## 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**

### 2008/03/19

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

Maximal Regularity for Mixed Order Systems

**Juergen Saal**(Department of Mathematics and Statistics, University of Konstanz)Maximal Regularity for Mixed Order Systems

[ Abstract ]

In classical boundary value problems the related symbols are homogeneous in space and time. This allows for the application of a standard compactness argument in order to obtain the important maximal regularity. However, quasilinear systems arising e.g. from free boundary problems are in general of mixed order. In other words the related symbols are of intricate structure and in particular highly inhomogeneous. Therefore, the standard compactness argument fails. The purpose of this talk is to introduce the Newton polygon method, which gives a systematic approach to such mixed order systems and to demonstrate its strength by applications to the Stefan problem and a free boundary problem for the Navier-Stokes equations.

In classical boundary value problems the related symbols are homogeneous in space and time. This allows for the application of a standard compactness argument in order to obtain the important maximal regularity. However, quasilinear systems arising e.g. from free boundary problems are in general of mixed order. In other words the related symbols are of intricate structure and in particular highly inhomogeneous. Therefore, the standard compactness argument fails. The purpose of this talk is to introduce the Newton polygon method, which gives a systematic approach to such mixed order systems and to demonstrate its strength by applications to the Stefan problem and a free boundary problem for the Navier-Stokes equations.

### 2008/01/30

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

Carleman estimates for parabolic equations, a Stokes system and the Navier-Stokes equations and applications to the control problem

**Oleg Yu. Emanouilov**(Colorado State University)Carleman estimates for parabolic equations, a Stokes system and the Navier-Stokes equations and applications to the control problem

[ Abstract ]

We prove a new Carleman estimate for general linear second order parabolic equation with nonhomogeneous boundary conditions. On basis of this estimate we obtain an improved Carleman estimate for the Stokes system and a system of parabolic equations with a parameter which can be viewed as an approximation of the Stokes system. We will discuss the applications to the control problem for these systems.

We prove a new Carleman estimate for general linear second order parabolic equation with nonhomogeneous boundary conditions. On basis of this estimate we obtain an improved Carleman estimate for the Stokes system and a system of parabolic equations with a parameter which can be viewed as an approximation of the Stokes system. We will discuss the applications to the control problem for these systems.

### 2007/09/05

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

Reguarity of Weak Solutions to the Navier-Stokes System beyond Serrin's Criterion

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

**Reinhard Farwig**(Darmstadt University of Technology)Reguarity of Weak Solutions to the Navier-Stokes System beyond Serrin's Criterion

[ Abstract ]

Consider a weak instationary solution $u(x,t)$ of the Navier-Stokes equations in a domain $\\Omega \\subset \\mathbb{R}^3$ in the sense of Leray-Hopf. As is well-known, $u$ is is unique and regular if $u\\in L^s(0,T;L^q(\\Omega))$ satisfies the {\\it strong energy inequality} and $s,q$ satisfy Serrin's condition $\\frac{2}{s} + \\frac{3}{q}=1$, $s>2,\\, q>3$. Now consider $u$ such that $$u\\in L^r(0,T;L^q(\\Omega))\\quad \\mbox{ where }\\quad \\frac{2}{r} + \\frac{3}{q}>1$$ and has a sufficiently small norm in $L^r(0,T;L^q(\\Omega))$. Then we will prove that $u$ is regular. Similar results of local rather than global type in space will be proved provided that $u$ satisfies the {\\it localized energy inequality}. Finally H\\"older continuity of the kinetic energy in time will imply regularity.

The proofs use local in time regularity results which are based on the {\\it theory of very weak solutions} and on uniqueness arguments for weak solutions.

[ Reference URL ]Consider a weak instationary solution $u(x,t)$ of the Navier-Stokes equations in a domain $\\Omega \\subset \\mathbb{R}^3$ in the sense of Leray-Hopf. As is well-known, $u$ is is unique and regular if $u\\in L^s(0,T;L^q(\\Omega))$ satisfies the {\\it strong energy inequality} and $s,q$ satisfy Serrin's condition $\\frac{2}{s} + \\frac{3}{q}=1$, $s>2,\\, q>3$. Now consider $u$ such that $$u\\in L^r(0,T;L^q(\\Omega))\\quad \\mbox{ where }\\quad \\frac{2}{r} + \\frac{3}{q}>1$$ and has a sufficiently small norm in $L^r(0,T;L^q(\\Omega))$. Then we will prove that $u$ is regular. Similar results of local rather than global type in space will be proved provided that $u$ satisfies the {\\it localized energy inequality}. Finally H\\"older continuity of the kinetic energy in time will imply regularity.

The proofs use local in time regularity results which are based on the {\\it theory of very weak solutions} and on uniqueness arguments for weak solutions.

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

### 2007/07/04

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

The Lipschitz truncation method

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

**Lars Diening**(Universitat Freiburg)The Lipschitz truncation method

[ Abstract ]

We study the existence of weak solutions to the incompressible $p$-Navier Stokes equations. This system can be used to describe the flow of honey, ketchup, blood, suspensions, polymers, and glaciers. We are interested in small values of $p$, where the method of monotone operators fails. We establish weak solutions by means of the Lipschitz truncation technique, where Sobolev Functions are approximated by Lipschitz functions in a special way. We apply the technique also to electrorheological fluids, where the exponent $p$ depends on the electric field.

[ Reference URL ]We study the existence of weak solutions to the incompressible $p$-Navier Stokes equations. This system can be used to describe the flow of honey, ketchup, blood, suspensions, polymers, and glaciers. We are interested in small values of $p$, where the method of monotone operators fails. We establish weak solutions by means of the Lipschitz truncation technique, where Sobolev Functions are approximated by Lipschitz functions in a special way. We apply the technique also to electrorheological fluids, where the exponent $p$ depends on the electric field.

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

### 2007/06/13

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

Steady Water Waves with Vorticity

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

**Walter Strauss**(Brown University)Steady Water Waves with Vorticity

[ Abstract ]

Consider a classical 2D water wave under the influence of gravity with an arbitrary vorticity function. Assume such a wave is traveling at a constant speed over a flat bed. Then there exist many families of such waves of large amplitude. The proof is based on elliptic PDEs, bifurcation and degree theory. I will also exhibit some recent numerical computations. If the vorticity is sufficiently large, the first stagnation point occurs not at the crest (as with irrotational flows) but on the bed directly below the crest. For variable vorticity the first stagnation point can occur in the interior of the fluid.

[ Reference URL ]Consider a classical 2D water wave under the influence of gravity with an arbitrary vorticity function. Assume such a wave is traveling at a constant speed over a flat bed. Then there exist many families of such waves of large amplitude. The proof is based on elliptic PDEs, bifurcation and degree theory. I will also exhibit some recent numerical computations. If the vorticity is sufficiently large, the first stagnation point occurs not at the crest (as with irrotational flows) but on the bed directly below the crest. For variable vorticity the first stagnation point can occur in the interior of the fluid.

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

### 2007/03/22

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

A semidiscrete scheme for the Perona Malik equation

http://coe.math.sci.hokudai.ac.jp/

**Matteo Novaga**(Hokkaido University / Universita di Pisa)A semidiscrete scheme for the Perona Malik equation

[ Abstract ]

We discuss the convergence of the spatial semidiscrete scheme for the one-dimensional Perona-Malik equation. If the initial datum is 1-Lipschitz out of a finite number of jump points, we haracterize the problem satisfied by the limit solution. In the general case, we construct a solution by a careful inspection of the behaviour of the approximating solutions in a space-time neighbourhood of the jump points.

[ Reference URL ]We discuss the convergence of the spatial semidiscrete scheme for the one-dimensional Perona-Malik equation. If the initial datum is 1-Lipschitz out of a finite number of jump points, we haracterize the problem satisfied by the limit solution. In the general case, we construct a solution by a careful inspection of the behaviour of the approximating solutions in a space-time neighbourhood of the jump points.

http://coe.math.sci.hokudai.ac.jp/

### 2007/01/17

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

Fast Singular Oscillating Limits of Hydrodynamic PDEs: application to 3D Euler, Navier-Stokes and MHD equations

http://coe.math.sci.hokudai.ac.jp/

**Alex Mahalov**(Department of Mathematics and Statistics, Department of Mechanical and Aerospace Engineering, Program in Environmental Fluid Dynamics, Arizona State University )Fast Singular Oscillating Limits of Hydrodynamic PDEs: application to 3D Euler, Navier-Stokes and MHD equations

[ Abstract ]

Methods of harmonic analysis and dispersive properties are applied

to 3d hydrodynamic equations to obtain long-time and/or global existence results to the Cauchy problem for special classes of 3d initial data. Smoothness assumptions for initial data are the same as in local existence theorems. Techniques for fast singular oscillating limits are used and large and/or infinite time regularity is obtained by bootstrapping from global regularity of the limit equations.

The latter gain regularity from 3d nonlinear cancellation of oscillations.

Applications include Euler, Navier-Stokes, Boussinesq and MHD equations, in infinite, periodic and bounded cylindrical domains.

[ Reference URL ]Methods of harmonic analysis and dispersive properties are applied

to 3d hydrodynamic equations to obtain long-time and/or global existence results to the Cauchy problem for special classes of 3d initial data. Smoothness assumptions for initial data are the same as in local existence theorems. Techniques for fast singular oscillating limits are used and large and/or infinite time regularity is obtained by bootstrapping from global regularity of the limit equations.

The latter gain regularity from 3d nonlinear cancellation of oscillations.

Applications include Euler, Navier-Stokes, Boussinesq and MHD equations, in infinite, periodic and bounded cylindrical domains.

http://coe.math.sci.hokudai.ac.jp/

### 2006/11/01

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

A case study in petroleum industry: Mathematical modeling and numerical simulation in spontaneous potential well-logging

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

**Tan Yongji**(School of Mathematical Science, Fudan University )A case study in petroleum industry: Mathematical modeling and numerical simulation in spontaneous potential well-logging

[ Abstract ]

Spontaneous well-logging is an important technique in petroleum exploitation. The potential field is of strong discontinuity on the interface since the spontaneous potential differences. It causes difficulty in mathematical analysis and numerical computing.

New mathematical model and numerical method is designed to overcome the difficulty and good results is obtained.

[ Reference URL ]Spontaneous well-logging is an important technique in petroleum exploitation. The potential field is of strong discontinuity on the interface since the spontaneous potential differences. It causes difficulty in mathematical analysis and numerical computing.

New mathematical model and numerical method is designed to overcome the difficulty and good results is obtained.

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

### 2006/10/30

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

Inverse Problems and Index Formulae for Dirac Operators

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

**Matti Lassas**(Helsinki University of Technology, Institute of Mathematics)Inverse Problems and Index Formulae for Dirac Operators

[ Abstract ]

We consider a selfadjoin Dirac-type operator $D_P$ on a vector bundle $V$ over a compact Riemannian manifold $(M, g)$ with a nonempty boundary.

The operator $D_P$ is specified by a boundary condition $P(u|_{\\partial M})=0$ where $P$ is a projector which may be a non-local, i.e. a pseudodifferential operator.

We assume the existence of a chirality operator which decomposes $L2(M, V)$ into two orthogonal subspaces $X_+ \\oplus X_-$.

In the talk we consider the reconstruction of $(M, g)$, $V$, and $D_P$ from the boundary data on $\\partial M$.

The data used is either the Cauchy data, i.e. the restrictions to $\\partial M \\times R_+$ of the solutions to the hyperbolic Dirac equation, or the boundary spectral data, i.e. the set of the eigenvalues and the boundary values of the eigenfunctions of $D_P$. We obtain formulae for the index and prove uniqueness results for the inverse boundary value problems. We apply the obtained results to the classical Dirac-type operator in $M\\times \\C4$, $M \\subset \\R3$. The presented results have been done in collaboration with Yaroslav Kurylev (Loughborough, UK).

[ Reference URL ]We consider a selfadjoin Dirac-type operator $D_P$ on a vector bundle $V$ over a compact Riemannian manifold $(M, g)$ with a nonempty boundary.

The operator $D_P$ is specified by a boundary condition $P(u|_{\\partial M})=0$ where $P$ is a projector which may be a non-local, i.e. a pseudodifferential operator.

We assume the existence of a chirality operator which decomposes $L2(M, V)$ into two orthogonal subspaces $X_+ \\oplus X_-$.

In the talk we consider the reconstruction of $(M, g)$, $V$, and $D_P$ from the boundary data on $\\partial M$.

The data used is either the Cauchy data, i.e. the restrictions to $\\partial M \\times R_+$ of the solutions to the hyperbolic Dirac equation, or the boundary spectral data, i.e. the set of the eigenvalues and the boundary values of the eigenfunctions of $D_P$. We obtain formulae for the index and prove uniqueness results for the inverse boundary value problems. We apply the obtained results to the classical Dirac-type operator in $M\\times \\C4$, $M \\subset \\R3$. The presented results have been done in collaboration with Yaroslav Kurylev (Loughborough, UK).

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

### 2006/09/27

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

Integral operators in the weighted Lebesgue spaces with a variable exponent

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

**Professor Vakhtang Kokilashvili**(A. Razmadze Mathematical Institute, Georgian Academy of Science)Integral operators in the weighted Lebesgue spaces with a variable exponent

[ Abstract ]

We present a boundedness criteria of the maximal functions and the singular integral operators defined on Carleson curves in the weighted Lebesgue spaces with a variable exponent. There are also given the weighted estimates for the generalized singular integrals raised in the theory of generalized analytic functions of I.N.Vekua and the weighted Sobolev theorems for potentials on Carleson curves. The weight functions may be of power function type as well as oscillating type. The certain version of a Muckenhoupt-type condition for a variable exponent will be considered.

We also expect to treat two-weight problems for the classical integral operators in the variable Lebesgue spaces and to give some applications of the obtained results to the summability problems of Fourier series in two-weighted setting.

[ Reference URL ]We present a boundedness criteria of the maximal functions and the singular integral operators defined on Carleson curves in the weighted Lebesgue spaces with a variable exponent. There are also given the weighted estimates for the generalized singular integrals raised in the theory of generalized analytic functions of I.N.Vekua and the weighted Sobolev theorems for potentials on Carleson curves. The weight functions may be of power function type as well as oscillating type. The certain version of a Muckenhoupt-type condition for a variable exponent will be considered.

We also expect to treat two-weight problems for the classical integral operators in the variable Lebesgue spaces and to give some applications of the obtained results to the summability problems of Fourier series in two-weighted setting.

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

### 2006/07/12

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

Analysis of a crystal growth model

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

**Piotr Rybka**(Warsaw University)Analysis of a crystal growth model

[ Abstract ]

We are concerned with mathematical model of a single crystal growing from vapor. Mathematically this is an exterior, one-phase Stefan-type problem with Gibbs-Thomson law. We restrict our attention to an idealization of a ice crystal, i.e. our evolving free boundary is a circular cylinder. The system under consideration consists of an equation for the motion of the free boundary (the crystal surface) coupled to the quasi-steady approximation of the diffusion equation for the supersaturation of vapor. We present analysis of the system, we show well-posedness and draw the phase portrait, we use here the fact that we need just to variable to describe evolution of a cylinder.

We are mostly concerned with the shape-persitency problem of the

evolution. The problem is, the Gibbs-Thomson relation is in fact a

driven, weighted, mean, singular curvature flow and it is not obvious that the shape of the initial interface will persists throughout the evolution or even for some time. In order to solve this problem we show existence of the region in the phase plane which is a neighborhood of a unique steady state, such that in this region the shape of the cylinder is preserved. However, this set is not invariant with respect to dynamics of the problem.

It is a very interesting question what happens to surface of our crystal at the boundary of the shape-persitency (or shape stability) region. This problem in its full generality is open. However, we give some insight by studying the Gibbs-Thomson relation with a given driving, which inherits properties of the coupling to the diffusion field. We study the resulting driven weighted mean curvature flow for graphs and some special closed Lipschitz curves. We show well-posedness of the problem, but mainly we exhibit the phenomenon of bending flat parts of the curve, which grow ``too big''.

[ Reference URL ]We are concerned with mathematical model of a single crystal growing from vapor. Mathematically this is an exterior, one-phase Stefan-type problem with Gibbs-Thomson law. We restrict our attention to an idealization of a ice crystal, i.e. our evolving free boundary is a circular cylinder. The system under consideration consists of an equation for the motion of the free boundary (the crystal surface) coupled to the quasi-steady approximation of the diffusion equation for the supersaturation of vapor. We present analysis of the system, we show well-posedness and draw the phase portrait, we use here the fact that we need just to variable to describe evolution of a cylinder.

We are mostly concerned with the shape-persitency problem of the

evolution. The problem is, the Gibbs-Thomson relation is in fact a

driven, weighted, mean, singular curvature flow and it is not obvious that the shape of the initial interface will persists throughout the evolution or even for some time. In order to solve this problem we show existence of the region in the phase plane which is a neighborhood of a unique steady state, such that in this region the shape of the cylinder is preserved. However, this set is not invariant with respect to dynamics of the problem.

It is a very interesting question what happens to surface of our crystal at the boundary of the shape-persitency (or shape stability) region. This problem in its full generality is open. However, we give some insight by studying the Gibbs-Thomson relation with a given driving, which inherits properties of the coupling to the diffusion field. We study the resulting driven weighted mean curvature flow for graphs and some special closed Lipschitz curves. We show well-posedness of the problem, but mainly we exhibit the phenomenon of bending flat parts of the curve, which grow ``too big''.

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

### 2006/06/28

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

Uniqueness of Constant Anisotropic Mean Curvature Immersion of Sphere $S^2$ In $\\Bbb E^3$

**Jian Zhai**(Zhejiang University)Uniqueness of Constant Anisotropic Mean Curvature Immersion of Sphere $S^2$ In $\\Bbb E^3$

[ Abstract ]

We prove that the constant anisotropic mean curvature immersion of sphere $S^2$ in $\\Bbb E^3$ is unique, provided that the energy density function $\\gamma$ satisfies some reasonable assumptions.

We prove that the constant anisotropic mean curvature immersion of sphere $S^2$ in $\\Bbb E^3$ is unique, provided that the energy density function $\\gamma$ satisfies some reasonable assumptions.

### 2006/06/14

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

Bounds on eigenvalues of Dirichlet aplacian

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

**Qing-Ming Cheng**(Saga University)Bounds on eigenvalues of Dirichlet aplacian

[ Abstract ]

In this talk, I shall consider the eigenvalue problem of the Dirichlet Laplacian. I shall mention the Weyl asymptotic formula,

Polya conjecture and its partial solution. Furthermore, I shall talk about Bochner-Kac problem.

For universal inequalities for eigenvalues, I shall consider

conjectures of Payne, Polya and Weinberger and their development. In the final, I shall talk the universal bounds for eigenvalues as main part of my talk, which is my recent joint work with rofessor Yang.

[ Reference URL ]In this talk, I shall consider the eigenvalue problem of the Dirichlet Laplacian. I shall mention the Weyl asymptotic formula,

Polya conjecture and its partial solution. Furthermore, I shall talk about Bochner-Kac problem.

For universal inequalities for eigenvalues, I shall consider

conjectures of Payne, Polya and Weinberger and their development. In the final, I shall talk the universal bounds for eigenvalues as main part of my talk, which is my recent joint work with rofessor Yang.

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

### 2006/06/07

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

On phase boundary motion by surface diffusion with triple junction

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/022.html

**高坂 良史**(室蘭工業大学)On phase boundary motion by surface diffusion with triple junction

[ Abstract ]

The phase boundary motion by a geometrical evolution law in a bounded domain is studied in this talk. We consider the surface diffusion flow equation, which has the gradient flow structure with respect to $H^{-1}$-inner product and the area-preserving property. This equation was derived by Mullins to model the motion of interfaces in the case that the motion of interfaces is governed purely by mass diffusion within the interfaces. We study the three-phase problem with triple junction in a bounded domain and analyze the stability of the stationary solutions for this problem.

[ Reference URL ]The phase boundary motion by a geometrical evolution law in a bounded domain is studied in this talk. We consider the surface diffusion flow equation, which has the gradient flow structure with respect to $H^{-1}$-inner product and the area-preserving property. This equation was derived by Mullins to model the motion of interfaces in the case that the motion of interfaces is governed purely by mass diffusion within the interfaces. We study the three-phase problem with triple junction in a bounded domain and analyze the stability of the stationary solutions for this problem.

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/022.html

### 2006/01/18

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

Analyticity of the interface of the classical two-phase Stefan problem

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

**Juergen Saal**(TU Darmstadt)Analyticity of the interface of the classical two-phase Stefan problem

[ Abstract ]

The Stefan problem is a model for phase transitions in liquid-solid systems, as e.g. ice surrounded by water, and accounts for heat diffusion and exchange of latent heat in a homogeneous medium.

The strong formulation of this model corresponds to a free boundary problem involving a parabolic diffusion equation for each phase and a transmission condition prescribed at the interface separating the phases.

We prove that under mild regularity assumptions on the initial data the two-phase classical Stefan problem admits a unique solution that is analytic in space and time.

The result is based on $L_p$ maximal regularity for a linearized problem, which is proved first, and the implicit function theorem.

[ Reference URL ]The Stefan problem is a model for phase transitions in liquid-solid systems, as e.g. ice surrounded by water, and accounts for heat diffusion and exchange of latent heat in a homogeneous medium.

The strong formulation of this model corresponds to a free boundary problem involving a parabolic diffusion equation for each phase and a transmission condition prescribed at the interface separating the phases.

We prove that under mild regularity assumptions on the initial data the two-phase classical Stefan problem admits a unique solution that is analytic in space and time.

The result is based on $L_p$ maximal regularity for a linearized problem, which is proved first, and the implicit function theorem.

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

### 2006/01/11

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

On Fluid Mechanics Formulation of Monge-Kantorovich Mass Transfer Problem

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

Local and Global Exact Controllability of Evolution Equations

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

**伊東 一文**(North Carolina State University) 10:30-11:30On Fluid Mechanics Formulation of Monge-Kantorovich Mass Transfer Problem

[ Abstract ]

The Monge-Kantorovich mass transfer problem is equivalently formulated as an optimal control problem for the mass transport equation. The equivalency of the two problems is establish using the Lax-Hopf formula and the optimal control theory arguments. Also, it is shown that the optimal solution to the equivalent control problem is given in a gradient form in terms of the potential solution to the Monge-Kantorovich problem. It turns out

that the control formulation is a dual formulation of the Kantrovich distance problem via the Hamilton-Jacobi equations.

[ Reference URL ]The Monge-Kantorovich mass transfer problem is equivalently formulated as an optimal control problem for the mass transport equation. The equivalency of the two problems is establish using the Lax-Hopf formula and the optimal control theory arguments. Also, it is shown that the optimal solution to the equivalent control problem is given in a gradient form in terms of the potential solution to the Monge-Kantorovich problem. It turns out

that the control formulation is a dual formulation of the Kantrovich distance problem via the Hamilton-Jacobi equations.

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

**Oleg Yu. Imanuvilov**(Colorado State University) 11:45-12:45Local and Global Exact Controllability of Evolution Equations

[ Abstract ]

We discuss rcent global and local controlability results for the Navier-Stokes system and Bousinesq system. The control is acting on the part of the boundary or locally distributed over subdomain.

[ Reference URL ]We discuss rcent global and local controlability results for the Navier-Stokes system and Bousinesq system. The control is acting on the part of the boundary or locally distributed over subdomain.

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

### 2005/11/09

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

Asymptotic solutions and Aubry sets for Hamilton-Jacobi equations

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

**藤田安啓**(富山大学)Asymptotic solutions and Aubry sets for Hamilton-Jacobi equations

[ Abstract ]

In this talk, we consider the asymptotic behavior of the viscosity solution of the Cauchy problem for the Hamilton-Jacobi equation $u_t + \\alpha x\\cdot Du + H(Du) =f(x)$ in ${\\rm I}\\!{\\rm R}^N \\times (0,\\infty)$, where $\\alpha$ is a positive constant and $H$ is a convex function on ${\\rm I} \\!{\\rm R}^N$. We show that, under some assumptions, $u(x,t) - ct - v(x)$ converges to $0$ locally uniformly in ${\\rm I}\\!{\\rm R}^N$ as $t \\to \\infty$, where $c$ is a constant and $v$ is a viscosity solution of the Hamilton-Jacobi equation $c + \\alpha x\\cdot Dv + H(Dv) = f(x)$ in ${\\rm I}\\!{\\rm R}^N$. A set in ${\\rm I}\\!{\\rm R}^N$, which is called the {\\it Aubry set}, gives a concrete representation of the viscosity solution $v$. We also discuss convergence rates of this asymptotic behavior. This is a joint work with Professors H. Ishii and P. Loreti.

[ Reference URL ]In this talk, we consider the asymptotic behavior of the viscosity solution of the Cauchy problem for the Hamilton-Jacobi equation $u_t + \\alpha x\\cdot Du + H(Du) =f(x)$ in ${\\rm I}\\!{\\rm R}^N \\times (0,\\infty)$, where $\\alpha$ is a positive constant and $H$ is a convex function on ${\\rm I} \\!{\\rm R}^N$. We show that, under some assumptions, $u(x,t) - ct - v(x)$ converges to $0$ locally uniformly in ${\\rm I}\\!{\\rm R}^N$ as $t \\to \\infty$, where $c$ is a constant and $v$ is a viscosity solution of the Hamilton-Jacobi equation $c + \\alpha x\\cdot Dv + H(Dv) = f(x)$ in ${\\rm I}\\!{\\rm R}^N$. A set in ${\\rm I}\\!{\\rm R}^N$, which is called the {\\it Aubry set}, gives a concrete representation of the viscosity solution $v$. We also discuss convergence rates of this asymptotic behavior. This is a joint work with Professors H. Ishii and P. Loreti.

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

### 2005/10/26

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

On Mullins-Sekerka as singular limit of Cahn-Hilliard, some mathematical progress and open problems

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

**利根川吉廣**(北海道大学)On Mullins-Sekerka as singular limit of Cahn-Hilliard, some mathematical progress and open problems

[ Abstract ]

The Cahn-Hilliard equation and its variants have been widely used in materials science community to model coarse graining phenomena in mesoscopic scale. The equation has a parameter corresponding the order of thickness of phase boundaries. When the parameter is close to zero, the phase boundary and the chemical potential field are known to evolve by the so-called Mullins-Sekerka problem. The rigorous justification for the latter statement is known only for short-time so far. I describe some recent progress as well as some difficulties on the long-time case, relateing my recent works and those by M. Roeger and R. Schaetzle.

[ Reference URL ]The Cahn-Hilliard equation and its variants have been widely used in materials science community to model coarse graining phenomena in mesoscopic scale. The equation has a parameter corresponding the order of thickness of phase boundaries. When the parameter is close to zero, the phase boundary and the chemical potential field are known to evolve by the so-called Mullins-Sekerka problem. The rigorous justification for the latter statement is known only for short-time so far. I describe some recent progress as well as some difficulties on the long-time case, relateing my recent works and those by M. Roeger and R. Schaetzle.

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

### 2005/09/28

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

The Navier-Stokes flow in the exterior of a rotating obstacle

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

**Matthias Geissert**(ダルムシュタット工科大学)The Navier-Stokes flow in the exterior of a rotating obstacle

[ Abstract ]

We show the existence of solutions of the Navier-Stokes flow in the exterior of a rotating obstacle. In the first step we transform the Navier-Stokes equations to a problem in a time independent domain. In this talk we present two different change of coordinates to do this. Finally, we discuss the advantages of both approaches and show the local existence and uniqueness of mild and strong $L^p$ solutions.

[ Reference URL ]We show the existence of solutions of the Navier-Stokes flow in the exterior of a rotating obstacle. In the first step we transform the Navier-Stokes equations to a problem in a time independent domain. In this talk we present two different change of coordinates to do this. Finally, we discuss the advantages of both approaches and show the local existence and uniqueness of mild and strong $L^p$ solutions.

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

### 2005/07/20

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

Equivalence between the boundary Harnack principle and the Carleson estimate

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

**相川弘明**(島根大学)Equivalence between the boundary Harnack principle and the Carleson estimate

[ Abstract ]

Both the boundary Harnack principle and the Carleson estimate describe the boundary behavior of positive harmonic functions vanishing on a portion of the boundary. These notions are inextricably related and have been obtained simultaneously for domains with specific geometrical conditions. The main aim of this talk is to show that the boundary Harnack principle and the Carleson estimate are equivalent for arbitrary domains.

[ Reference URL ]Both the boundary Harnack principle and the Carleson estimate describe the boundary behavior of positive harmonic functions vanishing on a portion of the boundary. These notions are inextricably related and have been obtained simultaneously for domains with specific geometrical conditions. The main aim of this talk is to show that the boundary Harnack principle and the Carleson estimate are equivalent for arbitrary domains.

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

### 2005/07/13

10:30-11:30 Room #122 (Graduate School of Math. Sci. Bldg.)

On classical solutions of the compressible Navier-Stokes equation with nonnegative density

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

**Yonggeun Cho**(北海道大学)On classical solutions of the compressible Navier-Stokes equation with nonnegative density

[ Abstract ]

In this talk, we discuss a recent progress on the regularity of solution of compressible Navier-Stokes equations with nonnegative density. The nonnegativity of density cauases a problem in using the usual parabolicity of momentum equations and hence in general makes it hard to gain a regularity of solution. To overcome the difficulty, we develop a natural compatibility condition. Then observing a smoothing effect for positive time, we obtain classical solutions of the compressible Navier-Stokes equations.

[ Reference URL ]In this talk, we discuss a recent progress on the regularity of solution of compressible Navier-Stokes equations with nonnegative density. The nonnegativity of density cauases a problem in using the usual parabolicity of momentum equations and hence in general makes it hard to gain a regularity of solution. To overcome the difficulty, we develop a natural compatibility condition. Then observing a smoothing effect for positive time, we obtain classical solutions of the compressible Navier-Stokes equations.

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

### 2005/06/15

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

Singular and fractional integral operators on function spaces related to Morrey spaces

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

**中井英一**(大阪教育大学)Singular and fractional integral operators on function spaces related to Morrey spaces

[ Abstract ]

It is known that the Hardy-Littlewood maximal operator, singular integral operators and fractional integral operators are bounded on L^p spaces and Morrey spaces. We extend the boundedness to generalized Morrey spaces, Orlicz-Morrey spaces, etc.

[ Reference URL ]It is known that the Hardy-Littlewood maximal operator, singular integral operators and fractional integral operators are bounded on L^p spaces and Morrey spaces. We extend the boundedness to generalized Morrey spaces, Orlicz-Morrey spaces, etc.

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

### 2005/06/08

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

Weighted Hardy spaces on an interval and Jacobi series

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

**宮地晶彦**(東京女子大学)Weighted Hardy spaces on an interval and Jacobi series

[ Abstract ]

For the classical Hardy class consisting of functions holomorphic in the unit disc, the Burkholder-Gundy-Silverstein theorem gives a characterization of the class in terms of certain maximal functions. We give a variant of this theorem related to weighted Hardy spaces on the interval(0,$\\pi$) and generalized holomorphic functions efined through ultraspherical (Gegenbauer) polynomials.

[ Reference URL ]For the classical Hardy class consisting of functions holomorphic in the unit disc, the Burkholder-Gundy-Silverstein theorem gives a characterization of the class in terms of certain maximal functions. We give a variant of this theorem related to weighted Hardy spaces on the interval(0,$\\pi$) and generalized holomorphic functions efined through ultraspherical (Gegenbauer) polynomials.

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

### 2005/06/01

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

Annihilation of wave fronts of a reaction-diffusion equation

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

**Jong-Shenq-Guo**(国立台湾師範大学)Annihilation of wave fronts of a reaction-diffusion equation

[ Abstract ]

We shall present some recent results on the existence and uniqueness of 2-front entire solutions of a reaction-diffusion equation. These entire solutions behave as two opposite wave fronts approaching each other from both sides of the x-axis and then annihilating in a finite time.

[ Reference URL ]We shall present some recent results on the existence and uniqueness of 2-front entire solutions of a reaction-diffusion equation. These entire solutions behave as two opposite wave fronts approaching each other from both sides of the x-axis and then annihilating in a finite time.

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

### 2005/05/25

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

Some regularity results for Stefan equation

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

Nonlinear elliptic systems with general growth

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

**Vincenzo Vespri**(Dipartimento di Matematica Ulisse Dini Viale Morgagni) 10:30-11:30Some regularity results for Stefan equation

[ Abstract ]

We consider the eqation $\\beta (u)_t = A(u)$ where $A$ is an elliptic operator and $\\beta$ is a maximal graph. Under suitable hypothesis on $\\beta$ and $A$ we prove the continuity of local solutions extendind some techniques introduced in the 80's.

[ Reference URL ]We consider the eqation $\\beta (u)_t = A(u)$ where $A$ is an elliptic operator and $\\beta$ is a maximal graph. Under suitable hypothesis on $\\beta$ and $A$ we prove the continuity of local solutions extendind some techniques introduced in the 80's.

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html

**Paolo Marcellini**(Università degli Studi di Firenze) 11:45-12:45Nonlinear elliptic systems with general growth

[ Abstract ]

We prove \\textit{local Lipschitz-continuity} and, as a consequence, $C^{k}$%\\textit{\\ and }$C^{\\infty }$\\textit{\\ regularity} of \\textit{weak} solutions $u$ for a class of \\textit{nonlinear elliptic differential systems} of the form $\\sum_{i=1}^{n}\\frac{\\partial }{\\partial x_{i}}a_{i}^{\\alpha}(Du)=0,\\;\\alpha =1,2\\dots m$. The \\textit{growth conditions} on the dependence of functions $a_{i}^{\\alpha }(\\cdot )$ on the gradient $Du$ are so mild to allow us to embrace growths between the \\textit{linear} and the \\textit{exponential} cases, and they are more general than those known in the literature.

[ Reference URL ]We prove \\textit{local Lipschitz-continuity} and, as a consequence, $C^{k}$%\\textit{\\ and }$C^{\\infty }$\\textit{\\ regularity} of \\textit{weak} solutions $u$ for a class of \\textit{nonlinear elliptic differential systems} of the form $\\sum_{i=1}^{n}\\frac{\\partial }{\\partial x_{i}}a_{i}^{\\alpha}(Du)=0,\\;\\alpha =1,2\\dots m$. The \\textit{growth conditions} on the dependence of functions $a_{i}^{\\alpha }(\\cdot )$ on the gradient $Du$ are so mild to allow us to embrace growths between the \\textit{linear} and the \\textit{exponential} cases, and they are more general than those known in the literature.

http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/index.html