Senjo Shimizu

A Decay Property of the Fourier Transform and its Application to the Stokes Problem

Abstract. We prove an optimal relationship between the regularity of a function and the asymptotic behavior of its Fourier transform. As an application of this result we show Lp-estimates for the Stokes semigroup in

On the Lp-Lq maximal regularity of the Neumann problem for the Stokes equations in a bounded domain

\Bbb R^n

On a resolvent estimate of the interface problem for the Stokes system in a bounded domain

and

Report on a local in time solvability of free surface problems for the Navier–Stokes equations with surface tension

{\Bbb R}_+^n

The Limiting Absorption Principle for Elastic Wave Propagation Problems in Perturbed Stratified Media ℝ3

when

Qualitative Behaviour of Incompressible Two-Phase Flows with Phase Transitions: The Case of Non-Equal Densities

1\leqq p\leqq \infty

Local Solvability of Free Boundary Problems for the Two-phase Navier-Stokes Equations with Surface Tension in the Whole Space

.

On Incompressible Two-Phase Flows with Phase Transitions and Variable Surface Tension

Abstract In this paper, we prove the Lp-Lq maximal regularity of solutions to the Neumann problem for the Stokes equations with non-homogeneous boundary condition and divergence condition in a bounded domain. The result was first stated by Solonnikov V. A. Solonnikov, On the transient motion of an isolated volume of viscous incompressible fluid, Math. USSR Izvest. 31 (1988), 381–405., but he assumed that p = q > 3 and considered only the finite time interval case. In this paper, we consider not only the case: 1 < p, q < ∞ but also the infinite time interval case. Especially, we obtain the Lp-Lq maximal regularity theorem with exponential stability on the infinite time interval. Our method can be applied to any initial boundary value problem for the equation of parabolic type with suitable boundary condition which generates an analytic semigroup, for example the Stokes equation with non-slip, slip or Robin boundary conditions.

Global Existence of Solutions to 2-D Navier–Stokes Flow with Non-decaying Initial Data in Exterior Domains

Abstract Obtained is the L p estimate of solutions to the resolvent problem for the Stokes system with interface condition in a bounded domain in R n . It is the first step to consider the free boundary value problem.

Modeling of Two-Phase Flows With and Without Phase Transitions

We consider the free boundary problem of the Navier–Stokes equation with surface tension. Our initial domain Ω is one of a bounded domain, an exterior domain, a perturbed half-space or a perturbed layer in ℝ n (n ≥ 2). We report a local in time unique existence theorem in the space with some T > 0, 2 < p < ∞ and n < q < ∞ for any initial data which satisfy compatibility condition. Our theorem can be proved by the standard fixed point argument based on the L p –L q maximal regularity theorem for the corresponding linearized equations. Our results cover the cases of a drop problem and an ocean problem that were studied by Solonnikov (Solvability of the evolution problem for an isolated mass of a viscous incompressible capillary liquid, Zap. Nauchn. Sem. (LOMI) 140 (1984) pp. 179–186 (in Russian) (English transl.: J. Soviet Math. 32 (1986), pp. 223–238)), Solonnikov (Unsteady motion of a finite mass of fluid, bounded by a free surface, Zap. Nauchn. Sem. (LOMI) 152 (1986), pp. 137–157 (in Russian) (English transl.: J. Soviet Math. 40 (1988), pp. 672–686)), Solonnikov (On nonstationary motion of a finite isolated mass of self-gravitating fluid, Algebra Anal. 1 (1989), pp. 207–249 (in Russian) (English transl.: Leningrad Math. J. 1 (1990), pp. 227–276)), Solonnikov (Solvability of the problem of evolution of a viscous incompressible fluid bounded by a free surface on a finite time interval, Algebra Anal. 3 (1991), pp. 222–257 (in Russian) (English transl.: St. Petersburg Math. J. 3 (1992) 189–220)), Beale (Large time regularity of viscous surface waves, Arch. Rat. Mech. Anal. 84 (1984), pp. 307–352) and Tani (Small-time existence for the three-dimensional incompressible Navier–Stokes equations with a free surface, Arch. Rat. Mech. Anal. 133 (1996), pp. 299–331).