## PDE Real Analysis Seminar

Seminar information archive ～09/14｜Next seminar｜Future seminars 09/15～

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

### 2005/05/18

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

A model of damage evolution in viscous locking material.

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

**剣持信幸**(千葉大学)A model of damage evolution in viscous locking material.

[ Abstract ]

A model problem, describing the damage evolution for instance in some composite materials, is considered. The model is a system of nonlinear PDEs, which are kinetic equations for the displacement and damage quantity in the material. They are both heavily nonlinear parabolic equations, and one of them is of degenerate type. In this talk, the existence of a global in time solution is shown with some key ideas.

[ Reference URL ]A model problem, describing the damage evolution for instance in some composite materials, is considered. The model is a system of nonlinear PDEs, which are kinetic equations for the displacement and damage quantity in the material. They are both heavily nonlinear parabolic equations, and one of them is of degenerate type. In this talk, the existence of a global in time solution is shown with some key ideas.

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

### 2005/04/20

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

Small modifications of quadrature domains around a cusp

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

**酒井 良**(都立大学)Small modifications of quadrature domains around a cusp

[ Abstract ]

A flow which is produced by injection of fluid into the narrow gap between two parallel planes is called a Hele-Shaw flow. We regard the flow as an increasing family of plane domains and discuss the case that the initial domain has a cusp on the boundary. We give sufficient conditions for the cusp to be a laminar-flow point.

[ Reference URL ]A flow which is produced by injection of fluid into the narrow gap between two parallel planes is called a Hele-Shaw flow. We regard the flow as an increasing family of plane domains and discuss the case that the initial domain has a cusp on the boundary. We give sufficient conditions for the cusp to be a laminar-flow point.

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

### 2005/03/23

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

Pseudodifferential Boundary Value Problems with Non-Smooth Coefficients

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

**Helmut Abels**(Max Planck Institute)Pseudodifferential Boundary Value Problems with Non-Smooth Coefficients

[ Abstract ]

We discuss an operator class that models elliptic differential boundary value problems as well as their solution operators and is closed under compositions. It was introduced by Boutet de Monvel in 1971 and provides a powerful tool to calculate with symbols associated to these operators. But the standard calculus and most of its further developments require that the symbols have smooth coefficient in the space and phase variable. We present some results which extend the calculus to symbols which have limited smoothness in the space variable. Such an extension is nescessary to apply the calculus to nonlinear partial differential boundary value problems and free boundary value problems.

[ Reference URL ]We discuss an operator class that models elliptic differential boundary value problems as well as their solution operators and is closed under compositions. It was introduced by Boutet de Monvel in 1971 and provides a powerful tool to calculate with symbols associated to these operators. But the standard calculus and most of its further developments require that the symbols have smooth coefficient in the space and phase variable. We present some results which extend the calculus to symbols which have limited smoothness in the space variable. Such an extension is nescessary to apply the calculus to nonlinear partial differential boundary value problems and free boundary value problems.

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

### 2005/03/02

10:30-11:30 Room #270 (Graduate School of Math. Sci. Bldg.)

The maximum principle in unbounded domains

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

Aubry set and applications

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

**Italo Capuzzo-Dolcetta**(Universita di Roma) 10:30-11:30The maximum principle in unbounded domains

[ Abstract ]

The issue of the talk is the validity of the Weak Maximum Principle for functions u satisfying a second-order partial differential inequality of the form

(*) F(x,u,Du,D^2u) ≧ 0

in a domain A of the n-dimensional euclidean space.

The main result presented in the lecture is that for bounded above upper semicontinuous functions verifying

(*) in the viscosity sense, the inequality u≦ 0 on the boundary ∂A is propagated in the interior of the domain itself, under suitable conditions on F and A.

These conditions include ellipticity of F, a general geometric condition on the (possibly) unbounded domain A and a joint requirement involving the spread of A and the decay of first order terms at infinity.

This result, contained in I.C.D, A.Leoni, A.Vitolo "The Alexandrov-Bakelman-Pucci weak Maximum Principle for fully nonlinear equations in unbounded domains", to appear in Comm.in PDE's, extends previous results due to X.Cabré and L.Caffarelli-X.Cabré.

In the second part of the talk we present different versions of Weak Maximum Principle, namely for solutions growing exponentially fast of (*) in narrow domains and for solutions of

(**) F(x,u,Du,D^2u) + c(x)u ≧ 0

(c changing sign) in domains of small measure.

[ Reference URL ]The issue of the talk is the validity of the Weak Maximum Principle for functions u satisfying a second-order partial differential inequality of the form

(*) F(x,u,Du,D^2u) ≧ 0

in a domain A of the n-dimensional euclidean space.

The main result presented in the lecture is that for bounded above upper semicontinuous functions verifying

(*) in the viscosity sense, the inequality u≦ 0 on the boundary ∂A is propagated in the interior of the domain itself, under suitable conditions on F and A.

These conditions include ellipticity of F, a general geometric condition on the (possibly) unbounded domain A and a joint requirement involving the spread of A and the decay of first order terms at infinity.

This result, contained in I.C.D, A.Leoni, A.Vitolo "The Alexandrov-Bakelman-Pucci weak Maximum Principle for fully nonlinear equations in unbounded domains", to appear in Comm.in PDE's, extends previous results due to X.Cabré and L.Caffarelli-X.Cabré.

In the second part of the talk we present different versions of Weak Maximum Principle, namely for solutions growing exponentially fast of (*) in narrow domains and for solutions of

(**) F(x,u,Du,D^2u) + c(x)u ≧ 0

(c changing sign) in domains of small measure.

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

**Antonio Siconolfi**(Universita di Roma) 11:45-12:45Aubry set and applications

[ Abstract ]

For given Hamiltonian H(x, p) continuous and quasiconvex in the second argument, defined in Rn × Rn or on the cotangent bundle of a compact boundaryless manifold, we consider the equation

H= c

with c critical value, i.e. for which the equation admits locally Lipschitzcontinuous a.e. subsolutions, but not strict subsolutions. We show that there is a subset of the state variable space, called Aubry set and denoted by A, where the obstruction to the existence of such subsolutions is concentrated. We give a metric characterization of A, and we discuss its main properties.

They are applied to a projection problem in a Banach space, to the study of the largetime behaviour of subsolutions to a timedependent HamiltonJacobi equation, and to construct a Lyapunov function for a perturbed dynamics, under suitable stability assumptions.

[ Reference URL ]For given Hamiltonian H(x, p) continuous and quasiconvex in the second argument, defined in Rn × Rn or on the cotangent bundle of a compact boundaryless manifold, we consider the equation

H= c

with c critical value, i.e. for which the equation admits locally Lipschitzcontinuous a.e. subsolutions, but not strict subsolutions. We show that there is a subset of the state variable space, called Aubry set and denoted by A, where the obstruction to the existence of such subsolutions is concentrated. We give a metric characterization of A, and we discuss its main properties.

They are applied to a projection problem in a Banach space, to the study of the largetime behaviour of subsolutions to a timedependent HamiltonJacobi equation, and to construct a Lyapunov function for a perturbed dynamics, under suitable stability assumptions.

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

### 2005/01/26

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

L^p-Theory of the Navier-Stokes flow past rotating or moving obstacles

**Matthias Hieber**(ダルムシュタット工科大学)L^p-Theory of the Navier-Stokes flow past rotating or moving obstacles

[ Abstract ]

In this talk we consider the equation of Navier-Stokes in the exterior of a rotating or moving domain. Using techniques from the analysis of Ornstein-Uhlenbeck operators it is shown that, after rewriting the problem on a fixed domain $\\Omega$, the solution of the linearized equation is governed by a $C_0$-semigroup on $L^p_\\sigma(\\Omega)$ for $1

In this talk we consider the equation of Navier-Stokes in the exterior of a rotating or moving domain. Using techniques from the analysis of Ornstein-Uhlenbeck operators it is shown that, after rewriting the problem on a fixed domain $\\Omega$, the solution of the linearized equation is governed by a $C_0$-semigroup on $L^p_\\sigma(\\Omega)$ for $1

[ Reference URL ]

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

### 2004/12/15

10:30-12:45 Room #122 (Graduate School of Math. Sci. Bldg.)

Hamilton-Jacobi-Bellman equations for optimal control of stochastic Navier-Stokes equations.

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

A GEOMETRICAL APPROACH TO FRONT PROPAGATION PROBLEMS IN BOUNDED DOMAINS WITH NEUMANN-TYPE BOUNDARY AND APPLICATIONS

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

**Andrzej Swiech**(ジョージア工科大学) 10:30-11:30Hamilton-Jacobi-Bellman equations for optimal control of stochastic Navier-Stokes equations.

[ Abstract ]

We consider a parameterized family of continuous functions, which containsas its members Bourbai's and Perkins's nowhere differentiable functions as well as the Cantor-Lebesgue singular functions.

[ Reference URL ]We consider a parameterized family of continuous functions, which containsas its members Bourbai's and Perkins's nowhere differentiable functions as well as the Cantor-Lebesgue singular functions.

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

**Francesca Da Lio**(Dipartimento di Matematica P. e A.Universit di Padova researcher) 11:45-12:45A GEOMETRICAL APPROACH TO FRONT PROPAGATION PROBLEMS IN BOUNDED DOMAINS WITH NEUMANN-TYPE BOUNDARY AND APPLICATIONS

[ Abstract ]

We talk about a new definition of weak solution for the global-in-time motion of a front in bounded domains with normal velocity depending not only on its curvature but also on the measure of the set it encloses and with a contact angle boundary condition. We apply this definition to study the asymptotic behaviour of the solutions of some local and nonlocal reaction-diffusion equations.

[ Reference URL ]We talk about a new definition of weak solution for the global-in-time motion of a front in bounded domains with normal velocity depending not only on its curvature but also on the measure of the set it encloses and with a contact angle boundary condition. We apply this definition to study the asymptotic behaviour of the solutions of some local and nonlocal reaction-diffusion equations.

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

### 2004/12/01

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

A remark on continuous, nowhere differentiable functions

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

**岡本 久**(京都大学)A remark on continuous, nowhere differentiable functions

[ Abstract ]

We consider a parameterized family of continuous functions, which containsas its members Bourbai's and Perkins's nowhere differentiable functions as well as the Cantor-Lebesgue singular functions.

[ Reference URL ]We consider a parameterized family of continuous functions, which containsas its members Bourbai's and Perkins's nowhere differentiable functions as well as the Cantor-Lebesgue singular functions.

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

### 2004/10/20

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

Some recent results on the Navier-Stokes equations

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

**Hermann Sohr**(University of Paderborn)Some recent results on the Navier-Stokes equations

[ Abstract ]

The aim of this talk is to explain some new results in particular on local regularity properties of Hopf type weak solutions to the Navier-Stokes equations for arbitrary domains. Further we explain a new existence result for nonhomogeneous data and a result for global regular solutions with "slightly" modified forces.

[ Reference URL ]The aim of this talk is to explain some new results in particular on local regularity properties of Hopf type weak solutions to the Navier-Stokes equations for arbitrary domains. Further we explain a new existence result for nonhomogeneous data and a result for global regular solutions with "slightly" modified forces.

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

### 2004/10/13

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

Existence, uniqueness, and continuous dependence of entropy solutions to hyperbolic systems

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

**Philippe G. LeFloch**(University of Paris 6)Existence, uniqueness, and continuous dependence of entropy solutions to hyperbolic systems

[ Abstract ]

I will review the well-posedness theory of nonlinear hyperbolic systems, in conservative or in non-conservative form, by focusing attention on the existence and properties of entropy solutions with sufficiently small total variation.

New results and perspectives on the following issues will be discussed: Glimm's existence theorem,

Bressan-LeFloch's uniqueness theorem,and the L1 continuous dependence property (established by Bressan, LeFloch, Liu, and Yang).

[ Reference URL ]I will review the well-posedness theory of nonlinear hyperbolic systems, in conservative or in non-conservative form, by focusing attention on the existence and properties of entropy solutions with sufficiently small total variation.

New results and perspectives on the following issues will be discussed: Glimm's existence theorem,

Bressan-LeFloch's uniqueness theorem,and the L1 continuous dependence property (established by Bressan, LeFloch, Liu, and Yang).

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

### 2004/09/29

10:30-11:30 Room #117 (Graduate School of Math. Sci. Bldg.)

Global Regularity of the 3D Navier-Stokes with Uniformly Large Initial Vorticity

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

**Alex Mahalov**(Arizona State University)Global Regularity of the 3D Navier-Stokes with Uniformly Large Initial Vorticity

[ Abstract ]

We prove existence on infinite time intervals of regular solutions to the 3D Navier-Stokes Equations for fully three-dimensional initial data characterized by uniformly large vorticity with periodic boundary conditions and in bounded cylindrical domains; smoothness assumptions for initial data are the same as in local existence theorems. There are no conditional assumptions on the properties of solutions at later times, nor are the global solutions close to any 2D manifold.

The global existence is proven using techniques of fast singular oscillating limits and the Littlewood-Paley dyadic decomposition. The approach is based on the computation of singular limits of rapidly oscillating operators and cancellation of oscillations in the nonlinear interactions for the vorticity field. With nonlinear averaging methods in the context of almost periodic functions, we obtain fully 3D limit resonant Navier-Stokes equations. Using Lemmas on restricted convolutions, we establish the global regularity of the latter without any restriction on the size of 3D initial data.

With strong convergence theorems, we bootstrap this into the global regularity of the 3D Navier-Stokes Equations with uniformly large initial vorticity.

[ Reference URL ]We prove existence on infinite time intervals of regular solutions to the 3D Navier-Stokes Equations for fully three-dimensional initial data characterized by uniformly large vorticity with periodic boundary conditions and in bounded cylindrical domains; smoothness assumptions for initial data are the same as in local existence theorems. There are no conditional assumptions on the properties of solutions at later times, nor are the global solutions close to any 2D manifold.

The global existence is proven using techniques of fast singular oscillating limits and the Littlewood-Paley dyadic decomposition. The approach is based on the computation of singular limits of rapidly oscillating operators and cancellation of oscillations in the nonlinear interactions for the vorticity field. With nonlinear averaging methods in the context of almost periodic functions, we obtain fully 3D limit resonant Navier-Stokes equations. Using Lemmas on restricted convolutions, we establish the global regularity of the latter without any restriction on the size of 3D initial data.

With strong convergence theorems, we bootstrap this into the global regularity of the 3D Navier-Stokes Equations with uniformly large initial vorticity.

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