ∞,∞) for β < γ. Finally, if we consider a "very weak" solution, called in the paper mild vanishing viscous, also in a γ-Hölder space, then there is no regularity constraint on b. In this case, there is somehow a mild infinite regularisation by turbulence. The vanishing viscosity allows to overwhelm the potential blowing up of the rough coefficients, this is what we call a regularisation by turbulence. However, we do not get uniqueness of the considered built solution; we even conjecture that there is no unique selection of the parabolic approximation in such a rough framework. Importantly, as a by-product of our analysis, we can give a meaning of a product of distributions. For b lying in a γ-Hölder space, we obtain the same condition as for the usual Bony's paraproduct; but in a weaker solution framework, the product is defined beyond the paraproduct condition and even with no constraint at all in the mild vanishing viscous context. We also obtain that the time averaging of the distributions product is γ-Hölder continuous. These strong results happens because one of the distribution is the gradient of a solution, in a certain sense, of the transport equation. Thanks to our analysis, we also get a Hölder control of a solution of the inviscid Burgers' equation. The vanishing viscous procedure seems to avoid the well-known time of the regularity blowing-up of the solution built by characteristics.
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