Nagoya Differintial Equations Seminar

Organizers: Mitsuru Sugimoto, Toshiaki Hishida, Kotaro Tsugawa, Jun Kato, Yutaka Terasawa

We consider some inverse acoustic scattering problems. For the purpose, we derive the factorization method, which is a sampling method for solving certain kinds of inverse problems where the shape and location of unknown objects have to be reconstructed. Here, we introduce new results related to the factorization method.

We study the Cauchy problem of the linear damped wave equation and give sharp $L^p$-$L^q$ estimates of the solution. This is an improvement of the so-called Matsumura estimates. Moreover, as its application, we consider the nonlinear problem with slowly decaying initial data, and determine the critical exponent. In particular, we prove that the small data global existence holds in the critical case if the initial data does not belong to $L^1$. This talk is based on a joint work with Masahiro Ikeda (RIKEN), Mamoru Okamoto (Shinshu University), and Takahisa Inui (Osaka University).

We study the periodic traveling wave solutions of the derivative nonlinear Schrödinger equation (DNLS). It is known that (DNLS) has two types of solitons on the whole line; one has exponential decay and the other has algebraic decay. The latter corresponds to the soliton for the massless case. In the new global results recently obtained by Fukaya, Hayashi and Inui, the properties of two-parameter of the solitons are essentially used in the proof, and especially the soliton for the massless case plays an important role. To investigate further properties of the solitons, we construct exact periodic traveling wave solutions which yield the solitons on the whole line including the massless case in the long-period limit. Moreover, we study the regularity of the convergence of these exact solutions in the long-period limit.

In this talk I will focus on the asymptotic behavior of nonsmooth radial solutions of semilinear Schrödinger equations with a barely supercritical nonlinearity (i.e a nonlinearity that grows faster than the critical power but not faster than a logarithm). It is known that we have scattering of smooth radial solutions of defocusing loglog energy-supercritical Schrödinger equations. I will recall the techniques used to prove this result. Then I will explain how we can use Jensen-type inequalities to prove scattering of nonsmooth radial solutions of defocusing loglog energy-supercritical Schrödinger equations.

In this talk, I will discuss about a critical exponent for semilinear wave equations with time-dependent damping. When the damping is “effective,” it is Fujita exponent which is known to be the one for semilinear heat equations. Recently, by showing a sub-critical blow-up result, I have introduced a new conjecture that it is Strauss exponent which is known to be one for semilinear wave equations as far as the damping is “scattering.” I will also discuss about other nonlinearities and an intermediate situation, namely, the scaling invariant case. All the results in this talk are joint works with Ning-An Lai (Lishui University, China).

In this talk we give a simple introduction of Gubinelli-Imkeller-Perkowski's paracontrolled calculus. (This is basically a survey talk, but at the end we may present our own result a little bit.) This theory solves many formerly ill-defined, but physically important stochastic PDEs and is now competing with Hairer's regularity structure theory. Fortunately, paracontrolled calculus is based on existing theories and therefore not too big. It uses Besov space theory, in particular, Bony's paradifferential calculus. To make our presentation clear to non-experts, we give up generality and focus on the most important example, namely, the 3D dynamic $\Phi^4$-model (also known as the 3D stochastic quantization equation). It is a singular SPDE on $(0, \infty) \times T^3$ and looks like this: $$\partial_t u= \triangle_x u -u^3 +\xi \quad(\mbox{with $u_0$ given}).$$ Here, $\xi$ is a space-time white noise and $T^3$ is the 3 dimensional torus.

We prove existence of weak solutions for a diffuse interface model for the flow of two viscous incompressible Newtonian fluids with different densities in a bounded domain in two and three space dimensions. In contrast to previous works, we study a model with a singular non-local free energy, which controls the fractional Sobolev norm of the volume fraction. We show existence of weak solutions for large times with the aid of an implicit time discretization. This talk is based on a joint work with Helmut Abels (Regensburg).

The primitive equations is considered to be a fundamental model for geophysical flows, e.g. the ocean and the atmosphere. We can formally derive this equations from a scaled Navier-Stokes equations. In this talk we will give mathematically rigorous justification of its derivations in general settings.

In Damage and Fracture Mechanics, the degree of damage in a test specimen is often represented in terms of a phase-field, whose evolution is usually described by a gradient flow of a free energy. On the other hand, due to strongly irreversible characteristics of damaging phenomena, the phase-field (i.e., the degree of damage) is supposed to be monotone in time. Accordingly, evolution laws of such phase-fields are often given in terms of gradient flows with constraints. This talk is concerned with an Allen-Cahn type equation with the positive-part function, which is a typical example of constrained gradient flows and would be a good test bed to develop techniques for dealing with gradient flows with constraints and to investigate influence of such constraints upon properties and behaviors of solutions. In this talk, we shall discuss well-posedness of the Allen-Cahn type equation in (possibly) unbounded domains by developing (re)formulations of the equation as well as energy techniques based on subdifferential calculus. Moreover, we shall also overview qualitative properties and asymptotic behaviors of solutions and some of them appear to be peculiar and different from classical Allen-Cahn equations.

In this lecture, based on joint work with Peter Hintz, I will discuss Kerr-de Sitter black holes, which are rotating black holes in a universe with a positive cosmological constant, i.e. they are explicit solutions (in 3+1 dimensions) of Einstein's equations of general relativity. They are parameterized by their mass and angular momentum. I will first discuss the geometry of these black holes as well as that of the underlying de Sitter space, and then talk about the stability question for these black holes in the initial value formulation. Namely, appropriately interpreted, Einstein's equations can be thought of as quasilinear wave equations, and then the question is if perturbations of the initial data produce solutions which are close to, and indeed asymptotic to, a Kerr-de Sitter black hole, typically with a different mass and angular momentum. In the last part of the talk I will discuss analytic aspects of the stability problem, in particular showing that Kerr-de Sitter black holes with small angular momentum are stable in this sense.

Explicit and partly sharp estimates are given of integrals over the square of Bessel functions with an integrable weight which can be singular at the origin. They are uniform with respect to the order of the Bessel functions and provide explicit bounds for the $L^{2}$ restrictions of Fourier transforms onto spheres in ${\bf R}^{n}$ which are independent of the radius of the sphere. This talk is based on a joint work with Hubert Kalf (Munich Univ.) and Osanobu Yamada (Ritsumeikan University).

We consider the semilinear heat equation with supercritical power nonlinearity, and show the existence of backward self-similar solutions by an ODE shooting method. As an application, we will construct peaking solutions by connecting a backward self-similar solution with a forward self-similar solution. In particular, we show the existence of incomplete blow-up solutions with blow-up profile above the singular steady state. This talk is based on a joint work with Takasi Senba (Fukuoka University).

In this talk, we develop the R-boundedness for the generalized Stokes resolvent problem in an infinite layer, with Neumann boundary condition on both upper and lower boundary. This has not been proved for such a boundary condition, while it has been proved for Neumann and Dirichlet boundary condition on upper and lower boundary, respectively. As an application, we also establish the local well-posedness for the incompressible Navier-Stokes equation in an infinite layer with a free surface for both upper and lower boundaries.

We consider a free boundary problem of compressible and incompressible two-phase flows with surface tensions and phase transitions in bounded regions. The compressible and incompressible fluids are described by the Navier–Stokes–Korteweg equations and the Navier–Stokes equations, respectively. The purpose of this talk is to show that our model is consistent with the second law of thermodynamics and can be considered to be an extension of the Navier-Stokes-Fourier system. In addition, we show a local and global in time unique existence theorem for the free boundary problem under the assumption that the initial data are near the equilibrium.