KRM
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
KRM
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
DCDS
Global existence and uniqueness for a hyperbolic system with free boundary
Tong Yang Fahuai Yi
In this paper, we consider a $2\times 2$ hyperbolic system originates from the theory of phase dynamics. This one-phase problem can be obtained by using the Catteneo-Fourier law which is a variant of the standard Fourier law in one dimensional space. A new classical existence and uniqueness result is established by some a priori estimates using the characteristic method. The convergence of the solutions to the one of classical Stefan problems is also obtained.
keywords: Stefan problem classical solution. Hyperbolic system
DCDS
Asymptotics toward strong rarefaction waves for $2\times 2$ systems of viscous conservation laws
Tong Yang Huijiang Zhao
This paper concerns the time asymptotic behavior toward large rarefaction waves of the solution to general systems of $2\times 2$ hyperbolic conservation laws with positive viscosity coefficient $B(u)$

$u_t+F(u)_x=(B(u)u_x)_x,\quad u\in R^2,\qquad $ ($*$)

$u(0,x)=u_0(x)\rightarrow u_\pm\quad$ as $x\rightarrow \pm\infty.$

Assume that the corresponding Riemann problem

$u_t+F(u)_x=0,$

$ u(0,x)=u^r_0(x)=u_-,\quad x<0, and u_+,\quad x>0$

can be solved by one rarefaction wave. If $u_0(x)$ in ($*$) is a small perturbation of an approximate rarefaction wave constructed in Section 2, then we show that the Cauchy problem ($*$) admits a unique global smooth solution $u(t,x)$ which tends to $ u^r(t,x)$ as the $t$ tends to infinity. Here, we do not require $|u_+ - u_-|$ to be small and thus show the convergence of the corresponding global smooth solutions to strong rarefaction waves for $2\times 2$ viscous conservation laws.

keywords: Strong rarefaction waves $2\times 2$ viscous conservation laws energy method strongly coupling condition.
KRM
Regularity of solutions for spatially homogeneous Boltzmann equation without angular cutoff
Zhaohui Huo Yoshinori Morimoto Seiji Ukai Tong Yang
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
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
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
KRM
Stability of the nonrelativistic Vlasov-Maxwell-Boltzmann system for angular non-cutoff potentials
Renjun Duan Shuangqian Liu Tong Yang Huijiang Zhao
Although there recently have been extensive studies on the perturbation theory of the angular non-cutoff Boltzmann equation (cf. [4] and [17]), it remains mathematically unknown when there is a self-consistent Lorentz force coupled with the Maxwell equations in the nonrelativistic approximation. In the paper, for perturbative initial data with suitable regularity and integrability, we establish the large time stability of solutions to the Cauchy problem of the Vlasov-Maxwell-Boltzmann system with physical angular non-cutoff intermolecular collisions including the inverse power law potentials, and also obtain as a byproduct the convergence rates of solutions. The proof is based on a new time-velocity weighted energy method with two key technical parts: one is to introduce the exponentially weighted estimates into the non-cutoff Boltzmann operator and the other to design a delicate temporal energy $X(t)$-norm to obtain its uniform bound. The result also extends the case of the hard sphere model considered by Guo [Invent. Math. 153(3): 593--630 (2003)] to the general collision potentials.
keywords: non-cutoff potentials time-velocity weight. energy method The Vlasov-Maxwell-Boltzmann system
DCDS-S
Global existence of weak solutions to the three-dimensional Prandtl equations with a special structure
Cheng-Jie Liu Ya-Guang Wang Tong Yang
The global existence of weak solutions to the three space dimensional Prandtl equations is studied under some constraint on its structure. This is a continuation of our recent study on the local existence of classical solutions with the same structure condition. It reveals the sufficiency of the monotonicity condition on one component of the tangential velocity field and the favorable condition on pressure in the same direction that leads to global existence of weak solutions. This generalizes the result obtained by Xin-Zhang [14] on the two-dimensional Prandtl equations to the three-dimensional setting.
keywords: 3D Prandtl equations weak solutions favorable pressure. monotonic velocity field global existence
KRM
Editorial
Pierre Degond Seiji Ukai Tong Yang
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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