## PDE Real Analysis Seminar

Seminar information archive ～03/04｜Next seminar｜Future seminars 03/05～

Date, time & place | Tuesday 10:30 - 11:30 056Room #056 (Graduate School of Math. Sci. Bldg.) |
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**Seminar information archive**

### 2016/07/12

10:20-11:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Special cases of the planar least gradient problem (English)

**Piotr Rybka**(University of Warsaw)Special cases of the planar least gradient problem (English)

[ Abstract ]

We study the least gradient problem in two special cases:

(1) the natural boundary conditions are imposed on a part of the strictly convex domain while the Dirichlet data are given on the rest of the boundary; or

(2) the Dirichlet data are specified on the boundary of a rectangle. We show existence of solutions and study properties of solution for special cases of the data. We are particularly interested in uniqueness and continuity of solutions.

We study the least gradient problem in two special cases:

(1) the natural boundary conditions are imposed on a part of the strictly convex domain while the Dirichlet data are given on the rest of the boundary; or

(2) the Dirichlet data are specified on the boundary of a rectangle. We show existence of solutions and study properties of solution for special cases of the data. We are particularly interested in uniqueness and continuity of solutions.

### 2016/07/12

14:20-15:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Global Strong $L^p$ Well-Posedness of the 3D Primitive Equations (English)

**Amru Hussein**(TU Darmstadt)Global Strong $L^p$ Well-Posedness of the 3D Primitive Equations (English)

[ Abstract ]

Primitive Equations are considered to be a fundamental model for geophysical flows. Here, the $L^p$ theory for the full primitive equations, i.e. the three dimensional primitive equations coupled to the equation for temperature and salinity, is developed. This set of equations is globally strongly well-posed for arbitrary large initial data lying in certain interpolation spaces, which are explicitly characterized as subspaces of $H^2/p$, $p$, $1 < p < \infty$, satisfying certain boundary conditions. Thus, the general $L^p$ setting admits rougher data than the usual $L^2$ theory with initial data in $H^1$.

In this study, the linearized Stokes type problem plays a prominent role, and it turns out that it can be treated efficiently using perturbation methods for $H^\infty$-calculus.

Primitive Equations are considered to be a fundamental model for geophysical flows. Here, the $L^p$ theory for the full primitive equations, i.e. the three dimensional primitive equations coupled to the equation for temperature and salinity, is developed. This set of equations is globally strongly well-posed for arbitrary large initial data lying in certain interpolation spaces, which are explicitly characterized as subspaces of $H^2/p$, $p$, $1 < p < \infty$, satisfying certain boundary conditions. Thus, the general $L^p$ setting admits rougher data than the usual $L^2$ theory with initial data in $H^1$.

In this study, the linearized Stokes type problem plays a prominent role, and it turns out that it can be treated efficiently using perturbation methods for $H^\infty$-calculus.

### 2016/07/12

12:10-12:50 Room #056 (Graduate School of Math. Sci. Bldg.)

The role of convection in some Keller-Segel models (English)

**Elio Espejo**(National University of Colombia)The role of convection in some Keller-Segel models (English)

[ Abstract ]

An interesting problem in reaction-diffusion equations is the understanding of the role of convection in phenomena like blow-up or convergence. I will discuss this problem through some Keller-Segel type models arising in mathematical biology and show some recent results.

An interesting problem in reaction-diffusion equations is the understanding of the role of convection in phenomena like blow-up or convergence. I will discuss this problem through some Keller-Segel type models arising in mathematical biology and show some recent results.

### 2016/07/12

11:20-12:00 Room #056 (Graduate School of Math. Sci. Bldg.)

The total variation flow in $H^{−s}$ (English)

**Monika Muszkieta**(Wroclaw University of Science and Technology)The total variation flow in $H^{−s}$ (English)

[ Abstract ]

In the talk, we consider the 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 introduce a numerical scheme for an approximate solution to this flow which has been derived based on the primal-dual approach and discuses some issues concerning its convergence. We also show and compare results of numerical experiments obtained by application of this scheme for a simple initial data and different values of the index $s$.

This is a join work with Y. Giga.

In the talk, we consider the 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 introduce a numerical scheme for an approximate solution to this flow which has been derived based on the primal-dual approach and discuses some issues concerning its convergence. We also show and compare results of numerical experiments obtained by application of this scheme for a simple initial data and different values of the index $s$.

This is a join work with Y. Giga.

### 2016/04/27

15:00-16:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Fourier transform versus Hilbert transform (English)

http://u.math.biu.ac.il/~liflyand/

**Elijah Liflyand**(Bar-Ilan University, Israel)Fourier transform versus Hilbert transform (English)

[ Abstract ]

We present several results in which the interplay between the Fourier transform and the Hilbert transform is of special form and importance.

1. In 50-s (Kahane, Izumi-Tsuchikura, Boas, etc.), the following problem in Fourier Analysis attracted much attention: Let $\{a_k\},$ $k=0,1,2...,$ be the sequence of the Fourier coefficients of the absolutely convergent sine (cosine) Fourier series of a function $f:\mathbb T=[-\pi,\pi)\to \mathbb C,$ that is $\sum |a_k|<\infty.$ Under which conditions on $\{a_k\}$ the re-expansion of $f(t)$ ($f(t)-f(0)$, respectively) in the cosine (sine) Fourier series will also be absolutely convergent?

We solve a similar problem for functions on the whole axis and their Fourier transforms. Generally, the re-expansion of a function with integrable cosine (sine) Fourier transform in the sine (cosine) Fourier transform is integrable if and only if not only the initial Fourier transform is integrable but also the Hilbert transform of the initial Fourier transform is integrable.

2. The following result is due to Hardy and Littlewood: If a (periodic) function $f$ and its conjugate $\widetilde f$ are both of bounded variation, their Fourier series converge absolutely.

We generalize the Hardy-Littlewood theorem (joint work with U. Stadtmüller) to the Fourier transform of a function on the real axis and its modified Hilbert transform. The initial Hardy-Littlewood theorem is a partial case of this extension, when the function is taken to be with compact support.

3. These and other problems are integrated parts of harmonic analysis of functions of bounded variation. We have found the maximal space for the integrability of the Fourier transform of a function of bounded variation. Along with those known earlier, various interesting new spaces appear in this study. Their inter-relations lead, in particular, to improvements of Hardy's inequality.

There are multidimensional generalizations of these results.

[ Reference URL ]We present several results in which the interplay between the Fourier transform and the Hilbert transform is of special form and importance.

1. In 50-s (Kahane, Izumi-Tsuchikura, Boas, etc.), the following problem in Fourier Analysis attracted much attention: Let $\{a_k\},$ $k=0,1,2...,$ be the sequence of the Fourier coefficients of the absolutely convergent sine (cosine) Fourier series of a function $f:\mathbb T=[-\pi,\pi)\to \mathbb C,$ that is $\sum |a_k|<\infty.$ Under which conditions on $\{a_k\}$ the re-expansion of $f(t)$ ($f(t)-f(0)$, respectively) in the cosine (sine) Fourier series will also be absolutely convergent?

We solve a similar problem for functions on the whole axis and their Fourier transforms. Generally, the re-expansion of a function with integrable cosine (sine) Fourier transform in the sine (cosine) Fourier transform is integrable if and only if not only the initial Fourier transform is integrable but also the Hilbert transform of the initial Fourier transform is integrable.

2. The following result is due to Hardy and Littlewood: If a (periodic) function $f$ and its conjugate $\widetilde f$ are both of bounded variation, their Fourier series converge absolutely.

We generalize the Hardy-Littlewood theorem (joint work with U. Stadtmüller) to the Fourier transform of a function on the real axis and its modified Hilbert transform. The initial Hardy-Littlewood theorem is a partial case of this extension, when the function is taken to be with compact support.

3. These and other problems are integrated parts of harmonic analysis of functions of bounded variation. We have found the maximal space for the integrability of the Fourier transform of a function of bounded variation. Along with those known earlier, various interesting new spaces appear in this study. Their inter-relations lead, in particular, to improvements of Hardy's inequality.

There are multidimensional generalizations of these results.

http://u.math.biu.ac.il/~liflyand/

### 2016/03/16

16:00-17:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Fluids, vortex membranes, and skew-mean-curvature flows (English)

**Boris Khesin**(University of Toronto)Fluids, vortex membranes, and skew-mean-curvature flows (English)

[ Abstract ]

We show that an approximation of the hydrodynamical Euler equation describes the skew-mean-curvature flow on vortex membranes in any dimension. This generalizes the classical binormal, or vortex filament, equation in 3D. We present a Hamiltonian framework for dynamics of higher-dimensional vortex filaments and vortex sheets as singular 2-forms (Green currents) with support of codimensions 2 and 1, respectively.

We show that an approximation of the hydrodynamical Euler equation describes the skew-mean-curvature flow on vortex membranes in any dimension. This generalizes the classical binormal, or vortex filament, equation in 3D. We present a Hamiltonian framework for dynamics of higher-dimensional vortex filaments and vortex sheets as singular 2-forms (Green currents) with support of codimensions 2 and 1, respectively.

### 2016/01/26

10:30-11:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Hamilton-Jacobi equations for optimal control on 2-dimensional junction (English)

**Salomé Oudet**(University of Tokyo)Hamilton-Jacobi equations for optimal control on 2-dimensional junction (English)

[ Abstract ]

We are interested in infinite horizon optimal control problems on 2-dimensional junctions (namely a union of half-planes sharing a common straight line) where different dynamics and different running costs are allowed in each half-plane. As for more classical optimal control problems, ones wishes to determine the Hamilton-Jacobi equation which characterizes the value function. However, the geometric singularities of the 2-dimensional junction and discontinuities of data do not allow us to apply the classical results of the theory of the viscosity solutions.

We will explain how to skirt these difficulties using arguments coming both from the viscosity theory and from optimal control theory. By this way we prove that the expected equation to characterize the value function is well posed. In particular we prove a comparison principle for this equation.

We are interested in infinite horizon optimal control problems on 2-dimensional junctions (namely a union of half-planes sharing a common straight line) where different dynamics and different running costs are allowed in each half-plane. As for more classical optimal control problems, ones wishes to determine the Hamilton-Jacobi equation which characterizes the value function. However, the geometric singularities of the 2-dimensional junction and discontinuities of data do not allow us to apply the classical results of the theory of the viscosity solutions.

We will explain how to skirt these difficulties using arguments coming both from the viscosity theory and from optimal control theory. By this way we prove that the expected equation to characterize the value function is well posed. In particular we prove a comparison principle for this equation.

### 2016/01/19

10:30-11:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Well-posedness and stability of the full Ericksen-Leslie system for incompressible nematic liquid crystal flows

**Hao Wu**(Fudan University)Well-posedness and stability of the full Ericksen-Leslie system for incompressible nematic liquid crystal flows

[ Abstract ]

In this talk, the general Ericksen-Leslie (E-L) system modelling the incompressible nematic liquid crystal flow will be discussed.

We shall prove the well-posedness and long-time behavior of the E-L system under proper assumptions on the viscous Leslie coefficients.

In particular, we shall discuss the connection between Parodi's relation and stability of the E-L system.

In this talk, the general Ericksen-Leslie (E-L) system modelling the incompressible nematic liquid crystal flow will be discussed.

We shall prove the well-posedness and long-time behavior of the E-L system under proper assumptions on the viscous Leslie coefficients.

In particular, we shall discuss the connection between Parodi's relation and stability of the E-L system.

### 2015/09/29

10:30-11:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Nonlocal self-improving properties (English)

**Tuomo Kuusi**(Aalto University)Nonlocal self-improving properties (English)

[ Abstract ]

The classical Gehring lemma for elliptic equations with measurable coefficients states that an energy solution, which is initially assumed to be $W^{1,2}$-Sobolev regular, is actually in a better Sobolev space space $W^{1,q}$ for some $q>2$. This is a consequence of a self-improving property that so-called reverse Hölder inequality implies. In the case of nonlocal equations a self-improving effect appears: Energy solutions are also more differentiable. This is a new, purely nonlocal phenomenon, which is not present in the local case. The proof relies on a nonlocal version of the Gehring lemma involving new exit time and dyadic decomposition arguments. This is a joint work with G. Mingione and Y. Sire.

The classical Gehring lemma for elliptic equations with measurable coefficients states that an energy solution, which is initially assumed to be $W^{1,2}$-Sobolev regular, is actually in a better Sobolev space space $W^{1,q}$ for some $q>2$. This is a consequence of a self-improving property that so-called reverse Hölder inequality implies. In the case of nonlocal equations a self-improving effect appears: Energy solutions are also more differentiable. This is a new, purely nonlocal phenomenon, which is not present in the local case. The proof relies on a nonlocal version of the Gehring lemma involving new exit time and dyadic decomposition arguments. This is a joint work with G. Mingione and Y. Sire.

### 2015/07/14

10:30-11:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Viscosity solutions of Hamilton-Jacobi equations from a dynamical viewpoint (English)

**Lin Wang**(Tsinghua University)Viscosity solutions of Hamilton-Jacobi equations from a dynamical viewpoint (English)

[ Abstract ]

By establishing an implicit variational principle for contact Hamiltonian systems, we detect some properties of viscosity solutions of Hamilton-Jacobi equations of certain Hamilton-Jacobi equations depending on unknown functions, including large time behavior and regularity on certain sets. Besides, I will talk about some connections with contact geometry, thermodynamics and nonholonomic mechanics.

By establishing an implicit variational principle for contact Hamiltonian systems, we detect some properties of viscosity solutions of Hamilton-Jacobi equations of certain Hamilton-Jacobi equations depending on unknown functions, including large time behavior and regularity on certain sets. Besides, I will talk about some connections with contact geometry, thermodynamics and nonholonomic mechanics.

### 2015/05/19

10:30-11:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Convex bodies and geometry of some associated Minkowski functionals (日本語)

**Sumio Yamada**(Gakushuin University)Convex bodies and geometry of some associated Minkowski functionals (日本語)

[ Abstract ]

In this talk, we will investigate the construction of so-called Hilbert metric, as well as Funk metric, defined on convex set from a new variational viewpoint. The local and global aspects of the geometry of the resulting Minkowski functionals will be contrasted. As an application, some remarks on the Perron-Frobenius theorem will be made. Part of the project is a joint work with Athanase Papadopoulos (Strasbourg).

In this talk, we will investigate the construction of so-called Hilbert metric, as well as Funk metric, defined on convex set from a new variational viewpoint. The local and global aspects of the geometry of the resulting Minkowski functionals will be contrasted. As an application, some remarks on the Perron-Frobenius theorem will be made. Part of the project is a joint work with Athanase Papadopoulos (Strasbourg).

### 2015/01/20

10:30-11:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Maximum Principle and generalized principal eigenvalue for degenerate elliptic operators (English)

**Italo Capuzzo Dolcetta**(Università degli Studi di Roma "La Sapienza")Maximum Principle and generalized principal eigenvalue for degenerate elliptic operators (English)

[ Abstract ]

In my presentation I will report on a joint paper with H. Berestycki, A. Porretta and L. Rossi to appear shortly on JMPA.

We characterize the validity of the Maximum Principle in bounded domains for fully nonlinear degenerate elliptic operators in terms of the sign of a suitably defined generalized principal eigenvalue. Here, the maximum principle refers to the property of non-positivity of viscosity subsolutions of the Dirichlet problem.

The new notion of generalized principal eigenvalue that we introduce here allows us to deal with arbitrary type of degeneracy of the elliptic operators.

We further discuss the relations between this notion and other natural generalizations of the classical notion of principal eigenvalue, some of which have been previously introduced for particular classes of operators.

In my presentation I will report on a joint paper with H. Berestycki, A. Porretta and L. Rossi to appear shortly on JMPA.

We characterize the validity of the Maximum Principle in bounded domains for fully nonlinear degenerate elliptic operators in terms of the sign of a suitably defined generalized principal eigenvalue. Here, the maximum principle refers to the property of non-positivity of viscosity subsolutions of the Dirichlet problem.

The new notion of generalized principal eigenvalue that we introduce here allows us to deal with arbitrary type of degeneracy of the elliptic operators.

We further discuss the relations between this notion and other natural generalizations of the classical notion of principal eigenvalue, some of which have been previously introduced for particular classes of operators.

### 2015/01/13

10:30-11:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Global regular solutions to the Navier-Stokes equations which remain close to the two-dimensional solutions (English)

**Wojciech Zajączkowski**(Institute of Mathematics Polish Academy of Sciences)Global regular solutions to the Navier-Stokes equations which remain close to the two-dimensional solutions (English)

[ Abstract ]

We consider the motion of the Navier-Stokes equations in a cylinder with the Navier-boundary conditions. First we prove global existence of regular two-dimensional solutions non-decaying in time. Next we show stability of these solutions. In this way we have existence of global regular solutions which remain close to the two-dimensional solutions. We prove the results for nonvanishing external force in time.

We consider the motion of the Navier-Stokes equations in a cylinder with the Navier-boundary conditions. First we prove global existence of regular two-dimensional solutions non-decaying in time. Next we show stability of these solutions. In this way we have existence of global regular solutions which remain close to the two-dimensional solutions. We prove the results for nonvanishing external force in time.

### 2015/01/06

10:30-11:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Global existence and asymptotic behavior for some Keller-Segel systems coupled with Navier-Stokes equations (英語)

**Elio Eduardo Espejo**(National University of Colombia / Osaka University)Global existence and asymptotic behavior for some Keller-Segel systems coupled with Navier-Stokes equations (英語)

[ Abstract ]

There are plenty of examples in nature, where cells move in response to some chemical signal in the environment. Biologists call this phenomenon chemotaxis. In my talk I will approach the problem of describing mathematically the phenomenon of chemotaxis when it happens surrounded by a fluid. This is a new research topic bringing the attention of many scientists because it has given rise to many interesting questions having relevance in both biology and mathematics. In particular, I will present some new mathematical models arising from my current research that have given rise to Keller-Segel type systems coupled with Navier-Stokes systems. I will present some results of global existence and asymptotic behavior. Finally I will discuss some open problems.

There are plenty of examples in nature, where cells move in response to some chemical signal in the environment. Biologists call this phenomenon chemotaxis. In my talk I will approach the problem of describing mathematically the phenomenon of chemotaxis when it happens surrounded by a fluid. This is a new research topic bringing the attention of many scientists because it has given rise to many interesting questions having relevance in both biology and mathematics. In particular, I will present some new mathematical models arising from my current research that have given rise to Keller-Segel type systems coupled with Navier-Stokes systems. I will present some results of global existence and asymptotic behavior. Finally I will discuss some open problems.

### 2014/12/02

10:30-11:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Remark on single exponential bound of the vorticity gradient for the two-dimensional Euler flow around a corner (JAPANESE)

**Tsubasa Itoh**(Tokyo Institute of Technology)Remark on single exponential bound of the vorticity gradient for the two-dimensional Euler flow around a corner (JAPANESE)

[ Abstract ]

In this talk, the two dimensional Euler flow under a simple symmetry condition with hyperbolic structure in a unit square $D=\{(x_{1}, x_{2}): 0 < x_{1} + x_{2} < \sqrt{2},\ 0<-x_{1} + x_{2} < \sqrt{2}\}$ is considered.

It is shown that the Lipschitz estimate of the vorticity on the boundary is at most single exponential growth near the stagnation point.

(Joint work with Tsuyoshi Yoneda and Hideyuki Miura.)

In this talk, the two dimensional Euler flow under a simple symmetry condition with hyperbolic structure in a unit square $D=\{(x_{1}, x_{2}): 0 < x_{1} + x_{2} < \sqrt{2},\ 0<-x_{1} + x_{2} < \sqrt{2}\}$ is considered.

It is shown that the Lipschitz estimate of the vorticity on the boundary is at most single exponential growth near the stagnation point.

(Joint work with Tsuyoshi Yoneda and Hideyuki Miura.)

### 2014/09/17

16:30-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

On the abstract evolution equations of hyperbolic type (JAPANESE)

**Kentarou Yoshii**(Faculty of Science Division I, Tokyo University of Science)On the abstract evolution equations of hyperbolic type (JAPANESE)

[ Abstract ]

This talk deals with the abstract Cauchy problem for linear evolution equations of hyperbolic type in a Hilbert space. We will discuss the existence and uniqueness of its classical solution and apply the results to linear Schrödinger equations with time dependent potentials.

This talk deals with the abstract Cauchy problem for linear evolution equations of hyperbolic type in a Hilbert space. We will discuss the existence and uniqueness of its classical solution and apply the results to linear Schrödinger equations with time dependent potentials.

### 2014/06/24

10:30-11:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Sudden directional diffusion: counting and watching facets (ENGLISH)

**Piotr Rybka**(University of Warsaw)Sudden directional diffusion: counting and watching facets (ENGLISH)

[ Abstract ]

We study two examples of singular parabolic equations such that the diffusion is so strong that is leads to creation of facets. By facets we mean flat parts of the graphs of solutions with singular slopes. In one of the equations we study there are two singular slopes. The other equation has just one singular slope and the isotropic diffusion term. For both problems we watch and count facet.

For the system with two singular slopes a natural question arises if any solution may have an infinite number of oscillations. We also show that the solutions we constructed are viscosity solutions. This in turn gives estimates on the extinction time based on the comparison principle.

We study two examples of singular parabolic equations such that the diffusion is so strong that is leads to creation of facets. By facets we mean flat parts of the graphs of solutions with singular slopes. In one of the equations we study there are two singular slopes. The other equation has just one singular slope and the isotropic diffusion term. For both problems we watch and count facet.

For the system with two singular slopes a natural question arises if any solution may have an infinite number of oscillations. We also show that the solutions we constructed are viscosity solutions. This in turn gives estimates on the extinction time based on the comparison principle.

### 2014/04/15

10:30-11:30 Room #056 (Graduate School of Math. Sci. Bldg.)

An application of weighted Hardy spaces to the Navier-Stokes equations (JAPANESE)

**Yohei Tsutsui**(The University of Tokyo)An application of weighted Hardy spaces to the Navier-Stokes equations (JAPANESE)

[ Abstract ]

The purpose of this talk is to investigate decay orders of the L^2 energy of solutions to the incompressible homogeneous Navier-Stokes equations on the whole spaces by the aid of the theory of weighted Hardy spaces. The main estimates are two weighted inequalities for heat semigroup on weighted Hardy spaces and a weighted version of the div-curl lemma due to Coifman-Lions-Meyer-Semmes. It turns out that because of the use of weighted Hardy spaces, our decay orders of the energy can be close to the critical one of Wiegner.

The purpose of this talk is to investigate decay orders of the L^2 energy of solutions to the incompressible homogeneous Navier-Stokes equations on the whole spaces by the aid of the theory of weighted Hardy spaces. The main estimates are two weighted inequalities for heat semigroup on weighted Hardy spaces and a weighted version of the div-curl lemma due to Coifman-Lions-Meyer-Semmes. It turns out that because of the use of weighted Hardy spaces, our decay orders of the energy can be close to the critical one of Wiegner.

### 2013/11/13

10:30-11:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Eigenvalue Constraints and Regularity of Q-tensor Navier-Stokes Dynamics (ENGLISH)

**Mark Wilkinson**(École normale supérieure - Paris)Eigenvalue Constraints and Regularity of Q-tensor Navier-Stokes Dynamics (ENGLISH)

[ Abstract ]

The Q-tensor is a traceless and symmetric 3x3 matrix that describes the small-scale structure in nematic liquid crystals. In order to be physically meaningful, its eigenvalues should be bounded below by -1/3 and above by 2/3. This constraint raises questions regarding the physical predictions of theories which employ the Q-tensor; it also raises analytical issues in both static and dynamic Q-tensor theories of nematic liquid crystals. John Ball and Apala Majumdar recently constructed a singular map on traceless, symmetric matrices that penalises unphysical Q-tensors by giving them an infinite energy cost. In this talk, I shall present some mathematical results for a coupled Navier-Stokes system modelling nematic dynamics into which this map is built, including the existence, regularity and so-called `strict physicality' of its weak solutions.

The Q-tensor is a traceless and symmetric 3x3 matrix that describes the small-scale structure in nematic liquid crystals. In order to be physically meaningful, its eigenvalues should be bounded below by -1/3 and above by 2/3. This constraint raises questions regarding the physical predictions of theories which employ the Q-tensor; it also raises analytical issues in both static and dynamic Q-tensor theories of nematic liquid crystals. John Ball and Apala Majumdar recently constructed a singular map on traceless, symmetric matrices that penalises unphysical Q-tensors by giving them an infinite energy cost. In this talk, I shall present some mathematical results for a coupled Navier-Stokes system modelling nematic dynamics into which this map is built, including the existence, regularity and so-called `strict physicality' of its weak solutions.

### 2013/11/12

10:30-11:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Optimal initial values and regularity conditions of Besov space type for weak solutions to the Navier-Stokes system (ENGLISH)

**Reinhard Farwig**(Technische Universität Darmstadt)Optimal initial values and regularity conditions of Besov space type for weak solutions to the Navier-Stokes system (ENGLISH)

[ Abstract ]

In a joint work with H. Sohr (Paderborn) and W. Varnhorn (Kassel) we discuss the optimal condition on initial values for the instationary Navier-Stokes system in a bounded domain to get a locally regular solution in Serrin's class.

Then this result based on a description in Besov spaces will be used at all or almost all instants to prove new conditional regularity results for weak solutions.

In a joint work with H. Sohr (Paderborn) and W. Varnhorn (Kassel) we discuss the optimal condition on initial values for the instationary Navier-Stokes system in a bounded domain to get a locally regular solution in Serrin's class.

Then this result based on a description in Besov spaces will be used at all or almost all instants to prove new conditional regularity results for weak solutions.

### 2013/10/22

10:30-11:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Fractional harmonic maps and applications (ENGLISH)

**Armin Schikorra**(MPI for Mathematics in the Sciences, Leipzig)Fractional harmonic maps and applications (ENGLISH)

[ Abstract ]

Fractional harmonic mappings are critical points of a generalized Dirichlet Energy where the gradient is replaced with a (non-local) differential operator.

I will present aspects of the regularity theory of (non-local) fractional harmonic maps into manifolds, which extends (and contains) the theory of (poly-)harmonic mappings.

I also will mention, how one can show regularity for critical points of the Moebius (Knot-) Energy, applying the techniques developed in this theory.

Fractional harmonic mappings are critical points of a generalized Dirichlet Energy where the gradient is replaced with a (non-local) differential operator.

I will present aspects of the regularity theory of (non-local) fractional harmonic maps into manifolds, which extends (and contains) the theory of (poly-)harmonic mappings.

I also will mention, how one can show regularity for critical points of the Moebius (Knot-) Energy, applying the techniques developed in this theory.

### 2013/10/18

10:30-11:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Some applications of partial balayage (ENGLISH)

**Björn Gustavsson**(KTH Royal Institute of Technology)Some applications of partial balayage (ENGLISH)

[ Abstract ]

Partial balayage is a rather recent tool in potential theory. One of its origins is the construction of quadrature domains for subharmonic functions by Makoto Sakai in the 1970's. It also gives a convenient way of describing weak solutions to a moving boundary problem for Hele-Shaw flow (Laplacian growth), and recently Stephen Gardiner and Tomas Sjödin have used partial balayage to make progress on an inverse problem in potential theory. I plan to discuss some of these, and related, matters.

Partial balayage is a rather recent tool in potential theory. One of its origins is the construction of quadrature domains for subharmonic functions by Makoto Sakai in the 1970's. It also gives a convenient way of describing weak solutions to a moving boundary problem for Hele-Shaw flow (Laplacian growth), and recently Stephen Gardiner and Tomas Sjödin have used partial balayage to make progress on an inverse problem in potential theory. I plan to discuss some of these, and related, matters.

### 2013/07/23

10:30-11:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Analysis of the Simplified Ericksen-Leslie Model for Liquid Crystals (ENGLISH)

**Matthias Hieber**(Technische Universität Darmstadt)Analysis of the Simplified Ericksen-Leslie Model for Liquid Crystals (ENGLISH)

[ Abstract ]

Consider the Ericksen-Leslie model for the flow of liquid crystals in a bounded domain $\\Omega \\subset \\R^n$. In this talk we discuss various simplifications of the general model and describe a dynamic theory for the simplified equations by analyzing it as a quasilinear system. In particular, we show the existence of a unique, global, strong solutions to this system provided the initial data are close to an equilibrium or the solution is eventually bounded in the norm of the underlying state space. In this case the solution converges exponentially to an equilibrium. Moreover, the solution is shown to be real analytic, jointly in time and space.

We further analyze a non-isothermal extension of this model safisfying the first and second law of thermodynamics and show that results of the above type hold as well in this setting.

This is joint work with M. Nesensohn, J. Prüss and K. Schade.

Consider the Ericksen-Leslie model for the flow of liquid crystals in a bounded domain $\\Omega \\subset \\R^n$. In this talk we discuss various simplifications of the general model and describe a dynamic theory for the simplified equations by analyzing it as a quasilinear system. In particular, we show the existence of a unique, global, strong solutions to this system provided the initial data are close to an equilibrium or the solution is eventually bounded in the norm of the underlying state space. In this case the solution converges exponentially to an equilibrium. Moreover, the solution is shown to be real analytic, jointly in time and space.

We further analyze a non-isothermal extension of this model safisfying the first and second law of thermodynamics and show that results of the above type hold as well in this setting.

This is joint work with M. Nesensohn, J. Prüss and K. Schade.

### 2013/04/19

10:30-11:30 Room #056 (Graduate School of Math. Sci. Bldg.)

An approximation scheme for the anisotropic and the planar crystalline curvature flow (JAPANESE)

**Katsuyuki Ishii**(Kobe University)An approximation scheme for the anisotropic and the planar crystalline curvature flow (JAPANESE)

[ Abstract ]

In 2004 Chambolle proposed an algorithm for the mean curvature flow based on a variational problem. Since then, some extensions of his algorithm have been studied.

In this talk we would like to discuss the convergence of the anisotropic variant of his algorithm by use of the anisotropic signed distance function. An application to the approximation for the planar motion by crystalline curvature is also discussed.

In 2004 Chambolle proposed an algorithm for the mean curvature flow based on a variational problem. Since then, some extensions of his algorithm have been studied.

In this talk we would like to discuss the convergence of the anisotropic variant of his algorithm by use of the anisotropic signed distance function. An application to the approximation for the planar motion by crystalline curvature is also discussed.

### 2013/03/15

10:30-11:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Geometric rigidity for incompatible fields and an application to strain-gradient plasticity (ENGLISH)

**Caterina Zeppieri**(Universität Münster)Geometric rigidity for incompatible fields and an application to strain-gradient plasticity (ENGLISH)

[ Abstract ]

Motivated by the study of nonlinear plane elasticity in presence of edge dislocations, in this talk we show that in dimension two the Friesecke, James, and Müller Rigidity Estimate holds true also for matrix-fields with nonzero curl, modulo an error depending on the total mass of the curl.

The above generalised rigidity is then used to derive a strain-gradient model for plasticity from semi-discrete nonlinear dislocation energies by Gamma-convergence.

The above results are obtained in collaboration with S. Müller (University of Bonn, Germany) and L. Scardia (University of Glasgow, UK).

Motivated by the study of nonlinear plane elasticity in presence of edge dislocations, in this talk we show that in dimension two the Friesecke, James, and Müller Rigidity Estimate holds true also for matrix-fields with nonzero curl, modulo an error depending on the total mass of the curl.

The above generalised rigidity is then used to derive a strain-gradient model for plasticity from semi-discrete nonlinear dislocation energies by Gamma-convergence.

The above results are obtained in collaboration with S. Müller (University of Bonn, Germany) and L. Scardia (University of Glasgow, UK).