KRM
Uniqueness of solutions for the non-cutoff Boltzmann equation with soft potential
Radjesvarane Alexandre Yoshinori Morimoto Seiji Ukai Chao-Jiang Xu Tong Yang
Kinetic & Related Models 2011, 4(4): 919-934 doi: 10.3934/krm.2011.4.919
In this paper, we consider the Cauchy problem for the non-cutoff Boltzmann equation in the soft potential case. By using a singular change of velocity variables before and after collision, we prove the uniqueness of weak solutions to the Cauchy problem in the space of functions with polynomial decay in the velocity variable.
keywords: uniqueness of solution. Boltzmann equation singular change of velocity variables
KRM
Bounded solutions of the Boltzmann equation in the whole space
Radjesvarane Alexandre Yoshinori Morimoto Seiji Ukai Chao-Jiang Xu Tong Yang
Kinetic & Related Models 2011, 4(1): 17-40 doi: 10.3934/krm.2011.4.17
We construct bounded classical solutions of the Boltzmann equation in the whole space without specifying any limit behaviors at the spatial infinity and without assuming the smallness condition on initial data. More precisely, we show that if the initial data is non-negative and belongs to a uniformly local Sobolev space in the space variable and a standard Sobolev space with Maxwellian type decay property in the velocity variable, then the Cauchy problem of the Boltzmann equation possesses a unique non-negative local solution in the same function space, both for the cutoff and non-cutoff collision cross section with mild singularity. The known solutions such as solutions on the torus (space periodic solutions) and in the vacuum (solutions vanishing at the spatial infinity), and solutions in the whole space having a limit equilibrium state at the spatial infinity are included in our category.
keywords: local existence locally uniform Sobolev space spatial behavior at infinity pseudo-differential calculus. Boltzmann equation
KRM
Regularity of solutions for spatially homogeneous Boltzmann equation without angular cutoff
Zhaohui Huo Yoshinori Morimoto Seiji Ukai Tong Yang
Kinetic & Related Models 2008, 1(3): 453-489 doi: 10.3934/krm.2008.1.453
The spatially homogeneous Boltzmann equation without angular cutoff is discussed on the regularity of solutions for the modified hard potential and Debye-Yukawa potential. When the angular singularity of the cross section is moderate, any weak solution having the finite mass, energy and entropy lies in the Sobolev space of infinite order for any positive time, while for the general potentials, it lies in the Schwartz space if it has moments of arbitrary order. The main ingredients of the proof are the suitable choice of the mollifiers composed of pseudo-differential operators and the sharp estimates of the commutators of the Boltzmann collision operator and pseudo-differential operators. The method developed here also provides some new estimates on the collision operator.
keywords: angular non-cutoff commutator. regularity Boltzmann equation pseudo-differential operators
DCDS
Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff
Yoshinori Morimoto Seiji Ukai Chao-Jiang Xu Tong Yang
Discrete & Continuous Dynamical Systems - A 2009, 24(1): 187-212 doi: 10.3934/dcds.2009.24.187
Most of the work on the Boltzmann equation is based on the Grad's angular cutoff assumption. Even though the smoothing effect from the singular cross-section without the angular cutoff corresponding to the grazing collision is expected, there is no general mathematical theory especially for the spatially inhomogeneous case. As a further study on the problem in the spatially homogeneous situation, in this paper, we will prove the Gevrey smoothing property of the solutions to the Cauchy problem for Maxwellian molecules without angular cutoff by using pseudo-differential calculus. Furthermore, we apply similar analytic techniques for the Sobolev space regularity to the nonlinear equation, and prove the smoothing property of solutions for the spatially homogeneous nonlinear Boltzmann equation with the Debye-Yukawa potential.
keywords: Debye-Yukawa potential Gevrey hypoellipticity Boltzmann equation non-cutoff cross-sections.
KRM
A remark on Cannone-Karch solutions to the homogeneous Boltzmann equation for Maxwellian molecules
Yoshinori Morimoto
Kinetic & Related Models 2012, 5(3): 551-561 doi: 10.3934/krm.2012.5.551
The purpose of this paper is to extend the result concerning the existence and the uniqueness of infinite energy solutions, given by Cannone-Karch, of the Cauchy problem for the spatially homogeneous Boltzmann equation of Maxwellian molecules without Grad's angular cutoff assumption in the mild singularity case, to the strong singularity case. This extension follows from a simple observation of the symmetry on the unit sphere for the Bobylev formula which is the Fourier transform of the Boltzmann collision term.
keywords: Boltzmann equation uniqueness of solution non-cutoff Maxwellian molecules.
KRM
Phase space analysis and functional calculus for the linearized Landau and Boltzmann operators
Nicolas Lerner Yoshinori Morimoto Karel Pravda-Starov Chao-Jiang Xu
Kinetic & Related Models 2013, 6(3): 625-648 doi: 10.3934/krm.2013.6.625
We prove that the linearized non-cutoff Boltzmann operator with Maxwellian molecules is exactly equal to a fractional power of the linearized Landau operator which is the sum of the harmonic oscillator and the spherical Laplacian. This result allows to display explicit sharp coercive estimates satisfied by the linearized non-cutoff Boltzmann operator for both Maxwellian and non-Maxwellian molecules.
keywords: spectral analysis anisotropy microlocal analysis. Boltzmann operator Landau operator
KRM
Local existence with mild regularity for the Boltzmann equation
Radjesvarane Alexandre Yoshinori Morimoto Seiji Ukai Chao-Jiang Xu Tong Yang
Kinetic & Related Models 2013, 6(4): 1011-1041 doi: 10.3934/krm.2013.6.1011
Without Grad's angular cutoff assumption, the local existence of classical solutions to the Boltzmann equation is studied. There are two new improvements: the index of Sobolev spaces for the solution is related to the parameter of the angular singularity; moreover, we do not assume that the initial data is close to a global equilibrium. Using the energy method, one important step in the analysis is the study of fractional derivatives of the collision operator and related commutators.
keywords: existence of solution energy estimates fractional derivatives. Boltzmann equation
KRM
A remark on the ultra-analytic smoothing properties of the spatially homogeneous Landau equation
Yoshinori Morimoto Karel Pravda-Starov Chao-Jiang Xu
Kinetic & Related Models 2013, 6(4): 715-727 doi: 10.3934/krm.2013.6.715
We consider the non-linear spatially homogeneous Landau equation with Maxwellian molecules in a close-to-equilibrium framework and show that the Cauchy problem for the fluctuation around the Maxwellian equilibrium distribution enjoys a Gelfand-Shilov regularizing effect in the class $S_{1/2}^{1/2}(\mathbb{R}^d)$, implying the ultra-analyticity and the production of exponential moments of the fluctuation, for any positive time.
keywords: ultra-analyticity smoothing effect. Gelfand-Shilov regularity Landau equation

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