## 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**

### 2012/12/05

10:30-11:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Convergence to Equilibrium of Bounded Solutions with Application of Lojasiewicz-Simon's inequality (ENGLISH)

**Jie Jiang**(Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences)Convergence to Equilibrium of Bounded Solutions with Application of Lojasiewicz-Simon's inequality (ENGLISH)

[ Abstract ]

In this talk, we present the application of Lojasiewicz-Simon's inequality to the study on convergence of bounded global solutions to some evolution equations. We take a semi-linear parabolic initial-boundary problem as an example. With the help of Lojasiewicz-Simon's inequality we prove that the bounded global solution will converge to an equilibrium as time goes to infinity provided the nonlinear term is analytic in the unknown function. We also present the application of Lojasiewicz-Simon's inequality to the asymptotic behavior studies on phase-field models with Cattaneo law and chemotaxis models with volume-filling effect.

In this talk, we present the application of Lojasiewicz-Simon's inequality to the study on convergence of bounded global solutions to some evolution equations. We take a semi-linear parabolic initial-boundary problem as an example. With the help of Lojasiewicz-Simon's inequality we prove that the bounded global solution will converge to an equilibrium as time goes to infinity provided the nonlinear term is analytic in the unknown function. We also present the application of Lojasiewicz-Simon's inequality to the asymptotic behavior studies on phase-field models with Cattaneo law and chemotaxis models with volume-filling effect.

### 2012/11/21

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

Shape Optimization And Asymptotic For The Twisted Dirichlet Eigenvalue (ENGLISH)

**Giovanni Pisante**(Seconda Università degli Studi di Napoli)Shape Optimization And Asymptotic For The Twisted Dirichlet Eigenvalue (ENGLISH)

[ Abstract ]

Aim of the talk is to discuss some recent results obtained with G. Croce and A. Henrot on a generalization of the functional defining the first twisted eigenvalue.

We look at the set functional defined by minimizing a Rayleigh quotient involving Lebesgue norms with different exponents p and q among functions satisfying a zero boundary condition as well as a zero mean condition of order q.

First under suitable conditions on p and q, that ensure the existence of a minimizing function, we investigate the validity of an isoperimetric type inequality of the Reyleigh-Faber-Krahn type.

Then we study the limit of the functional for p and q tending to 1 and to infinity and discuss the relation with the limits of the second eigenvalues of the p-laplacian operator.

Aim of the talk is to discuss some recent results obtained with G. Croce and A. Henrot on a generalization of the functional defining the first twisted eigenvalue.

We look at the set functional defined by minimizing a Rayleigh quotient involving Lebesgue norms with different exponents p and q among functions satisfying a zero boundary condition as well as a zero mean condition of order q.

First under suitable conditions on p and q, that ensure the existence of a minimizing function, we investigate the validity of an isoperimetric type inequality of the Reyleigh-Faber-Krahn type.

Then we study the limit of the functional for p and q tending to 1 and to infinity and discuss the relation with the limits of the second eigenvalues of the p-laplacian operator.

### 2012/06/20

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

On the Navier-Stokes Cauchy problem with nondecaying data (ENGLISH)

**Paolo Maremonti**(Seconda Università degli Studi di Napoli)On the Navier-Stokes Cauchy problem with nondecaying data (ENGLISH)

[ Abstract ]

We prove the well posedeness of the Navier-Stokes Cauchy problem for nondecaying initial data u_0 \\in C (R^n) \\cap L^\\infty (R^n), n >= 3. This problem is studied by Giga, Inui and Matsui for n >= 3, and Giga, Matsui and Sawada in the two dimensional case. The aims of our paper are slight different since we also find pointwise estimates for the pressure field. Via a uniqueness theorem, we give a sort of structure theorem to GIM solutions.

We prove the well posedeness of the Navier-Stokes Cauchy problem for nondecaying initial data u_0 \\in C (R^n) \\cap L^\\infty (R^n), n >= 3. This problem is studied by Giga, Inui and Matsui for n >= 3, and Giga, Matsui and Sawada in the two dimensional case. The aims of our paper are slight different since we also find pointwise estimates for the pressure field. Via a uniqueness theorem, we give a sort of structure theorem to GIM solutions.

### 2012/05/23

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

$L^p$ Estimates of the Vector Fields and their Applications (ENGLISH)

**Xingfei Xiang**(East China Normal University)$L^p$ Estimates of the Vector Fields and their Applications (ENGLISH)

[ Abstract ]

For $1< p < \\infty$, the estimates of $W^{1,p}$ norm of the vector fields in bounded domains in $\\mathbb R^3$ in terms of their divergence and curl have been well studied. In this talk, we shall present the $L^{{3}/{2}}$ estimates of vector fields with the $L^1$ norm of the $\\curl$ in bounded domains. By a similar discussion, we establish the $L^p$ estimates of the vector fields for $1 < p < \\infty$. As an application of the $L^p$ estimates, the Global $\\dv-\\curl$ lemma in Sobolev spaces of negative indices is given.

For $1< p < \\infty$, the estimates of $W^{1,p}$ norm of the vector fields in bounded domains in $\\mathbb R^3$ in terms of their divergence and curl have been well studied. In this talk, we shall present the $L^{{3}/{2}}$ estimates of vector fields with the $L^1$ norm of the $\\curl$ in bounded domains. By a similar discussion, we establish the $L^p$ estimates of the vector fields for $1 < p < \\infty$. As an application of the $L^p$ estimates, the Global $\\dv-\\curl$ lemma in Sobolev spaces of negative indices is given.

### 2012/04/04

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

Multi linear formulation of differential geometry and matrix regularizations (ENGLISH)

**Jens Hoppe**(Sogang University / KTH Royal Institute of Technology)Multi linear formulation of differential geometry and matrix regularizations (ENGLISH)

[ Abstract ]

We prove that many aspects of the differential geometry of embedded Riemannian manifolds can be formulated in terms of multi linear algebraic structures on the space of smooth functions. In particular, we find algebraic expressions for Weingarten's formula, the Ricci curvature and the Codazzi-Mainardi equations.

For matrix analogues of embedded surfaces we define discrete curvatures and Euler characteristics, and a non-commutative Gauss–Bonnet theorem is shown to follow. We derive simple expressions for the discrete Gauss curvature in terms of matrices representing the embedding coordinates, and a large class of explicit examples is provided.

We prove that many aspects of the differential geometry of embedded Riemannian manifolds can be formulated in terms of multi linear algebraic structures on the space of smooth functions. In particular, we find algebraic expressions for Weingarten's formula, the Ricci curvature and the Codazzi-Mainardi equations.

For matrix analogues of embedded surfaces we define discrete curvatures and Euler characteristics, and a non-commutative Gauss–Bonnet theorem is shown to follow. We derive simple expressions for the discrete Gauss curvature in terms of matrices representing the embedding coordinates, and a large class of explicit examples is provided.

### 2012/03/21

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

The asymptotic behaviors of the solutions of Poisson-Boltzmann type of equations (ENGLISH)

**Chiun-Chang Lee**(National Taiwan University)The asymptotic behaviors of the solutions of Poisson-Boltzmann type of equations (ENGLISH)

[ Abstract ]

Understanding the existence of electrical double layers around particles in the colloidal dispersion (system) is a crucial phenomenon of the colloid science. The Poisson-Boltzmann (PB) equation is one of the most widely used models to describe the equilibrium phenomenon of an electrical double layer in colloidal systems. This motivates us to study the asymptotic behavior for the boundary layer of the solutions of the PB equation. In this talk, we introduce the precise asymptotic formulas for the slope of the boundary layers with the exact leading order term and the second-order term. In particular, these formulas show that the mean curvature of the boundary exactly appears in the second-order term. This part is my personal work.

On the other hand, to study how the ionic concentrations and ionic valences affect the difference between the boundary and interior potentials in an electrolyte solution, we also introduce a modified PB equation - New Poisson-Boltzmann (PB_n) equation - joint works with Prof. Tai-Chia Lin and Chun Liu and YunKyong Hyon. We give a specific formula showing the influence of these crucial physical quantities on the potential difference in an electrolyte solution. This cannot be found in the PB equation.

Understanding the existence of electrical double layers around particles in the colloidal dispersion (system) is a crucial phenomenon of the colloid science. The Poisson-Boltzmann (PB) equation is one of the most widely used models to describe the equilibrium phenomenon of an electrical double layer in colloidal systems. This motivates us to study the asymptotic behavior for the boundary layer of the solutions of the PB equation. In this talk, we introduce the precise asymptotic formulas for the slope of the boundary layers with the exact leading order term and the second-order term. In particular, these formulas show that the mean curvature of the boundary exactly appears in the second-order term. This part is my personal work.

On the other hand, to study how the ionic concentrations and ionic valences affect the difference between the boundary and interior potentials in an electrolyte solution, we also introduce a modified PB equation - New Poisson-Boltzmann (PB_n) equation - joint works with Prof. Tai-Chia Lin and Chun Liu and YunKyong Hyon. We give a specific formula showing the influence of these crucial physical quantities on the potential difference in an electrolyte solution. This cannot be found in the PB equation.

### 2012/03/14

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

Exponential convergence to equilibria for a general model in hydrodynamics (ENGLISH)

**Jürgen Saal**(Technische Universität Darmstadt)Exponential convergence to equilibria for a general model in hydrodynamics (ENGLISH)

[ Abstract ]

We present a thorough analysis of the Navier-Stokes-Nernst-Planck-Poisson equations. This system describes the dynamics of charged particles dispersed in an incompressible fluid.

In contrast to existing literature and in view of its physical relevance, we also allow for different diffusion coefficients of the charged species.

In addition, the commonly assumed electro-neutrality condition is not required by our approach.

Our aim is to present results on local and global well-posedness as well as exponential stability of equilibria. The results are obtained jointly with Dieter Bothe and Andre Fischer at the Center of Smart Interfaces at TU Darmstadt.

We present a thorough analysis of the Navier-Stokes-Nernst-Planck-Poisson equations. This system describes the dynamics of charged particles dispersed in an incompressible fluid.

In contrast to existing literature and in view of its physical relevance, we also allow for different diffusion coefficients of the charged species.

In addition, the commonly assumed electro-neutrality condition is not required by our approach.

Our aim is to present results on local and global well-posedness as well as exponential stability of equilibria. The results are obtained jointly with Dieter Bothe and Andre Fischer at the Center of Smart Interfaces at TU Darmstadt.

### 2012/02/01

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

Energetic variational approach: generalized diffusion, stochastic differential equations and optimal transport (ENGLISH)

**Chun Liu**(University of Tokyo / Penn State University)Energetic variational approach: generalized diffusion, stochastic differential equations and optimal transport (ENGLISH)

[ Abstract ]

In the talk, I will explore the general framework of energetic variational approaches, which are the direct consequences of classical isothermal thermodynamics, and their particular applications in generalized diffusion problems. In particular, we reveal the roles of different stochastic integrations (Ito's form, Stratonovich's form and other possible forms) and the Wasserstein metric and the procedure of optimal transport in the context of general framework of theories of linear responses.

In the talk, I will explore the general framework of energetic variational approaches, which are the direct consequences of classical isothermal thermodynamics, and their particular applications in generalized diffusion problems. In particular, we reveal the roles of different stochastic integrations (Ito's form, Stratonovich's form and other possible forms) and the Wasserstein metric and the procedure of optimal transport in the context of general framework of theories of linear responses.

### 2011/10/31

13:30-14:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Stationary Weak Solutions of the Navier-Stokes Equations Past an Obstacle (ENGLISH)

**Horst Heck**

(Technische Universität Darmstadt)Stationary Weak Solutions of the Navier-Stokes Equations Past an Obstacle (ENGLISH)

[ Abstract ]

Consider the stationary Navier-Stokes equations in an exterior smooth domain $\\Omega$. If the flow condition $u_\\infty$ for $u$ at infinity is non-zero and the external force $f\\in \\dot H^{-1}_2(\\Omega)$ is given Leray constructed a weak solution $u\\in \\dot H^1_2(\\Omega)$, the homogeneous Sobolev space, with $u-u_\\infty\\in L^6(\\Omega)$.

We show that if in addition $f\\in \\dot H^{-1}_q(\\Omega)$ for some $q\\in (4/3,4)$ then the weak solution has the property $u-u_\\infty\\in L^{4q/(4-q)}(\\Omega)$.

This additional integrability implies that $u$ satisfies the energy identity. Further consequences are uniqueness results for small $u_\\infty$ and $f$ and continuous dependence of the solution with respect to $u_\\infty$.

The presented results are joint work with Hyunseok Kim and Hideo Kozono.

Consider the stationary Navier-Stokes equations in an exterior smooth domain $\\Omega$. If the flow condition $u_\\infty$ for $u$ at infinity is non-zero and the external force $f\\in \\dot H^{-1}_2(\\Omega)$ is given Leray constructed a weak solution $u\\in \\dot H^1_2(\\Omega)$, the homogeneous Sobolev space, with $u-u_\\infty\\in L^6(\\Omega)$.

We show that if in addition $f\\in \\dot H^{-1}_q(\\Omega)$ for some $q\\in (4/3,4)$ then the weak solution has the property $u-u_\\infty\\in L^{4q/(4-q)}(\\Omega)$.

This additional integrability implies that $u$ satisfies the energy identity. Further consequences are uniqueness results for small $u_\\infty$ and $f$ and continuous dependence of the solution with respect to $u_\\infty$.

The presented results are joint work with Hyunseok Kim and Hideo Kozono.

### 2011/07/06

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

Trace inequality and Morrey spaces (JAPANESE)

**Hitoshi Tanaka**(University of Tokyo)Trace inequality and Morrey spaces (JAPANESE)

### 2011/04/20

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

Stochastic power law fluids (JAPANESE)

http://www.math.kyoto-u.ac.jp/~nobuo/

**Yoshida, Nobuo**(Department of Mathematics, Kyoto University)Stochastic power law fluids (JAPANESE)

[ Abstract ]

This talk is based in part on a joint work with Yutaka Terasawa.

We consider a SPDE (stochastic partial differential equation) which describes the velocity field of a viscous, incompressible non-Newtonian fluid subject to a random force.

Here, the extra stress tensor of the fluid is given by a polynomial of degree $p-1$ of the rate of strain tensor, while the colored noise is considered as a random force.

We first investigate the existence and the uniqueness of weak solutions to this SPDE.

We next turn to the special case: $p \\in [1 + {d \\over 2},{2d\\overd-2})$,

where $d$ is the dimension of the space. We prove there that the Galerkin scheme approximates the velocity field in a strong sense. As a consequence, we establish the energy equality for the velocity field.

[ Reference URL ]This talk is based in part on a joint work with Yutaka Terasawa.

We consider a SPDE (stochastic partial differential equation) which describes the velocity field of a viscous, incompressible non-Newtonian fluid subject to a random force.

Here, the extra stress tensor of the fluid is given by a polynomial of degree $p-1$ of the rate of strain tensor, while the colored noise is considered as a random force.

We first investigate the existence and the uniqueness of weak solutions to this SPDE.

We next turn to the special case: $p \\in [1 + {d \\over 2},{2d\\overd-2})$,

where $d$ is the dimension of the space. We prove there that the Galerkin scheme approximates the velocity field in a strong sense. As a consequence, we establish the energy equality for the velocity field.

http://www.math.kyoto-u.ac.jp/~nobuo/

### 2011/01/26

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

)

Quenching Problem Arising in Micro-electro Mechanical Systems

(JAPANESE)

**Jong-Shenq Guo**(Department of Mathematics, Tamkang University)

Quenching Problem Arising in Micro-electro Mechanical Systems

(JAPANESE)

[ Abstract ]

In this talk, we shall present some recent results on quenching problems which arise in Micro-electro Mechanical Systems.

We shall also give some open problems in this research area.

In this talk, we shall present some recent results on quenching problems which arise in Micro-electro Mechanical Systems.

We shall also give some open problems in this research area.

### 2010/05/26

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

Hokkaido University)

A SELECTION CRITERION FOR SOLUTIONS OF A

SYSTEM OF EIKONAL EQUATIONS

(ENGLISH)

**Giovanni Pisante**(Department of MathematicsHokkaido University)

A SELECTION CRITERION FOR SOLUTIONS OF A

SYSTEM OF EIKONAL EQUATIONS

(ENGLISH)

[ Abstract ]

We deal with the system of eikonal equations |ðu/ðx1|=1, |ðu/ðx2|=1 in a planar Lipschitz domain with zero boundary condition. Exploiting the classical pyramidal construction introduced by Cellina, it is easy to prove that there exist infinitely many Lipschitz solutions. Then, the natural problem that has arisen in this framework is to find a way to select and characterize a particular meaningful class of solutions.

We propose a variational method to select the class of solutions which minimize the discontinuity set of the gradient. More precisely we select an optimal weighted measure for the jump set of the second derivatives of a given solution v of the system and we prove the existence of minimizers of the corresponding variational problem.

We deal with the system of eikonal equations |ðu/ðx1|=1, |ðu/ðx2|=1 in a planar Lipschitz domain with zero boundary condition. Exploiting the classical pyramidal construction introduced by Cellina, it is easy to prove that there exist infinitely many Lipschitz solutions. Then, the natural problem that has arisen in this framework is to find a way to select and characterize a particular meaningful class of solutions.

We propose a variational method to select the class of solutions which minimize the discontinuity set of the gradient. More precisely we select an optimal weighted measure for the jump set of the second derivatives of a given solution v of the system and we prove the existence of minimizers of the corresponding variational problem.

### 2010/05/12

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

The University of Tokyo)

Exact controllability for incompressible fluids (ENGLISH)

**Jean-Pierre Puel**(Graduate School of Mathematical SciencesThe University of Tokyo)

Exact controllability for incompressible fluids (ENGLISH)

[ Abstract ]

After a short presentation of J.-M. Coron's results for Euler equation, we will give the good notions of controllability for Navier-Stokes equations, namely the exact controllability to trajectories.

We will outline the strategy for obtaining local results, based on a fixed point argument following the study of null controllability for the linearized problem. This is equivalent to an observability inequality for the adjoint system, which requires a global Carleman estimate for linearized Navier-Stokes equations. We will explain this estimate and the different steps for obtaining it along the lines of the articles by E.Fernadez-Cara, S.Guerrero, O.Imanuvilov and J.-P.Puel (JMPA, 2004) and M.Gonzalez-Burgos, S.Guerrero and J.-P.Puel (CPAA, 2009).

We will end up with some important open problems.

After a short presentation of J.-M. Coron's results for Euler equation, we will give the good notions of controllability for Navier-Stokes equations, namely the exact controllability to trajectories.

We will outline the strategy for obtaining local results, based on a fixed point argument following the study of null controllability for the linearized problem. This is equivalent to an observability inequality for the adjoint system, which requires a global Carleman estimate for linearized Navier-Stokes equations. We will explain this estimate and the different steps for obtaining it along the lines of the articles by E.Fernadez-Cara, S.Guerrero, O.Imanuvilov and J.-P.Puel (JMPA, 2004) and M.Gonzalez-Burgos, S.Guerrero and J.-P.Puel (CPAA, 2009).

We will end up with some important open problems.

### 2010/04/28

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

)

GLOBAL AND SINGULAR SOLUTIONS TO SOME

HYDRODYNAMIC EVOLUTION EQUATIONS

**Marcus Wunsch**(Kyoto University)

GLOBAL AND SINGULAR SOLUTIONS TO SOME

HYDRODYNAMIC EVOLUTION EQUATIONS

[ Abstract ]

The two-component Hunter-Saxton system is a recently derived system of evolution equations modeling, e.g., the nonlinear dynamics of nondissipative dark matter and the propagation of orientation waves in nematic liquid crystals. It is imbedded into a parameterized family of systems called the generalized Hunter-Saxton (2HS) system [2] reducing, if one component is omitted, to the generalized Proudman-Johnson(gPJ) equation [1] modeling three-dimensional vortex dynamics.

After demonstrating, by means of Kato's semigroup theory, the local-in-time existence of classical solutions, the blow-up scenarios for the 2HS system and the gPJ equation are described. The explicit construction of weak dissipative solutions for both models is discussed in detail.

Finally, global existence in time of these weak solutions is proved.

The two-component Hunter-Saxton system is a recently derived system of evolution equations modeling, e.g., the nonlinear dynamics of nondissipative dark matter and the propagation of orientation waves in nematic liquid crystals. It is imbedded into a parameterized family of systems called the generalized Hunter-Saxton (2HS) system [2] reducing, if one component is omitted, to the generalized Proudman-Johnson(gPJ) equation [1] modeling three-dimensional vortex dynamics.

After demonstrating, by means of Kato's semigroup theory, the local-in-time existence of classical solutions, the blow-up scenarios for the 2HS system and the gPJ equation are described. The explicit construction of weak dissipative solutions for both models is discussed in detail.

Finally, global existence in time of these weak solutions is proved.

### 2010/03/09

10:30-11:30 Room #123 (Graduate School of Math. Sci. Bldg.)

Shallow water waves with singularities

**Joachim Escher**(Leibniz University of Hanover)Shallow water waves with singularities

[ Abstract ]

The Degasperis-Procesi equation is a recently derived shallow water wave equation, which is - similar as the Camassa-Holm equation - embedded in a family of spatially periodic third order dispersive conservation laws.

The coexistence of globally in time defined classical solutions, wave breaking solutions, and spatially periodic peakons and shock waves is evidenced in the talk, and the precise blow-up scenario, including blow-up rates and blow-up sets, is discussed in detail. Finally several conditions on the initial profile are presented, which ensure the occurence of a breaking wave.

The Degasperis-Procesi equation is a recently derived shallow water wave equation, which is - similar as the Camassa-Holm equation - embedded in a family of spatially periodic third order dispersive conservation laws.

The coexistence of globally in time defined classical solutions, wave breaking solutions, and spatially periodic peakons and shock waves is evidenced in the talk, and the precise blow-up scenario, including blow-up rates and blow-up sets, is discussed in detail. Finally several conditions on the initial profile are presented, which ensure the occurence of a breaking wave.

### 2009/12/02

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

A hyperbolic fluid model based on Cattaneo's law

**Juergen Saal**(University of Konstanz)A hyperbolic fluid model based on Cattaneo's law

[ Abstract ]

In various applications a delay of the propagation speed of a fluid (temperature, ...) has been observed. Such phenomena cannot be described by standard parabolic models, whose derivation relies on Fourier's law (Paradoxon of infinite propagation speed).

One way to give account to these observations and which was successfully applied to several models, is to replace Fourier's law by the law of Cattaneo. In the case of a fluid, this leads to a hyperbolicly perturbed quasilinear Navier-Stokes system for which existence and uniqueness results will be presented.

In various applications a delay of the propagation speed of a fluid (temperature, ...) has been observed. Such phenomena cannot be described by standard parabolic models, whose derivation relies on Fourier's law (Paradoxon of infinite propagation speed).

One way to give account to these observations and which was successfully applied to several models, is to replace Fourier's law by the law of Cattaneo. In the case of a fluid, this leads to a hyperbolicly perturbed quasilinear Navier-Stokes system for which existence and uniqueness results will be presented.

### 2009/11/25

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

Recent results on weak and strong solutions of the Navier-Stokes equations

**Hermann Sohr**(University Paderborn)Recent results on weak and strong solutions of the Navier-Stokes equations

[ Abstract ]

Our purpose is to develop the optimal initial value condition for the existence of a unique local strong solution of the Navier-Stokes equations in a smooth bounded domain.

This condition is not only sufficient

- there are several well-known sufficient conditions in this context

- but also necessary, and yields therefore the largest possible class of such strong solutions.

As an application we obtain several extensions of Serrin's regularity condition. A restricted result also holds for completely general domains. Furthermore we extend the well-known class of Leray-Hopf weak solutions with zero boundary conditions and zero divergence to a larger class with corresponding nonzero conditions.

Our purpose is to develop the optimal initial value condition for the existence of a unique local strong solution of the Navier-Stokes equations in a smooth bounded domain.

This condition is not only sufficient

- there are several well-known sufficient conditions in this context

- but also necessary, and yields therefore the largest possible class of such strong solutions.

As an application we obtain several extensions of Serrin's regularity condition. A restricted result also holds for completely general domains. Furthermore we extend the well-known class of Leray-Hopf weak solutions with zero boundary conditions and zero divergence to a larger class with corresponding nonzero conditions.

### 2009/10/07

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

Wiener measure and Feynman-Kac formula on the Heisenberg group

**劉和平(Liu Heping)**(Beijing University)Wiener measure and Feynman-Kac formula on the Heisenberg group

[ Abstract ]

It is well known that the Feynman-Kac formula on the Euclidean space gives the solution of Schrodinger equation by the Wiener integral. We will discuss the Wiener measure and Feynman-Kac formula on the Heisenberg group. The results hold on the H-type groups.

It is well known that the Feynman-Kac formula on the Euclidean space gives the solution of Schrodinger equation by the Wiener integral. We will discuss the Wiener measure and Feynman-Kac formula on the Heisenberg group. The results hold on the H-type groups.

### 2009/06/24

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

Monge-Ampere equations, the Bellman Function Technique, and Muckenhoupt weights

**Winston Ou**(Scripps College / currently visiting assistant professor at Keio University)Monge-Ampere equations, the Bellman Function Technique, and Muckenhoupt weights

[ Abstract ]

In the last few years several classical results in harmonic analysis (in particular, the study of $A_\\infty$ weights have been sharpened with the use of a version of the Bellman function method (promulgated by Nazarov, Treil, and Volberg in the 90's) that involves recognizing the Bellman function as the solution of a Monge-Ampere PDE (the method was introduced by Vasyunin in 2003). We will give a sketch of the modified technique, outline some recent work-in-progress (with Slavin and Wall) using the technique in $A_\\infty$, and then present a few related problems.

In the last few years several classical results in harmonic analysis (in particular, the study of $A_\\infty$ weights have been sharpened with the use of a version of the Bellman function method (promulgated by Nazarov, Treil, and Volberg in the 90's) that involves recognizing the Bellman function as the solution of a Monge-Ampere PDE (the method was introduced by Vasyunin in 2003). We will give a sketch of the modified technique, outline some recent work-in-progress (with Slavin and Wall) using the technique in $A_\\infty$, and then present a few related problems.

### 2009/06/17

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

Variational problems for anisotropic surface energies

**小磯深幸**(奈良女子大学理学部/JSTさきがけ)Variational problems for anisotropic surface energies

[ Abstract ]

A surface energy is anisotropic if it depends on the direction of the surface. The minimizer of an anisotropic surface energy among all closed surfaces enclosing a fixed volume is called the Wulff shape. We will discuss the characterization of the Wulff shape, the uniqueness and stability of solutions to variational problems for anisotropic surface energy with several boundary conditions.

A surface energy is anisotropic if it depends on the direction of the surface. The minimizer of an anisotropic surface energy among all closed surfaces enclosing a fixed volume is called the Wulff shape. We will discuss the characterization of the Wulff shape, the uniqueness and stability of solutions to variational problems for anisotropic surface energy with several boundary conditions.

### 2009/05/20

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

Stability of the Burgers vortex

**前川泰則**(神戸大学)Stability of the Burgers vortex

[ Abstract ]

The Burgers vortex is an exact vortex solution to the three dimensional stationary Navier-Stokes equations for viscous incompressible fluids in the presence of an axisymmetric background straining flow. In this talk we discuss the stability of the Burgers vortex with respect to two or three dimensional perturbation flows.

The Burgers vortex is an exact vortex solution to the three dimensional stationary Navier-Stokes equations for viscous incompressible fluids in the presence of an axisymmetric background straining flow. In this talk we discuss the stability of the Burgers vortex with respect to two or three dimensional perturbation flows.

### 2009/04/22

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

From Codazzi-Mainardi to Cauchy-Riemann

**Wilhelm Klingenberg**(University of Durham)From Codazzi-Mainardi to Cauchy-Riemann

[ Abstract ]

In joint work with Brendan Guilfoyle we established an upper bound for the winding number of the principal curvature foliation at any isolated umbilic of a surface in Euclidean three-space. In our talk, we will focus on the analytic core of the problem. Here is a model of the triaxial ellipsoid with its curvature foliation and one umbilic on the right.

In joint work with Brendan Guilfoyle we established an upper bound for the winding number of the principal curvature foliation at any isolated umbilic of a surface in Euclidean three-space. In our talk, we will focus on the analytic core of the problem. Here is a model of the triaxial ellipsoid with its curvature foliation and one umbilic on the right.

### 2008/10/15

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

Asymptotic behavior of solutions for BCF model describing crystal surface growth

**八木厚志**(大阪大学)Asymptotic behavior of solutions for BCF model describing crystal surface growth

[ Abstract ]

This talk is concerned with the initial-boundary value problem for a nonlinear parabolic equation which was presented Johnson et al. for describing the process of growth of a crystal surface on the

basis of the BCF theory. We will investigate asymptotic behavior of solutions by construct exponential attractors and a Lyapunov function and by examining a structure of the $\\omega$ limit set.

This talk is concerned with the initial-boundary value problem for a nonlinear parabolic equation which was presented Johnson et al. for describing the process of growth of a crystal surface on the

basis of the BCF theory. We will investigate asymptotic behavior of solutions by construct exponential attractors and a Lyapunov function and by examining a structure of the $\\omega$ limit set.

### 2008/06/04

16:00-18:15 Room #056 (Graduate School of Math. Sci. Bldg.)

Inverse Obstacle Recovery when the boundary condition is also unknown

The Inverse Scattering Problem for an Isotropic Medium

**William Rundell**(Department of Mathematics, Texas A&M University) 16:00-17:00Inverse Obstacle Recovery when the boundary condition is also unknown

[ Abstract ]

We consider the inverse problem of recovering the shape, location

and surface properties of an object where the surrounding medium

is both conductive and homogeneous. It is assumed that the physical situation is modeled by either harmonic functions or solutions of the Helmholtz equation and that the boundary condition on the obstacle is one of impedance type. We measure either Cauchy data, on an accessible part of the exterior boundary or the far field pattern resulting from a plane wave. Given sets of Cauchy data pairs we wish to recover both the shape and location of the unknown obstacle together with its impedance.

It turns out this adds considerable complexity to the analysis. We give a local injectivity result and use two different algorithms

to investigate numerical reconstructions. The setting is in two space dimensions, but indications of possible extensions (and difficulties) to three dimensions are provided. We also look at the case of a nonlinear impedance function.

We consider the inverse problem of recovering the shape, location

and surface properties of an object where the surrounding medium

is both conductive and homogeneous. It is assumed that the physical situation is modeled by either harmonic functions or solutions of the Helmholtz equation and that the boundary condition on the obstacle is one of impedance type. We measure either Cauchy data, on an accessible part of the exterior boundary or the far field pattern resulting from a plane wave. Given sets of Cauchy data pairs we wish to recover both the shape and location of the unknown obstacle together with its impedance.

It turns out this adds considerable complexity to the analysis. We give a local injectivity result and use two different algorithms

to investigate numerical reconstructions. The setting is in two space dimensions, but indications of possible extensions (and difficulties) to three dimensions are provided. We also look at the case of a nonlinear impedance function.

**David Colton**(Department of Mathematical Sciences, University of Delaware) 17:15-18:15The Inverse Scattering Problem for an Isotropic Medium

[ Abstract ]

This talk is concerned with the inverse electromagnetic scattering problem for an isotropic inhomogeneous infinite cylinder. After formulating the direct scattering problem we proceed to the inverse scattering problem which is the main theme of this lecture. After discussing what is known about uniqueness for the inverse problem,we will proceed to the definition and properties of the far field operator. This leads to the study of a rather unusual spectral problem for partial differential equations called the interior transmission problem. We will state what is known about this problem including its role in determining lower bounds for the index of refraction from a knowledge of the far field pattern of the scattered wave, The talk is concluded by briefly considering the case of limited aperture data,in particular the use of the gap reciprocity method to determine the shape and location of buried objects. Numerical examples will be given as well as a number of open problems.

This talk is concerned with the inverse electromagnetic scattering problem for an isotropic inhomogeneous infinite cylinder. After formulating the direct scattering problem we proceed to the inverse scattering problem which is the main theme of this lecture. After discussing what is known about uniqueness for the inverse problem,we will proceed to the definition and properties of the far field operator. This leads to the study of a rather unusual spectral problem for partial differential equations called the interior transmission problem. We will state what is known about this problem including its role in determining lower bounds for the index of refraction from a knowledge of the far field pattern of the scattered wave, The talk is concluded by briefly considering the case of limited aperture data,in particular the use of the gap reciprocity method to determine the shape and location of buried objects. Numerical examples will be given as well as a number of open problems.