<p>We prove that solutions to the Boltzmann equation without cut-off satisfying pointwise bounds on some observables (mass, pressure, and suitable moments) enjoy a uniform bound in <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5356_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">$L^\infty $</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>∞</mi> </msup> </math></EquationSource> </InlineEquation> in the case of hard potentials. As a consequence, we derive <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5356_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="28" /> </InlineMediaObject> <EquationSource Format="TEX">$C^{\infty }$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mi>∞</mi> </msup> </math></EquationSource> </InlineEquation> estimates and decay estimates for all derivatives, conditional to these macroscopic bounds. Our <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5356_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">$L^\infty $</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>∞</mi> </msup> </math></EquationSource> </InlineEquation> estimates are uniform in the limit <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5356_Article_IEq4.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">$s \nearrow 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>↗</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and hence we recover the same results also for the Landau equation.</p>

Regularity for the Boltzmann Equation Conditional to Pressure and Moment Bounds

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2025年7月2日

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Xavier Fernández-Real,
Xavier Ros-Oton,
Marvin Weidner

摘要

We prove that solutions to the Boltzmann equation without cut-off satisfying pointwise bounds on some observables (mass, pressure, and suitable moments) enjoy a uniform bound in $L^\infty $ $L^{\infty}$ in the case of hard potentials. As a consequence, we derive $C^{\infty }$ $C^{\infty}$ estimates and decay estimates for all derivatives, conditional to these macroscopic bounds. Our $L^\infty $ $L^{\infty}$ estimates are uniform in the limit $s \nearrow 1$ $s ↗ 1$ and hence we recover the same results also for the Landau equation.