关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

In this chapter we introduce some basic notation and definitions. We discuss a few general results on interpolation spaces. The most important one is the Aronszajn-Gagliardo theorem.

General Properties of Interpolation Spaces

Preface 1. Fourier series and Fourier transforms 2. Distributions 3. Elliptic equations (fundamental theory) 4. Initial value problems (Cauchy problems) 5. Evolution equations 6. Hyperbolic equations 7. Semi-linear hyperbolic equations 8. Green's functions and spectra Supplementary remarks Guide to the literature Bibliography Symbols Index.

The theory of partial differential equations

In this paper, we study a hydrodynamic system describing fluids with viscoelastic properties After a brief examination of the relations between several models, we shall concentrate on a few analytical issues concerning them In particular, we establish local existence and global existence (with small initial data) of classical solutions for an Oldroyd system without an artificially postulated damping mechanism

On hydrodynamics of viscoelastic fluids

https://ir.nuk.edu.tw/bitstream/310360000Q/10885/2/main-fibering.pdf

The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions

We consider the equations of motion of compressible viscous and heat-conductive gases in an exterior domain in ℝ3. We give the L_q−L_p estimates for solutions to the linearized equations and show an optimal decay estimate for solutions to the nonlinear problem.

Decay estimates of solutions for the equations of motion of compressible viscous and heat-conductive gases in an exterior domain in ℝ3

We investigate the Lp–Lq estimate of solutions to the Cauchy problem of linear viscoelastic equation, especially, the diffusion wave property of linear viscoelastic equation like the Navier–Stokes equation in the compressible fluid case, which was studied by D. Hoff and K. Zumbrum and Tai-P. Liu and W. Wang. Copyright © 2000 John Wiley & Sons, Ltd.

On the rate of decay of solutions to linear viscoelastic equation

The global existence of smooth solutions to the equations of nonlinear hyperbolic system of 2nd order with third order viscosity is shown for small and smooth initial data in a bounded domain ofn-dimensional Euclidean space with smooth boundary. Dirichlet boundary condition is studied and the asymptotic behaviour of exponential decay type of solutions ast tending to ∞ is described. Time periodic solutions are also studied. As an application of our main theorem, nonlinear viscoelasticity, strongly damped nonlinear wave equation and acoustic wave equation in viscous conducting fluid are treated.

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Global existence and exponential stability of small solutions to nonlinear viscoelasticity

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.

Yoshihiro Shibata

Papers

Decay estimates of solutions for the equations of motion of compressible viscous and heat-conductive gases in an exterior domain in ℝ3

On the rate of decay of solutions to linear viscoelastic equation

Global existence and exponential stability of small solutions to nonlinear viscoelasticity

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

On the Oseen equation in the three dimensional exterior domains