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Regularity of renormalized solutions in the Boltzmann equation with long-range interactions. (English) Zbl 1234.35172

Summary: It is well-established that renormalized solutions of the Boltzmann equation enjoy some kind of regularity, or at least compactness, in the velocity variable when the angular collision kernel is nonintegrable. However, obtaining explicit estimates in convenient and natural functional settings proves rather difficult. In this work, we derive a velocity smoothness estimate from the a priori control of the renormalized dissipation. As a direct consequence of our result, we show that, in the presence of long-range interactions, any renormalized solution \(F(t, x, v)\) to the Boltzmann equation satisfies locally \(\frac{F}{1+F} \in W^{s,p}_{t,x,v}\) for every \(1 \leq p < \frac{D}{D-1}\) and for some \(s > 0\) depending on \(p\). We also provide an application of this new estimate to the hydrodynamic limit of the Boltzmann equation without cutoff.

MSC:

35Q20 Boltzmann equations
35B65 Smoothness and regularity of solutions to PDEs
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