2016-06-23

Navier-Stokes equations with external forces in time-weighted Besov spacesWe show existence theorem of global mild solutions with small initial data and external forces in the time-weighted Besov space which is an invariant space under the change of scaling. The result on local existence of solutions for large data is also discussed. Our method is based on the $L^p-L^q$ estimate on the Stokes equations in Besov spaces. Since we construct the global solution by means of the implicit function theorem, as a byproduct, its stability with respect to the given data is necessarily obtained. This is the joint work with Prof.Senjo Shimizu at Univ. Kyoto.

On regularity of some special solutions of 3D Navier-Stokes equations

We deal with regularity of some solutions of 3D Navier-Stokes equations. If solutions is of some special structure, the solutions becomes globally regular such as 2D or quasi-planar flow. We discuss these issues in more detail.

Quasi-optimal initial value conditions for the Navier-Stokes equations and questions of uniqueness

Consider weak solutions of the instationary Navier-Stokes system in a three-dimensional bounded domain $ \Omega $.
It is well-known that an initial value $u_0\in \mathcal D(A^{1/4})$ or even $u_0\in L^3_\sigma(\Omega)$, where $A=-P\Delta$ denotes the Stokes operator, admits a unique regular solution in Serrin's class $L^{s_q}(0,T;L^q(\Omega))$, $\frac2{s_q}+\frac3q = 1$, $2<{s_q}<\infty$, for some $T=T(u_0)$.
The optimal class of initial values $u_0 \in L^2_{\sigma}(\Omega)$ with this property was determined by H. Sohr, W. Varnhorn and the speaker in 2009 and is given by the Besov space $\mathbb B^{-1+3/q}_{q,s_q}(\Omega)$ of solenoidal vector fields $u_0 $, i.e., $u_0$ satisfies the condition
$ \int_0^\infty \big(\|e^{-\tau A}u_0\|_q\big)^{s_q}\, d\tau <\infty. $
This optimal condition can be weakened to a quasi-optimal condition on $u_0 \in L^2_{\sigma}(\Omega)$ with weighted finite integral
$ \int_0^\infty \big( \tau^\alpha\|e^{-\tau A}u_0\|_q\big)^s\, d\tau <\infty $
where $s>s_q$, $q>3$ satisfy $\frac2s+\frac3q = 1-2\alpha$, $0<\alpha<\frac12$. In the case $s=\infty$ the integral norm has to be replaced by the essential sup-norm. These conditions can be described by the scaling invariant Besov space
$\mathbb B^{-1+3/q}_{q,s}$, $q>3$, ${s_q}<$ $s$ $\leq\infty$. A weak solution with such an initial value is contained in an $L^s(L^q)$-space with time weight $\tau^\alpha$, still satisfies the energy equality on $[0,T(u_0))$ and Serrin's condition $u\in L^{s_q}(\varepsilon,T;L^q(\Omega))$, $\varepsilon>0$, but the classical Serrin weak-strong uniqueness theorem holds only under additional assumptions.
In this talk we present recent results obtained by R. Farwig, Y. Giga and Pen-Yuan Hsu on existence, uniqueness and continuity as well as stability in the space $C^0([0,T);\mathbb B^{-1+3/q}_{q,s})$.

Small remarks on the geometric regularity criterion for the Navier-Stokes equations

In this talk, I will review regularity criterion for the 3D Navier-Stokes equations. Also I show that if the volume of parallelepiped which is defined by three unit vectors--direction of the velocity, direction of the vorticity, and the direction of the curl of the vorticity --is small enough, then the solution to the 3D Navier-Stokes equations is classical.

Serrin type regularity criteria for 3D Navier-Stokes equations

We develop Ladyzhenskaya-Prodi-Serrin type regularity criteria for 3D incompressible Navier-Stokes equations in both a torus and the whole domain.
More precisely, for any N>0, let w_N be the sum of all spectral components of the velocity fields whose wave numbers |k_i|>N for all I=1, 2, 3. Then we show that the finiteness of the the Serrin type norm of w_N implies the regularity of the flow. The results in a torus is extended to the whole domain with extra efforts.

Two-phase flows with phase transitions

We consider models for compressible and incompressible two-phase flows with phase transitions. These are based on first principles, i.e., balance of mass, momentum, and energy. Performing a Hanzawa transform, the problem is transformed to a quasilinear parabolic two-phase problem with complicated transmission conditions on the interface in a fixed domain. We prove maximal Lp-regularity of the corresponding linearized problem, and then by a fixed point argument in a suitable space, we obtain local existence for the isothermal, compressible model with phase transitions.
This is a joint work with Prof. Jan Pr{¥"u}ss (Halle, Germany).

A new local regularity criterion for suitable weak solutions of the Navier--Stokes equations in terms of the velocity gradient

We study the partial regularity of suitable weak solutions to the three dimensional incompressible Navier--Stokes equations. There have been several attempts to refine the Caffarelli--Kohn--Nirenberg criterion (1982).
We present an improved version of the CKN criterion with a direct method, which also provides the quantitative relation in Seregin's criterion (2007).
This is a joint work with Prof. Hi Jun Choe and Prof. Joerg Wolf.

A cavity flow simulation for the interaction of flocking particles and an incompressible fluid

A numerical simulation is presented for the interaction of self-propelled particles and the viscous incompressible flows in a square laminar cavity. In recent researches (1; 2), new coupled kinetic-fluid model has been proposed for the interaction between Cucker-Smale(C-S) flocking particles and viscous incompressible fluid. The coupled system consists of the Kinetic Cucker-Smale equation and the incompressible Navier-Stokes equations, and these two systems are linked with the drag-force. The main theory of previous research is that the velocity of C-S particles and fluid velocities are aligned time-asymptotically under a specific conditions, the periodic boundary conditions with zero exterior force terms. Through this numerical simulations, we verified the property of velocity alignment for the interior C-S Navier-stokes with non-zero exterior force.

Fluid Computations using Free Discretizations

Computations in arbitrary domain are inevitable, particularly more for fluids. Through the celebration of 60th birthday of Professor Choe, we are going to talk about free discreization methods in arbitrary domains to solve flow problems of interest. In these potential methods, free nodes or even on free grids are available without degradation of accuracy.

2016-06-24

Viscous and inviscid fluid flowsWe discuss several recent results concerning stability of solutions and mutual relations between the compressible Euler system and the Navier-Stokes system. In particular, stability of strong solutions in the class of
very general so-called measure-valued solutions is considered. Finally, we
discuss the vanishing viscosity limit as a possible criterion of admissibility for the compressible Euler system.

Green tensor of the Stokes system and asymptotics of stationary Navier-Stokes flows in the half space

The estimates of Green tensor of the stationary Stokes system are refined in the half space and the spatial asymptotics of stationary solutions are analyzed for the incompressible Stokes and Navier-Stokes equations. The asymptotics of fast decaying flows is also discussed in the whole space as well as exterior domains. In the Appendix we show that there is no non-trivial axisymmetric self-similar solutions in the half space.

On works of Professor Hi Jun Choe

The works of Prof. Hi Jun Choe are briefly introduced.

On local Serrin type conditions for local weak solutions to the generalized Navier Stokes equations

We consider weak solutions to the equations of a non-Newtonian fluid in three spatial dimensions. In case if the deviatoric stress satisfies a power law we prove that the solution is locally strong if the velocity field satisfies a suitable integrability condition which is invariant under the natural scaling of the power law model. In particular this result generalizes the well known Serrin condition for the Navier-Stokes equations.

On the Existence of Global Finite Energy Solutions to the Multi-Dimensional Compressible Navier-Stokes System with Degenerate Viscosities.

In this talk, I will present some results on the existence of finite
energy weak solutions to the isentropic compressible Navier-Stokes systems density-dependent viscosities which degenerate at vacuum at
both 2-D and 3-D. The key is a deliberate construction of suitable approximate solutions satisfying various stability estimates. The results
in particular give a positive answer to a problems due to P. L. Lions. This is a joint work with Jing Li.

Regularity vs singularity in kinetic equations

We describe the structure of solutions of kinetic equations in domains with boundaries near the singular set. Representative equations are the Vlasov-Poisson, the kinetic Fokker-Planck equations. We discuss in particular regularity and singularity of the solutions of these equations with various boundary conditions.

Derivation of the Navier - Stokes - (Fourier)- Poisson system for an accretion disk

We study the 3-D compressible barotropic Navier-Stokes-(Fourier)- Poisson system describing the motion of a compressible rotating viscous fluid with renormalized gravitation confined to a straight layer, where the 'third" dimension is very small.We shall show that the weak solutions in the 3D domain converge to the strong solution of a rotating 2-D Navier-Stokes- (Fourier)- Poisson system, when we go with the "third" dimension to zero . It is a joint work with M. Pokorny and M. Caggio.

Initial and boundary values for $L^q_\alpha(L^p)$ solution of the Navier-Stokes equations in the half-space

In this paper, we study the initial and boundary value problem of the Navier-Stokes equations in the half-space.
We prove the existence of weak solution $u\in L^q_\alpha(0,\infty;L^p({\mathbb R}^n_+))$, $\alpha=\frac{1}{2}(1-\frac{n}{p}-\frac{2}{q})\geq 0$, $n < p < \infty$ with $\nabla u\in L^{\frac{q}{2}}_{loc}(0,\infty;L^{\frac{p}{2}}_{loc}({\mathbb R}^n_+))$ for the solenoidal initial data $h\in \dot{B}_{pq}^{-1+\frac{n}{p}}({\mathbb R}^n+)$ and the boundary data $ g\in L^q_\alpha(0,T;\dot B^{-\frac{1}{p}}_{pp}({\mathbb R}^{n-1}))$ when $\|h\|_{ \dot{B}_{pq}^{-1+\frac{n}{p}}({\mathbb R}^n_+)}+\| g\|_{ L^q_\alpha(0,T;\dot B^{-\frac{1}{p}}_{pp}({\mathbb R}^{n-1}))}$ is small enough. Moreover, the solution is unique in the class $L^q_\alpha(0,T;L^p({\mathbb R}^n_+))$ for any $T\leq \infty$ if $\alpha>0$ and for some $T<\infty$ if $\alpha=0$.

On heat equation with general random noise as the initial condition

In this talk we discuss the decreasing speed of the noise deviation and the average noise diffusion flux when the heat diffusion starts with a random noise field. Even with a very wild initial condition, the averaging effect caused by diffusion makes the noise deviation and the average noise diffusion flux inside the space domain decrease instantly. We are interested in the decreasing speeds of them. This question is inspired by D. Khoshnevisan's work that considered a white noise as the initial noise field. Our random noise models include less white, white, and whiter noises in one frame.

2016-06-25

Life span of solutions to nonlinear Schrodinger equations on torusWe give an explicit upper bound of life span of solutions to non gauge invariant nonlinear Schr"odinger equations on n-dimensional tours.
This talk is based on a recent joint work with Kazumasa Fujiwara.

Uniform Sobolev estimate for 2nd order differential operators

This talk is concerned with the uniform Sobolev type estimate which is independent of the first order and constant terms. We obtain a complete characterization of the Lebesgue spaces on which such uniform estimate holds. In case of non-elliptic operator the problem has been left open until recent. We also discuss connection to the Fourier restriction estimate and the boundedness of Bochner-Riesz type multiplier operators.

On the Hall-MHD equations

In this talk we discuss the Cauchy problem of the Hall-MHD system, where the "Hall term" is added to the usual incompressible MHD equations. After reviewing various recent results of the problem, we focus on the issue of the finite time singularity when the resistivity constant is zero.
In this case we show that there exists a smooth initial data for which either the Cauchy problem is locally ill-posed, or it is locally well-posed but the apparition of finite time singularity happens.

End-point maximal regularity for the Cauchy problem of the heat equations and some application

We consider maximal regularity for the Cauchy problem of the heat equation in the homogeneous and inhomogeneous Besov spaces. It is known that maximal regularity for the parabolic type equation is generally obtained in the UMD Banach space in the Lebesgue space. However the end point exponents one and infinity is usually excluded. Besides the UMD space is necessary reflexive, the general theory is not applicable for non-reflexive Banach spaces such as integrable space or bounded functions. We consider maximal regularity for the heat equation for the Cauchy problem of heat equation in the non-reflexive Besov spaces and discuss L-1 maximal regularity with some additional discussion.

Weak solutions to reaction diffusion equations - homogenization in isolated systems

Models involved by many components are of particular interests in ecology. We show a class of reaction diffusion equations whereby any solution is global-in-time weakly, that is, quasi-positive, quadratic, and dissipative. Provided with entropy, the solution converges to a unique spatially homogeneous stationary state. There is another class including Lotka-Volterra equation, provided with uniform boundedness in Zygmund norm of the solution in addition. In both cases, the solution is classical if the space dimension is less than 3. This work is supported by JSPS Core-to-Core project.

Entire solutions of quasilinear elliptic equations

I will discuss the existence of entire solutions of quasilinear elliptic equations and the asymptotic behavior near infinity. The asymptotic self-similarity is useful to characterize the solutions.

Geometric Capacity Theory

Here we introduce a new quantity that is suitable to study the regularity properties of domains.
Particularly we will show that if a domain is properly close in capacity of that of the half space, then the domain is close to be flat in Hausdorff distance.

Elliptic systems with mixed interactions

I would like to introduce some recent results on vector solutions and their asymptotic behavior for elliptic systems.

On the partial regularity of Navier-Stokes equations

In this talk, we overview partial regularity of Navier-Stokes equations.
We know minimal surface, harmonic map and free boundary are entangled with partial regularity and thereupon singularity analysis follows. We review quickly of known partial regularity results for various other fields and come back Navier-Stokes equations. We analyze local behavior of solution and discuss the very difficulty why singular set is hard to understand. Finally we discuss general nature of partial regularity questions for various fields.