Uniqueness of solutions for the non-cutoff Boltzmann equation with soft potential
Radjesvarane Alexandre Yoshinori Morimoto Seiji Ukai Chao-Jiang Xu Tong Yang
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
Bounded solutions of the Boltzmann equation in the whole space
Radjesvarane Alexandre Yoshinori Morimoto Seiji Ukai Chao-Jiang Xu Tong Yang
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
Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff
Yoshinori Morimoto Seiji Ukai Chao-Jiang Xu Tong Yang
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.
Phase space analysis and functional calculus for the linearized Landau and Boltzmann operators
Nicolas Lerner Yoshinori Morimoto Karel Pravda-Starov Chao-Jiang Xu
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
Gevrey regularizing effect of the Cauchy problem for non-cutoff homogeneous Kac's equation
Nadia Lekrine Chao-Jiang Xu
In this work, we consider a spatially homogeneous Kac's equation with a non cutoff cross section. We prove that the weak solution of the Cauchy problem is in the Gevrey class for positive time. This is a Gevrey regularizing effect for non smooth initial datum. The proof relies on the Fourier analysis of Kac's operators and on an exponential type mollifier.
keywords: Non-cutoff Kac's equation Boltzmann equation Gevrey regularizing effect Cauchy problem Fourier analysis.
Local existence with mild regularity for the Boltzmann equation
Radjesvarane Alexandre Yoshinori Morimoto Seiji Ukai Chao-Jiang Xu Tong Yang
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
A remark on the ultra-analytic smoothing properties of the spatially homogeneous Landau equation
Yoshinori Morimoto Karel Pravda-Starov Chao-Jiang Xu
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
Sharp regularity properties for the non-cutoff spatially homogeneous Boltzmann equation
Léo Glangetas Hao-Guang Li Chao-Jiang Xu
In this work, we study the Cauchy problem for the spatially homogeneous non-cutoff Boltzamnn equation with Maxwellian molecules. We prove that this Cauchy problem enjoys Gelfand-Shilov's regularizing effect, meaning that the smoothing properties are the same as the Cauchy problem defined by the evolution equation associated to a fractional harmonic oscillator. The power of the fractional exponent is exactly the same as the singular index of the non-cutoff collisional kernel of the Boltzmann equation. Therefore, we get the sharp regularity of solutions in the Gevrey class and also the sharp decay of solutions with an exponential weight. We also give a method to construct the solution of the Boltzmann equation by solving an infinite system of ordinary differential equations. The key tool is the spectral decomposition of linear and non-linear Boltzmann operators.
keywords: Boltzmann equation spectral decomposition Gelfand-Shilov class.
Propagation of Gevrey regularity for solutions of Landau equations
Hua Chen Wei-Xi Li Chao-Jiang Xu
By using the energy-type inequality, we obtain, in this paper, the result on propagation of Gevrey regularity for the solution of the spatially homogeneous Landau equation in the cases of Maxwellian molecules and hard potential.
keywords: Landau equation Boltzmann equation Gevrey regularity.
$C^\infty$ Local solutions of elliptical $2-$Hessian equation in $\mathbb{R}^3$
Guji Tian Qi Wang Chao-Jiang Xu
In this work, we study the existence of $C^{\infty}$ local solutions to $2$-Hessian equation in $\mathbb{R}^{3}$. We consider the case that the right hand side function $f$ possibly vanishes, changes the sign, is positively or negatively defined. We also give the convexities of solutions which are related with the annulation or the sign of right-hand side function $f$. The associated linearized operator are uniformly elliptic.
keywords: convexities. Elliptical $k$- Hessian equation existence of local solutions

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