We consider viscous axisymmetric incompressible magnetohydrodynamics (mhd) in a bounded cylinder with boundary conditions: normal components of velocity and magnetic fields, angular components of velocity and magnetic fields and tangent components of rotation of velocity and magnetic fields vanish on the boundary. First we prove global existence of regular solutions such that angular components of velocity vanish and only angular component of magnetic field is nonvanishing. Then we show stability of this solution in a set of all axisymmetric solutions. In this way we prove global existence of regular axisymmetric solutions which remain close for all time to the above special solutions.

We consider a special class of weak axi-symmetric solutions to the equations of magnetohydrodynamics for which the velocity field has only poloidal component and the magnetic field is toroidal. We prove local regularity for such solutions. The global strong solvability of the initial-boundary value problem for the corresponding system in a cylindrical domain with non-slip boundary conditions for the velocity on the cylindrical surface is established as well.

We consider magnetohydrodynamic equations in a torus located in a positive distance from the axis of symmetry. We prove existence and stability of global regular axially-symmetric solutions. In this way the existence of global regular solutions close to the axially-symmetric solutions for all times is shown.

This talk addresses a question concerning the behaviour of a sequence of global solutions to the Navier-Stokes equations, with the corresponding sequence of smooth initial data being bounded in the Lebesgue space $L_3$ or in weak Lebesgue space $L_{3,\infty}$. It is closely related to the question of what would be a reasonable definition of global weak solutions with a non-energy class of initial data, including the aforesaid Lebesgue spaces.

We investigate the problem of the existence of regular solutions to the three-dimensional MHD equations in cylindrical domains with perfectly conducting boundaries and under the Navier boundary conditions for the velocity field. We show that if the initial and external data are smooth enough and do not change too rapidly along the axis of the cylinder, then there exist a unique regular solution for any finite time.
This is a joint work with G. Ströhmer and W.M. Zajączkowski.

Consider weak solutions of the instationary Navier-Stokes system in a three-dimensional bounded domain $\Omega$. It is well-known that the optimal class of initial values $u_0$ to admit a local in time unique regular solution in Serrin's class $L^{s_q}(0,T;L^q(\Omega))$, $\frac{2}{s_q}+\frac{3}{q} = 1$, $2 < s_q < \infty$, is given by the Besov space $\mathbb B^{-1+3/q}_{q,s_q}(\Omega)$ of solenoidal vector fields $u_0 $, i.e., using the Stokes operator $A$, $u_0$ satisfies the condition $$ \int_0^\infty \big(\|e^{-\tau A}u_0\|_q\big)^{s_q}\, d\tau <\infty, $$ see results by H. Sohr, W. Varnhorn and the speaker (2009) (link). This optimal condition can be weakened to an almost 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 $[0,T(u_0))$, but the classical Serrin weak-strong uniqueness theorem holds only under additional assumptions. We present recent results obtained by R. Farwig, Y. Giga and Pen-Yuan Hsu (Tokyo University) in the framework of the IRSES project on existence, uniqueness and continuity as well as stability in the space $C^0([0,T);\mathbb B^{-1+3/q}_{q,s})$.

As one of the violent flow, tornadoes occur in many place of the world. In order to reduce the loss of human lives and material damage caused by tornadoes, there are many research methods. One of the effective methods is numerical simulation. The swirling structure is significant both in mathematical analysis and the numerical simulations of tornadoes. In this work [2] we try to clarify the swirling structure. More precisely, we do numerical computations on axi-symmetric Navier-Stokes flows with no-slip flat boundary. We compare a hyperbolic flow with swirl and one without swirl and observe some phenomenons occur only in the swirl case. Our main purpose in this work is to combine the point of view from mathematical analysis (especially regularity results) and numerical approach to observe phenomenons which related to the structure of tornadoes.

1. Y. Giga, P.-Y. Hsu and Y. Maekawa, A Liouville theorem for the planer Navier-Stokes equations with the no-slip boundary condition and its application to a geometric regularity criterion, Comm. Partial Differential Equations, 39 (2014), 1906 — 1935. (link)
2. P.-Y. Hsu, H. Notsu and T. Yoneda, A local analysis of the axi-symmetric Navier-Stokes flow near a saddle point and no-slip flat boundary, J. Fluid Mech. 794 (2016), 444 — 459. (link)

In my talk I will review the results on the question of uniqueness of zero trace solutions to $$ \Delta u +b\cdot \nabla u=0 \quad \text{in } \Omega $$ where $\Omega$ is an open bounded domain with regular boundary. Vector field $b$ is divergence free and of $L^p$ regularity. The question appears in problems of homogenization and turbulence. It was proved by Zhikov that for $1\leq p < 3/2$ solutions are nonunique, while for $p \geq 2$ they are unique. First I will show an alternative proof of uniqueness for $p\geq 2$ and than discuss some regularity problem which occurs when investigating the problem for $p$ between $3/2$ and $2$.

In contrast to linear fluid dynamics systems, providing an $L^q$ Calderón-Zygmund-type theory in the quasilinear (power-law type) case is difficult for two main reasons:

• For `high' $q$'s, due to lack of smoothing property of a homogenous problem (no Uhlenbeck-type structure).
• For `low' $q$'s, due to lack of existence theory below the duality exponent (no natural duality pairing).

The latter difficulty may be dealt with by means of a novel technique that uses linear Muckenhoupt-weighted theory in order to substitute the lacking natural duality pairing. Using this approach, I will present a `unified theory' (i.e. existence, uniqueness and optimal regularity) for the following stationary quasilinear Stokes system $$ \begin{aligned} -\operatorname{div} (\mathcal{A}(x, \epsilon(u))) +\nabla p &= \operatorname{div}f & &\text{ in }\Omega,\\ \operatorname{div} u&= 0 & &\text{ on }\partial \Omega \\ u&=0 & &\text{ on }\partial \Omega \end{aligned} $$ with assumptions on $\mathcal{A}$ that allow, among others, for a Carreau fluid.
This is a joint work with M. Bulíček and S. Schwarzacher.

Consider the two-dimensional Euler equations in $\Omega \subset \mathbb{R}^2$ $$ \begin{aligned} &u_t + u\cdot \nabla u + \nabla p =f, \\ &u\cdot n\vert_{\partial \Omega} = 0, \\ &\operatorname{div} u = 0,\\ &u(x,0) = u_0(x). \end{aligned} $$ We try to construct periodic solutions via the method of vanishing viscosity. Therefore we consider the corresponding Navier-Stokes equations $$ \begin{aligned} &u_t − \nu \Delta u + u\cdot \nabla u + \nabla p = f,\\ &u \cdot n\vert_{\partial \Omega} = 0, \\ &\operatorname{rot} u\vert_{\partial \Omega} =0,\\ &\operatorname{div} u = 0, \\ &u(x,0) = u_0(x). \end{aligned} $$ As this work is still in progress, we are trying to establish periodic solutions to the Navier-Stokes equations and a priori estimates to $u$ independent of $\nu$ to consider the limit $\nu \to 0$.
This project is a collaboration with Prof. Dr. Hideo Kozono.

In this talk, we consider the global well-posedness of the compressible Navier-Stokes-Korteweg system in $\mathbb{R}^N$. It is shown that this system admits a unique, global strong solution for small initial data in the $L_p$ in time and $L_q$ in space setting. For the purpose, we prove the maximal $L_p$-$L_q$ regularities and $L_p$-$L_q$ decay properties to the linearized equations. This talk is based on a joint work with Hirokazu Saito and Yoshihiro Shibata in Waseda University.

I would like to talk about the free boundary problem for the Navier-Stokes equations, where the reference body is an exterior domain in $\mathbb{R}^N$ ($N \geq 3$) and the surface tension is not taken into account. The local well-posedness is proved with the help of $L_p$-$L_q$ maximal regularity for the corresponding Stokes equations with free boundary condition in an exterior domain. The global well-posedness is proved by combination of the $L_p$-$L_q$ maximal regularity and $L_p$-$L_q$ decay estimate for the solution of slightly perturned Stokes equations with free boundary condition in an exterior domain, which follows from the spectral analysis to the corresponding resolvent problem. Since the polynomial decay of solutions in $L_q$ space norm is only obtained , it is crucial to use the $L_p$ summability in time with large exponent $p$. This is the advantage of using the $L_p$-$L_q$ maximal regularity with free choice of exponents $p$ and $q$.

I plan to talk about a free boundary problem for a drop of viscous incompressible fluid (the Navier-Stokes eqs without surface tension). The goal is to prove global in time existence of solutions for sufficiently small initial state. A difficulty lays in an external force, which although is small and rapidly vanishes in time may change in time total momenta of the fluid.
The project is realized in collaboration with Prof. Yoshihiro Shibata.

The problem on the evolution of a drop in an incompressible continuum is analyzed in the Sobolev--Slobodetskiĭ spaces $W_2^{l,l/2}$. A local existence theorem for the problem was proved in the case of non-negative surface tension in [1]. Global unique solvability is established for the problem with small data, the liquids being located in a container. The proof is based on an exponential global estimate for a generalized energy. An additional smoothness of the velocity vector--field which is necessary in proving process is obtained by ideas from the paper [2].

1. I.V. Denisova, Motion of a drop in the flow of a fluid (Russian), Dinamika Sploshn. Sredy 93—94 (1989), 32—37. English translation: Problem of the motion of two viscous incompressible fluids separated by a closed free interface, Acta Appl. Math., 37 (1994), 31—40. (link)
2. V.A. Solonnikov On an unsteady flow of a finite mass of a liquid bounded by a free surface, Boundary-value problems of mathematical physics and related problems of function theory. Part 18, Zap. Nauchn. Sem. LOMI, 152, "Nauka", Leningrad. Otdel., Leningrad (1986), 137—157. (pdf). English translation: Unsteady motion of a finite mass of fluid, bounded by a free surface, J. Soviet Math. 40 (1988), 672—686. (link)

The overview of results concerning the stationary Navier-Stokes system with non-homogeneous boundary conditions will be given. In details the stationary nonhomogeneous Navier-Stokes problem will be studied in a two dimensional symmetric domain with a semi-infinite outlet to infinity (for instance, paraboloid type or channel-like). Under the symmetry assumptions on the domain, boundary values and external force the existence of at least one weak symmetric solution will be proved without any restriction on the size of the fluxes, i.e. the fluxes of the boundary value over the inner and the outer boundaries may be arbitrarily large. The Dirichlet integral of the solution can be finite or infinite dependent on the geometry of the domain.

I will present some results related to the solvability of fractional diffusion equation with Caputo derivative. We shall consider the following problem $$ \left\{ \begin{aligned} \partial^{\alpha} u &= Lu + f & &\text{ in } \Omega^{T}\\ u &= 0 & &\text{ on } \partial \Omega \\ u &= u_{0} & & \text{ for } t=0, \\ \end{aligned} \right. $$ where $L$ is uniformly elliptic operator, $\alpha\in (0,1)$ and $\partial^{\alpha} u $ is Caputo fractional derivative define by formula $$ \partial^{\alpha} u (x,t) = \frac{1}{\Gamma(1- \alpha)}\int_{0}^{t}(t-\tau)^{-\alpha}u_{t}(x,\tau)\mathrm d\tau. $$

We are considering systems consisting of one or several barotropic gas balls in space. We show that an energy stability condition implies the non-linear stability of such systems and discuss how one can obtain this result by reformulating the non-linear equations by introducing an angle of rotation. This the leads to a different linearization that is more suitable for the nonlinear analysis.

We are interested in a system of equations describing stationary flow of a mixture of gases undergoing reversible chemical reactions which reads $$ \begin{aligned} &{\rm div} (\rho {\bf u}) = 0,\\ &{\rm div} (\rho {\bf u} \otimes {\bf u}) - {\rm div} \bf{S} + \nabla \pi =\rho {\bf f},\\ &{\rm div} (\rho E{\bf u} )+ {\rm div}(\pi{\bf u}) + {\rm div}\bf{Q}+ {\rm div} (\bf{S}{\bf u})=\rho{\bf f}\cdot{\bf u},\\ &{\rm div} (\rho Y_k {\bf u})+ {\rm div} {\bf f}_k = m_k{\bf w}_{k},\quad k\in \{1,\ldots,n\}, \end{aligned}\quad (1.1) $$ where ${\bf u}$ is the velocity of the fluid, $\rho$ is the density of the mixture which is a sum of species densities $\rho_k$ and $Y_k=\frac{\rho_k}{\rho}$ are the species mass fractions. Furthermore, $\bf{S}$ denotes the viscous stress tensor, $\pi$ the internal pressure of the fluid, $\bf{f}$ the external force, $E$ the specific total energy and $\bf{Q}$ the heat flux. The first three equations form the well known stationary compressible Navier-Stokes system and the equations $(1.1)_4$ describe the balance of masses of $n$ species. I will start with presentation of known results concerning existence of weak solutions to the system $(1.1).$ Then I will present some ideas of how to strengthen these results, which is a work in progress with M. Pokorný. It turn out that new pressure estimates developed recently for compressible Naver-Stokes system can be applied also to the mixture model $(1.1)$ and enable to extend the range of $\gamma$ in the pressure law $\pi(\rho)=\rho^\gamma + \rho \theta$. We also introduce a slightly more general definition of solutions, so called variational entropy solutions, which allow to reach the range $\gamma > 1$ and are weak solutions for sufficiently high $\gamma$.

In this talk, we would like to consider a resolvent problem on the upper half-space $\mathbb{R}_+^N=\{(x',x_N) \mid x'=(x_1,\dots,x_{N-1})\in\mathbb{R}^{N-1}, x_N>0\}$ $(N\geq 2)$ arising from a compressible fluid model of Korteweg type with free boundary condition. It is then proved that there exist $\mathcal{R}$-bounded solution operator families of the resolvent problem as follows: We first apply the partial Fourier transform with respect to the tangential variables $x'$ to the resolvent problem in order to obtain ordinary differential equations with respect to $x_N$ in the Fourier space. Secondly, we solve the ordinary differential equations. Thirdly, applying the inverse transform to the solution gives a solution of the resolvent problem and its representation formula. Finally, combining the representation formula with some technical lemmas shows the existence of $\mathcal{R}$-bounded solution operator families of the resolvent problem. In addition, we prove the maximal $L_p\text{-}L_q$ regularity theorem for a time-dependent linear problem associated with the resolvent problem as an application of the $\mathcal{R}$-bounded solution operator families.

We consider the Navier-Stokes equations in a domain with compact boundary and nonzero Dirichlet boundary data $\beta$. Assuming that $\beta(t)\to 0$ as $t\to \infty$ we proved in [1] that the solutions $v$ fulfills $\|v(t)\|_2\to 0$ as $t\to \infty$. Furthermore, we proved that the time-decay is exponentially if the domain is bounded and there exists a polynomial lower bound for the decay-rate if the domain is exterior.
Additionally, in [2] we proved that almost the same decay results can be obtained for an arbitrary turbulent solution. This can be shown by using the decay-results in [1], an existence theorem for strong solutions, and a uniqueness theorem of Serrin's type.
This is a result of a collaboration with Prof. Dr. Reinhard Farwig and Prof. Dr. Hideo Kozono.

1. R. Farwig, H. Kozono, H., D. Wegmann, Decay of non-stationary Navier-Stokes flow with nonzero Dirichlet boundary data, Indiana Univ. Math. J., accepted for publication (2015)
2. R. Farwig, H. Kozono, D. Wegmann, Existence of strong solutions and decay of turbulent solutions of Navier-Stokes flow with nonzero Dirichlet boundary data, in preparation (2016)

I will present some recent results concerning Vlasov-type kinetic equations describing collective motion of particles with non-local interaction. The nature of interactions may vary but the talk will be focused on the Cucker-Smale flocking model from 2007. The talk will be mostly based on ideas from papers by Boudin, Desvilletes, Grandmont and Moussa from 2008 (link) and Bae, Choi, Ha and Kang from 2012 (link) in which the authors coupled Vlasov-type equations with classical equations of hydrodynamics through a drag force.

The communication is concerned with the maximal regularity estimates in Hölder norms for parabolic and Stokes systems obtained on the basis of the Fourier multipliers theorem in anisotropic Hölder spaces due to O.A. Ladyzhenskaya and on K.K. Golovkin's results on equivalent norms in these spaces. They bring significant simplifications in the proof of classical solvability of some linear and non-linear evolution problems.

We discuss the crystalline mean curvature flow, a motion of a surface with normal velocity law $V = f(\nu, \kappa_\sigma)$, where $f$ is a given function of the normal vector $\nu$ and the so-called crystalline mean curvature $\kappa_\sigma$. This problem appears in models of crystal growth formulated as a gradient flow of the surface energy when the surface energy density is not differentiable. In general, a surface evolving under this flow develops flat parts, the facets of a crystal, on which the crystalline curvature is a nonlocal quantity. These facets are usually preserved by the flow, but they might also break or bend. Because of this phenomenon and other difficulties, even a local-in-time well-posedness had been open in dimensions higher than two except in special cases like convex initial data. We introduce a new notion of viscosity solutions for the level set formulation of the crystalline mean curvature flow. We prove a comparison principle, stability and global well-posedness of the initial value problem for arbitrary bounded crystals. One of the main new ideas is the introduction of faceted test functions with admissible facets compatible with the singularity of the surface energy density, and establishing that the class of such test functions is sufficiently large.
This talk is based on joint work with Mi-Ho Giga and Yoshikazu Giga from University of Tokyo.

In my talk, I will present some recent results on the $L^p$ theory of the time-periodically driven simplified Ericksen-Leslie model, which describes the flow of nematic liquid crystals, in the whole space, the half space and sufficiently regular bounded domains. Studying this model in a time-periodic setup is of particular physical interest, since most of the forces that drive the system in real life applications are triggered periodically, be it a mechanical pumping of the fluid as a whole or an electromagnetic pulsing in order to control the molecular orientation. A particular problem arises in the case of the unbounded domains, where we will be restricted to dimensions $n\geq 4$ due to a regularity loss in the steady-state part.
The key idea is to view the model as a time-periodic quasilinear parabolic equation, and use maximal $L^p$ regularity estimates for the linearized problem, which itself is a coupling of a Stokes part governing the flow with a heat equation controlling the molecular orientation. These estimates are obtained - depending on taste - either ``on foot'', i.e., by replacing the time axis with a torus, reformulating the linear problem on a locally compact abelian group and using Fourier techniques, or in an abstract way: In fact, I will show that for generators of $C_0$-semigroups a time-periodic variant of maximal $L^p$ regularity is equivalent to maximal $L^p$ regularity for the initial value problem. As it is a well-known fact that both the Stokes and the Laplace operator admit maximal $L^p$ regularity, we obtain the desired time-periodic estimates. Of course, one can also read this backwards and combine the abstract result with the ``on foot'' result to obtain a new and easy proof of the maximal $L^p$ regularity of the Stokes and Laplace operators.
The talk is based on a joint work with Yasunori Maekawa.

A prototype problem we study is $$ \begin{aligned} &u_t = (\operatorname{sgn} u_x)_x, \\ &u(x,0) = u_0(x), \end{aligned} \qquad x \in (0,L), t>0, $$ augmented with boundary conditions. This equation may be equivalently written as $$ \begin{aligned} &u_t = (dW/dp(u_x))_x, \\ &u(x,0) = u_0(x), \end{aligned} \qquad (*) $$ where $W(p) = \lvert p \rvert$. Hence, an easy generalization is to consider a piecewise linear and convex function $W$. We define the notion of viscosity solutions to equations like $(*).$ We show that weak solution to $(*)$, where $u_0$ belongs to $BV(0,L)$ are in fact viscosity solutions.
We recall that a comparison principle holds for viscosity solutions. We show that this tool is very useful to deduce properties of solutions.
This talk is based on joint project with Y.Giga, M.-H. Giga, M. Matusik, P.B. Mucha and A. Nakayasu.

We present a new type of a Harnack inequality for non-negative discrete supersolutions of fully nonlinear uniformly elliptic difference equations. This estimate applies to all supersolutions and has the Harnack constant depending on the graph distance on lattices. For the proof we modify the proof of the weak Harnack inequality. Important steps are construction and translation of a barrier function which is a solution of the Pucci equation except at the origin.

In this talk we discuss a very fundamental problem on the Stokes semigroup. It is well-known that the Stokes seimgroup in analytic in $L^p$ type space ($1 < p < \infty$) provided that the domain admits the $L^p$-Helmholtz decomposition, for example a bounded domain, an exterior domain. It is quite recent that $L^\infty$ theory is developed by Ken Abe and the author. We review this $L^infty$ theory with some emphasis of recent application to the Navier-Stokes equations by Ken Abe. It turns out this is very useful to get some geometric criteria of the Navier-Stokes flow in a domain. We also develop BMO theory for the Stokes semigroup. As an application we show that the existence of $L^p$ Helmholtz decomposition is NOT a necessary condition to get the analyticity of the Stokes semigroup in $L^p$ type space. In fact there is a sector-like planar smooth domain that does not admit Helmholtz decomposition in $L^p$ for some $p$ although there always admits $L^2$ Helmholtz decomposition. Even for such a domain we prove that the Stokes semigroup is analytic in the solenoidal $L^p$ space if it is given as a Lipschitz half-space with $C^3$ boundary. Unfortunately, the proof is quite involved. We first establish that there is a non-Helmholtz projection mapping $L^\infty $to BMO-type space of solenoidal fields. Then we prove that the Stokes semigroup is analytic in a solenoidal BMO-type space, provided that the domain is admissible in the sense of Ken Abe and the author. The BMO space on a domain introduced here is closely related to Miyachi's BMO. Interpolating $L^2$ results and BMO results we are able to establish the analyticity of the Stokes semigroup in the solenoidal $L^p$ space for $p, 2 < p < \infty$.
This result is a jointly obtained with Martin Bolkart (Darmstadt), Tatsu-Hiko Miura (Tokyo), Takuya Suzuki (Tokyo) and Yohei Tsutsui (Matsumoto).

In the talk, we consider the minimizing total variation flow in the Sobolev space $H^{−s}$. We explain the motivation to study this problem in the context of image processing applications and provide its rigorous interpretation under periodic boundary conditions. Furthermore, we explain how to construct a minimizing sequence based on the associated dual formulation and discuss some issues concerning its convergence. We also show and compare results of numerical experiments obtained by application of the proposed scheme for a simple initial data and different values of the index $s$.
This is a join work with Y. Giga (University of Tokyo) and P. Rybka (Warsaw University).

We study the evolutionary Hamilton-Jacobi equation on a complete geodesic metric space such as topological networks or post-critical finite fractals. The aim of this talk is to investigate asymptotic behavior of metric viscosity solutions introduced by Gangbo and Święch, and we will show large time behavior of the solution. To prove this result we will also establish some basic properties including stability and comparison principle to stationary equations.
This talk is based on the joint work with Tokinaga~Namba (U. Tokyo).

Given a closed countably rectifiable set of codimension one, with locally finite Hausdorff measure and suitable growth condition at infinity, we prove that there exists a mean curvature flow in the sense of Brakke starting from this set. The solution exists for all time or until it vanishes with certain continuity property in time. This existence result allows far richer class of initial sets not covered by the level set approach, such as 1 dimensional networks and 2 dimensional bubble clusters, and has numerous physical applications.
This is a joint work with Lami Kim.

The least gradient problem requires finding a minimizer of the following set $$ \left\{\int_{\Omega}|Du|: u \in BV(\Omega)\,,u|_{\partial \Omega} =f\right\}. $$ We establish connection between the planar least gradient solutions and variational problems in Free Material Design. We discuss also the following special cases:

• The trace $f$ is defined only on a piece of the boundary.
• The set $\Omega$ is a rectangle, as opposed to the existing theory in the literature for strictly convex domains.

We show that under some monotonicity conditions on the trace $f$ a solution to the least gradient problem exists, but may fail to be continuous and is not necessarily unique.
Results presented in this talk are joint work with W. Górny and P. Rybka at University of Warsaw.