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.
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
a unique non-negative local solution in the same function
both for the cutoff and non-cutoff collision cross section with
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.
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.
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
$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
Assume that the corresponding Riemann problem
$ 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.
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
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.
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
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.
Although there recently have been extensive studies on the
perturbation theory of the angular non-cutoff Boltzmann equation
(cf.  and ), 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
The global existence of weak solutions to the three space dimensional
Prandtl equations is studied under some constraint on its structure.
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.
the result obtained by Xin-Zhang  on the two-dimensional Prandtl equations to the three-dimensional setting.
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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