In this article we review some of the recent results on the Cauchy problem and the issue of finite time blow-up for the 3D incompressible Euler equations. We begin with the refinements Kato's local existence of classical solution and the finite time blow-up criterion by Beale, Kato and Majda, using the tools from harmonic analysis. Then, we present the developments of observation of depletion of nonlinearity of the vortex stretching term, first discovered by Constantin and Fefferman for the case of the Navier-Stokes equations. One consequence of this observation is the regularity control in terms of the direction field of the vorticity, and the other one is the regularity control in terms of the reduced number of vorticity components. We develop those idea both for the generalized Navier-Stokes equations and for the Euler equations. We also present some consequences of the dynamics of the deformation tensor. One of them leads to the spectral dynamics approaches to the Euler equations, which provides us with the local in time enstrophy growth and decay estimates, among others. The other one is characterizations of the set of points leading to the possible finite time singularities. Finally we present studies on the various model problems of the 3D Euler equations. More specifically, we introduce the 2D quasi-geostrophic equation and its one dimensional model equation, the 2D Boussinesq system and a modified Euler system. Finally we present recent result on the nonexistence of the self-similar blow-up.

The aim of this article is to establish a uniform pointwise decay estimate for the solution of the mixed problem for the linear wave equation in three space dimensions. We prove such an estimate by using the "cut-off method" as in the work of Shibata and Tsutsumi [23]. As an application, we treat the mixed problem for quadratically quasilinear wave equations exterior to a non-trapping obstacle.

An approach to obtaining dispersive estimates for certain multi-dimensional linear hyperbolic systems will be described. The key result is a local energy decay estimate which provides the intermediate link between the generalized Sobolev inequality and the energy inequality.

We consider $2 \times 2 \to q$ type bilinear restriction estimates for transverse subsets of the sphere, for all $q \gt \frac{n+2}{2}$. Moreover, we give its application to the wave-Schrödinger system.

In this survey note, we consider $2 \times 2$ systems of conservation laws with an umbilic point. At the umbilic point, the characteristic speeds coincide and the Jacobian matrix of the flux functions is diagonalizable. It is shown that if there are no undercompressive shocks with viscous shock profile then any compressive shock has a viscous shock profile and any overcompressive shock has a infinitely many viscous shock profiles.

This paper is concerned with the Helmholtz decomposition of vector fields on bounded domains and exterior domains, and provides the decompositions in Sobolev spaces and Besov spaces of order in certain interval. In particular, the decompositions in homogeneous Besov spaces $\dot{B}_{1,q}^{s}$ and $\dot{B}_{\infty,q}^{s}$ are given.

Consider the 3-dimensional Navier-Stokes fluid filling an infinite space exterior to a rotating body with constant angular velocity. By using the coordinate system attached to the body, the problem is reduced to an equivalent one in the fixed exterior domain. The reduced equation then involves an important drift operator, which is not subordinate to the usual Stokes operator and causes some difficulties. Based on [13], [26] and [12], this survey article addresses steady solutions to the reduced problem.

Non-selfadjoint perturbation of Schrödinger and wave equations is considered. In some cases, we can determine the behavior of solutions (energy decay or scattering) by only its spectral structure.

We shall prove a Gagliardo-Nirenberg type interpolation inequality with Lorentz space and BMO of functions of bounded mean oscillation in the critical case. Moreover, we obtain a Trudinger type inequality and a Brezis-Gallouet-Wainger type inequality as an application of the Gagliardo-Nirenberg type inequality.

We discuss $L^p - L^q$ type estimate of the Stokes semigroup and its application to the Navier-Stokes equations in a perturbed half-space and an aperture domain. Especially, we have the $L^p$-$L^q$ type estimate of the gradient of the Stokes semigroup for any $p$ and $q$ with $1 \le p \le q \lt \infty$, while the same estimate holds only for the exponents $p$ and $q$ with $1 \lt p \le q \le n$ in the exterior domain case, where $n$ denotes the space dimension. And therefore, we can get better results concerning the asymptotic behavior of solutions to the Navier-Stokes equations compared with the exterior domain case.

Our proof of the $L^p$-$L^q$ type estimate of the Stokes semigroup is based on the local energy decay estimate obtained by investigation of the asymptotic behavior of the Stokes resolvent near the origin. The order of asymptotic expansion of the Stokes resolvent near the origin is one half better compared with the exterior domain case, because we have the reflection principle on the boundary in the half-space case unlike the whole space case. And then, such better asymptotics near the boundary is also obtained in a perturbed half-space and an aperture domain by the perturbation argument. This is one of the reason why the result in our case is essentially better compared with the exterior domain case.

This note consists of the several types of bilinear estimates for the solutions of transport equations in one space dimension. As applications, the time local wellposedness results for the systems of these equations with quadratic interaction terms are shown.

We will consider the Cauchy problem for the incompressible homogeneous Navier-Stokes equations in the $d$-dimensional Eucledian space with initial data in uniformly local $L^p\ (L_{uloc}^{p})$ spaces where $p$ is larger than or equal to $d$. For the construction of the local mild solution of this, $L_{uloc}^{p} - L_{uloc}^{q}$ estimates for some convolution operators are important. So we explain these estimates here.

We consider the asymptotic behavior of the solution to the semi-classical Hartree equation in the limit of short wave length, with an initial data which causes focusing at a point. It is known that there exists a critical index which indicates whether or not the asymptotic behavior reflects the effect of the nonlinearity in the neighborhood of the caustic. Investigating time range where the nonlinear effect appears, we improve previous convergence results. In particular, we show that the solution behaves as a free solution before the caustic in some super critical cases.

We consider the two dimensional critical and super-critical dissipative quasi-geostrophic equations. We prove the local existence of a unique regular solution for arbitrary initial data in $B_{2,1}^{2-2\alpha}$ which is corresponding to the scaling invariant space of the equation.

We survey results on wave maps in gravitational theory. In the first part, we review global properties of cosmological spacetimes with two spacelike commutable Killing vector fields in the Einstein-Maxwell-dilaton-axion system, which is arising in the low energy effective superstring theory. It is shown that the dynamical evolution parts of this Einstein-matter equations become a system of wave map equations and global existence and inextendibility for the system are discussed. Asymptotic behavior of constant mean curvature foliations is also analyze. In the second part, we show a global existence theorem of a wave map on black hole background by using Choquet-Bruhat's arguments. Moreover, asymptotic decay property is shown by conformal transformation.

In this paper, we are concerned with the linearized problems of the stationary Navier-Stokes equations for viscous incompressible fluids in two dimensional exterior domain. Obtained is an error estimate between solutions of the Oseen equations and the Stokes equations near the boundary. Our main theorem implies that Stokes's linearization still works well even in a two dimensional exterior domain if we consider the neighbourhoods of the boundary. In order to prove such estimate, we mainly use classical hydrodynamic potential theory.

The Cauchy problem for a coupled system of Schrödinger and improved Boussinesq equations is studied. Local well-posedness is proved in $L^2 (\mathbf{R}^n)$ for $n \le 3$. Global well-posedness is proved in the energy space for $n \le 2$. Under smallness assumption on the Cauchy data, the local result in $L^2$ is proved for $n = 4$.

In present paper an attempt has been made to discuss weak-non linear waves through a two phase mixture of gas and dust particle, when particle volume fraction appears as an additional variable. Asymptotic method is used to find solution up to second order for high frequency harmonic waves.

We consider the scattering problem for the Klein-Gordon equation with cubic convolution nonlinearity. We present the method to prove the existence of the scattering operator on a neighborhood of 0 in the weighted Sobolev space $H^{s,\sigma} = (1- \Delta)^{-s/2}\ \langle x \rangle^{-\sigma}\ L_2 (\mathbb{R}^n)$. The method is based on the complex interpolation method of the weighted Sobolev spaces and the Strichartz estimates for the inhomogeneous Klein-Gordon equation.

We present some results concerning the asymptotic behavior of solutions to the fourth order nonlinear Schrödinger type equation. The nonlinear interaction is the power type one with degree three. We prove the existence of the modified wave operators for this equation.

We consider the Neumann problem for the Stokes equations with non-homogeneous boundary and divergence conditions in a bounded domain. We obtain a global in time $L_p$-$L_q$ maximal regularity theorem with exponential stability. To prove the $L_p$-$L_q$ maximal regularity, we use the Weis operator valued Fourier multiplier theorem.

We consider the initial value problem of the abstract wave equation with dissipation whose coefficient tends to 0 as $t \to \infty$. In the case that the coefficient of the dissipation is a positive constant, Ikehata–Nishihara and Chill–Haraux obtained the decay estimate of the difference between the solution of this equation and the solution of the corresponding abstract heat equation. In the case that $H = L^2 (\mathbb{R}^n)$ and $A$ is the Laplace operator and $b(t)$ is a positive valued monotone $C^2$ function satisfying the sufficient assumption, Wirth obtained the decay estimate of the difference between the solution of the dissipative wave equation and the solution of the corresponding heat equation. The purpose of this paper is to show the decay estimate of the difference between the solution of the abstract wave equation with decaying dissipative term and the solution of the corresponding abstract parabolic equation.

In this paper, we make a brief review on recent results concerning to the global existence and uniqueness of solution for thermo-elastic systems of shape memory alloys. The derivation of these systems is based on the Landau-Ginzburg theory using the shear strain (tensor) as an order parameter.