# sw NLPDE Z~i[

## ̃Z~i[̗\Schedules of future seminars

2017 N 10 27 @15:30  17:30
u
Vc L i򕌑wHwj
u
On the Ornstein-Uhlenbeck semigroup and its application to the fluid equations
uv|
The Cauchy problem of the incompressible Navier-Stokes equations is considered with linearly growing initial velocity. Using Ornstein-Uhlenbeck semigroup theories, the locally-in-time existence and uniqueness of mild solutions are established in the framework of Lebesgue spaces. Although the semigroup is not analytic, the mild solution is smooth, and then classical one. Moreover, for the rotating flows, the solution is real analytic in spatial variables. Some recent results for the primitive equations are also discussed.

2017 N 11 10 @15:30  17:30
u
X{ F isww@lԁEw _j
Yoshinori Morimoto (Professor Emeritus, Kyoto University)
u
Revisit on the spatially homogeneous Boltzmann equation for Debye-Yukawa potential
uv|
It is well-known that the kernel of the Boltzmann collision integral operator has a non-integrable fractional singularity with respect to the deviation angle by the collision, when the interactive potential of particles obeys the inverse power law, $\rho^{1-n}$, $n>2$, where $\rho$ is the distance between two particles. In 2009, S. Ukai, C.-J. Xu, T. Yang, and I proposed another kernel with much weaker singularity of logarithmic type, when the interaction is Debye-Yukawa type, $\rho^{-1}e^{-\rho^s}$, $0<s<2$, and we showed the smoothing effect of solutions to the Cauchy problem of the spatially homogeneous Boltzmann equation for a further simplified kernel that does not depend on the relative velocity of two particles. In this talk, we consider the same smoothing effect for a more physically rigorous model coming from the Debye-Yukawa type potential. The main results are based on the joint works with Shuaikun Wang and Tong Yang.