Regularizing Effect and Local Existence for the Non-Cutoff Boltzmann Equation

By Radjesvarane Alexandre, Yoshinori Morimoto, S. Ukai, Chao-Jiang Xu, Tong Yang. 2009

159 citations recorded by the source corpus.

The Boltzmann equation without Grad’s angular cutoff assumption is believed to have a regularizing effect on the solutions because of the non-integrable angular singularity of the cross-section. However, even though this has been justified satisfactorily for the spatially homogeneous Boltzmann equation, it is still basically unsolved for the spatially inhomogeneous Boltzmann equation. In this paper, by sharpening the coercivity and upper bound estimates for the collision operator, establishing the hypo-ellipticity of the Boltzmann operator based on a generalized version of the uncertainty principle, and analyzing the commutators between the collision operator and some weighted pseudo-differential operators, we prove the regularizing effect in all (time, space and velocity) variables on the solutions when some mild regularity is imposed on these solutions. For completeness, we also show that when the initial data has this mild regularity and a Maxwellian type decay in the velocity variable, there exists a unique local solution with the same regularity, so that this solution acquires the C∞ regularity for any positive time.

Subfields: Applied Mathematics, Mathematical Physics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/110343eb9713ccf52c8ce2488e7c8bda73bb4f55

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Luis Silvestre
2. C. Imbert
3. Clément Mouhot
4. Philip T. Gressman
5. Robert M. Strain
6. Tong Yang
7. Alexis Vasseur
8. Juan Luis V'azquez

In Mathematical Physics

1. C. Imbert
2. Luis Silvestre
3. Clément Mouhot
4. Robert M. Strain
5. Philip T. Gressman
6. Tong Yang
7. S. Ukai
8. Chao-Jiang Xu

In Mechanics of Materials

1. C. Imbert

Citations in the corpus

• The weak Harnack inequality for the Boltzmann equation without cut-off (2016, methodology): “Under these assumptions, they prove that f belongs to the Schwartz class for positive time in [7], [9], [28].”
• Global classical solutions of the Boltzmann equation without angular cut-off (2009, methodology): “In a very recent preprint from 2009, perhaps the first local existence theorem for large data and moderate angular singularities 0 s 1/2 has been shown by Alexandre-Morimoto-Ukai-Xu-Yang [8] for initial data which is somewhat smoother than ours, they also show the C∞t,x,v regularizing effect.”
• Harnack inequality for kinetic Fokker-Planck equations with rough coefficients and application to the Landau equation (2016, background): “Finally, we highlight the related results of regularisation for the Boltzmann equation without long-range interactions [19, 15, 16], and the related perturbative results for the Landau and (long-range interaction) Boltzmann equation [40, 38, 4, 5, 2, 62, 3].”
• Global Existence and Full Regularity of the Boltzmann Equation Without Angular Cutoff (2009, methodology): “In Sect. 6, the full regularity is proved along the approach of [ 12 ].”