Nagoya Differintial Equations Seminar

Date: Mondays, 16:00 ~ (90 min. ~ 120 min.)
Place: Rm. 453, Mathematics Bldg, Nagoya University
Organizers: Mitsuru Sugimoto, Toshiaki Hishida, Jun Kato, Yutaka Terasawa

In this talk we discuss the gradient estimates for heat equation in an exterior domain. Our results describe the time decay rates of the derivatives of solutions to the heat equation. As an application, we also consider the fractional Leibniz rule for the Dirichlet or Neumann Laplacian on the exterior domain. This is based on the joint work with Vladimir Georgiev (University of Pisa).

Motivated by the two-dimensional nucleation in crystal growth phenomena, we consider the initial-value problem of the level-set mean curvature flow equation with discontinuous source terms. We discuss uniqueness and existence of viscosity solutions and study the asymptotic shape of solutions. A game-theoretic representation of solutions is also established. Applying this formula, we study the asymptotic speed of solutions. This talk is based on a joint work with K. Misu (Hokkaido University).

In this talk, we consider the local well-posedness for higher order Benjamin-Ono type equations, especially fourth order equations. The proof is based on the energy method with correction terms. Our equations have at most three derivatives in nonlinear terms, so that we need to cancel out all derivative losses by introducing correction terms into the energy. We also employ the Bona-Smith approximation technique in order to show the continuity of the flow and the continuous dependence.

We analyze the asymptotic behavior of solutions to wave equations with the strong damping term. When we impose additional weighted $L^1$ conditions on the initial data, a lower bound for the $L^2$ difference between the solution and the leading term can be obtained.

We consider a Cauchy problem of the Boltzmann equation near the equilibrium without angular cutoff. In the known literature, $L^2$-based Sobolev and Besov spaces are used. In this talk, we will use the Wiener space, which is the set of functions whose Fourier coefficients absolutely converge. We will show the unique existence of a global solution for small data in this space. Also, we will see that the proof can be applied to a Cauchy problem of the Landau equation, which is closely related to the Boltzmann equation. This talk is based on a joint work with Renjun Duan (the Chinese University of Hong Kong), Shuangqian Liu (Jinan University), and Robert M. Strain (University of Pennsylvania).

We consider asymptotic behavior (at the infinity) of slowly decaying positive solutions of quasilinear ordinary differential equations with critical coefficient functions. Equations under consideration are generalizations of those which are satisfied by radially symmetric solutions of pseudo-Laplace equations. When the coefficient functions do not have critical behavior, slowly decaying solutions all have algebraic decays. On the other hand, slowly decaying solutions all have logarithmic decays when the coefficient functions have critical behavior.

We consider a semilinear heat equation without the self-similar structure. By focusing on some quasi-scaling property and its invariant integral, we develop a classification theory for the existence and nonexistence of local in time solutions, and then we discuss the existence of global in time solutions for small initial data. We also study the nonexistence of global in time solutions for nonnegative initial data. These results gives a generalization of the Fujita exponent for a semilinear heat equation with general nonlinearity. This talk is based on a joint work with N. Ioku (Ehime University).

We consider the compressible Navier-Stokes-Korteweg system describing the dynamics of a liquid-vapor mixture with diffuse interphase. The global solutions are established under linear stability conditions in critical Besov spaces. In particular, the sound speed may be greater than or equal to zero. By fully exploiting the parabolic property of the linearized system for all frequencies, we see that there is no loss of derivative usually induced by the pressure for the standard isentropic compressible Navier-Stokes system. This enables us to apply Banach's fixed point theorem to show the existence of global solution. Furthermore, we obtain the optimal decay rates of the global solutions in the $L^2(\mathbb{R}^d)$-framework.

We consider the stochastic quantization of the quantum field model with exponential interactions on the two-dimensional torus, which is called Hoegh-Krohn model. The model has been studied by Dirichlet forms. In this talk, we study the model by singular stochastic differential equations, which is recently developed. By the method, we construct the time-global solution and the invariant probability measure of the stochastic quantization, and see the relation to the process obtained by quasi-regular Dirichlet forms. This is a joint work with Masato Hoshino and Hiroshi Kawabi.

We present some results on the behaviour around the isolated singularity for solutions of nonlinear elliptic inequalities driven by the Laplace operator. We derive optimal conditions that imply either a blow-up or the existence of pointwise bounds for solutions. We obtain that whenever a pointwise bound exists, then an optimal bound is given by the fundamental solution of the Laplace operator. This situation changes in case of systems of inequalities where other types of optimal bounds may occur. The approach relies on integral representation of solutions combined with various nonlinear potential estimates. This talk is based on joint works with S. Taliaferro (Texas A&M University) and I. Verbitsky (Missouri University).

In this talk, we consider the Cauchy problem for semi-linear heat equations with exponential nonlinearity. The main purpose of this talk is to prove the existence of solutions lying on the borderline between global existence and blow-up infinite time. The existence has been shown for semi-linear heat equations with power type nonlinearity. We explain the main strategy to prove the existence. By using the definition of exponential function, we approximate the solution to exponential type equation by that of power type equation. Then we can use directly the knowledge for power type equation.

We discuss blow-up behavior for a semilinear heat equation with Sobolev supercritical power nonlinearity, focusing on the cases where the power is determined by Lepin or Joseph-Lundgren exponents. Based on matched asymptotic expansions and a priori estimates, I will introduce new examples of type II blow-up solutions satisfying various local-in-space estimates. In particular, several kinds of blow-up rates appear. They all differ from the ones in the previous studies. The construction also improves known results on classification of radial blow-up solutions.

The classical problem of a viscous flow past a fixed rigid obstacle is modeled by the steady incompressible Navier-Stokes equations. In this problem, it is known that the asymptotic behavior of the velocity field is described by the linearized problem around the velocity at infinity. The aim of this talk is to present the asymptotic behavior of the vorticity, which is surprisingly not characterized by the previous linearization in two dimensions, but by the linearization around a harmonic flow. If time permits, I will make a link to the Leray problem for large values of the fluxes, where a similar harmonic flow is also the main problem. This is joint work with Peter Wittwer.

In this talk we discuss dynamical stability of multiply covered circles under the surface diffusion flow. To this end we first establish a general form of the isoperimetric inequality for immersed closed curves under rotational symmetry, and then apply it to obtaining a certain class of rotationally symmetric initial curves from which solutions to the surface diffusion flow exist globally-in-time and converge to multiply covered circles. This talk is based on a joint work with Prof. Okabe at the Tohoku University.

In this talk, we consider Dirichlet forms given by two-dimensional space-time quantum fields with interactions of exponential type, called ${\exp}(\Phi)_{2}$-measures (i.e., Hoegh-Krohn's model) in Euclidean quantum field theory, in finite volume. We prove strong uniqueness of the corresponding Dirichlet operator and construct a unique solution of the modified-stochastic quantization equation under suitable conditions on the charge constant and the regularization parameter. This talk is based on a joint work with Sergio Albeverio, Stefan Mihalache and Michael Röckner.