Time-periodic solutions of the Boltzmann equation
Seiji Ukai
Discrete & Continuous Dynamical Systems - A 2006, 14(3): 579-596 doi: 10.3934/dcds.2006.14.579
The Boltzmann equation with a time-periodic inhomogeneous term is solved on the existence of a time-periodic solution that is close to an absolute Maxwellian and has the same period as the inhomogeneous term, under some smallness assumption on the inhomogeneous term and for the spatial dimension $n\ge 5$, and also for the case $n=3$ and $ 4$ with an additional assumption that the spatial integral of the macroscopic component of the inhomogeneous term vanishes. This solution is a unique time-periodic solution near the relevant Maxwellian and asymptotically stable in time. Similar results are established also with the space-periodic boundary condition. As a special case, our results cover the case where the inhomogeneous term is time-independent, proving the unique existence and asymptotic stability of stationary solutions. The proof is based on a combination of the contraction mapping principle and time-decay estimates of solutions to the linearized Boltzmann equation.
keywords: time-periodic solution asymptotic stability time-decay estimate. stationary solution Boltzmann equation
A symmetrization of the relativistic Euler equations with several spatial variables
Philippe G. LeFloch Seiji Ukai
Kinetic & Related Models 2009, 2(2): 275-292 doi: 10.3934/krm.2009.2.275
We consider the Euler equations governing relativistic compressible fluids evolving in the Minkowski spacetime with several spatial variables. We propose a new symmetrization which makes sense for solutions containing vacuum states and, for instance, applies to the case of compactly supported solutions which are important to model star dynamics. Then, relying on these symmetrization and assuming that the velocity does not exceed some threshold and remains bounded away from the light speed, we deduce a local-in-time existence result for solutions containing vacuum states. We also observe that the support of compactly supported solutions does not expand as time evolves.
keywords: symmetric hyperbolic system vacuum relativistic fluid local well-posedness theory. Euler equations
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
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
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
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.
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
Pierre Degond Seiji Ukai Tong Yang
Kinetic & Related Models 2008, 1(1): i-ii doi: 10.3934/krm.2008.1.1i
Kinetic theory is probably one of the most efficient and important theories allowing to bridge the microscopic and macroscopic descriptions of a variety of dynamical phenomena in many fields of science, technology, and more generally, in virtually all domains of knowledge. Originally rooted in the theory of rarefied gases since the seminal works of Boltzmann and Maxwell in the 19th century, followed by landmarks established by Hilbert, Chapman and Enskog, Carleman, Grad, and more recent mathematicians, kinetic theory has expanded to many new areas of applications, ranging from physics to economics and social sciences including especially modern fields such as biology, epidemiology, and genetics.

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Stationary solutions to the exterior problems for the Boltzmann equation, I. Existence
Tong Yang Seiji Ukai Huijiang Zhao
Discrete & Continuous Dynamical Systems - A 2009, 23(1&2): 495-520 doi: 10.3934/dcds.2009.23.495
The exterior problem arising from the study of a flow past an obstacle is one of the most classical and important subjects in gas dynamics and fluid mechanics. The point of this problem is to assign the bulk velocity at infinity, which is not a trivial driving force on the flow so that some non-trivial solution profiles persist. In this paper, we consider the exterior problem for the Boltzmann equation when the Mach number of the far field equilibrium state is small. The result here generalizes the previous one by Ukai-Asano on the same problem to more general boundary conditions by crucially using the velocity average argument.
keywords: stationary solutions velocity average argument Boltzmann equation exterior problems
Exterior Problem of Boltzmann Equation with Temperature Difference
Seiji Ukai Tong Yang Huijiang Zhao
Communications on Pure & Applied Analysis 2009, 8(1): 473-491 doi: 10.3934/cpaa.2009.8.473
The existence of stationary solution to an exterior domain of the Boltzmann equation was first studied by S. Ukai and K. Asano in [25, 27] and was recently generalized by S. Ukai, T. Yang, and H. J. Zhao in [29] to more general boundary conditions. We note, however, that the results obtained in [25, 29] require that the temperature of the far field Maxwellian is the same as the one of the Maxwellian preserved by the boundary conditions. The main purpose of this paper is to discuss the case when these two temperatures are different. The analysis is based on some new estimates on the linearized collision operator and the method introduced in [25, 27, 29].
keywords: exterior problem stationary solutions Boltzmann equation temperature difference

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