Yoshikazu Giga

Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations

where Vu is the (spatial) gradiant of u. Here VM/|VW| is a unit normal to a level surface of u, so div(Vw/|Vw|) is its mean curvature unless Vu vanishes on the surface. Since ut/\Vu\ is a normal velocity of the level surface, (1.3) implies that a level surface of solution u of (1.3) moves by its mean curvature unless Vu vanishes on the surface. We thus call (1.3) the mean curvature flow equation in this paper. The motion of a closed (hyper)surface in R by its mean curvature has been studied by many authors [1], [3], [4], [8], [10], [12], [14], [15]. Such a motion is also important in the singular perturbation theory related to

Abstract Lp estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains

Abstract We apply an abstract perturbation theorem to derive global in time Lq estimates for the Cauchy problem and Lq − Ls estimates for the nonstationary Stokes equations in exterior domains. This will be applied to obtain various new global in time estimates for weak solutions of the Navier-Stokes equations.

Nonlinear Partial Differential Equations

Nonlinear partial differential equations (PDE’s) play a central role in the modelling of a great number of phenomena, ranging from Theoretical Physics, Astrophysics and Chemistry to Economy, Medicine and Population Dynamics. Among the phenomena encountered, the diffusion processes play a fundamental role. In the last 25 years a great deal of work has been devoted to semi-linear equations. In this type of equations, the interaction between a linear partial differential operator and the superlinear reaction term (source or absorption) can be understood, at least in part, thanks to the linear theory. One of the main observations, valid in the most interesting cases is the existence of critical exponents, for example the Fujita exponents for nonlinear heat equation, the Pohozaev exponent, the Sobolev exponent. In general, the first results (blow-up, global estimates, decay estimate), were proved up to a critical exponent by more or less easy applications of linear energy estimates, linked to ODE techniques. Then the study of what happens if the exponent is critical or even supercritical involves a very delicate analysis, often based upon very sharp linear estimates or even, in some cases a completely new approach. The main areas of research in the current proposal include the description of singular phenomenon: blow-up, singularities, problems with singular measure data in a large class of reaction diffusion equations, with quasi-linear or fully nonlinear diffusion and strong reaction.

Navier-stokes flow in r3 with measures as initial vorticity and morrey spaces

On considere les equations de Navier-Stokes dans R 3 quand des profils initiaux de tourbillons sont des annneaux et des filaments

A bound for global solutions of semilinear heat equations

We show that global positive solutions of the initial-boundary value problem forut=Δu+up are bounded, provided thatp>1 is subcritical. Our bound depends only on sup norm of the initial data and is useful to classify initial data by the asymptotic behavior of the solutions as time tend to infinity.

Mean curvature flow through singularities for surfaces of rotation

In this paper, we study generalized “viscosity” solutions of the mean curvature evolution which were introduced by Chen, Giga, and Goto and by Evans and Spruck. We devote much of our attention to solutions whose initial value is a compact, smooth, rotationally symmetric hypersurface given by rotating a graph around an axis. Our main result is the regularity of the solution except at isolated points in spacetime and estimates on the number of such points.

Global Existence of Two-Dimensional Navier—Stokes Flow with Nondecaying Initial Velocity

Abstract. A global-in-time unique smooth solution is constructed for the Cauchy problem of the Navier—Stokes equations in the plane when initial velocity field is merely bounded not necessary square-integrable. The proof is based on a uniform bound for the vorticity which is only valid for planar flows. The uniform bound for the vorticity yields a coarse globally-in-time a priori estimate for the maximum norm of the velocity which is enough to extend a local solution. A global existence of solution for a q-th integrable initial velocity field is also established when

On lower semicontinuity of a defect energy obtained by a singular limit of the Ginzburg–Landau type energy for gradient fields

q > 2

EQUATIONS WITH SINGULAR DIFFUSIVITY

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On estimates in Hardy spaces for the Stokes flow in a half space

A defect energy J β , which measures jump discontinuities of a unit length gradient field, is studied. The number β indicates the power of the jumps of the gradient fields that appear in the density of J β . It is shown that J β for β = 3 is lower semicontinuous (on the space of unit gradient fields belonging to BV) in L 1 -convergence of gradient fields. A similar result holds for the modified energy , which measures only a particular type of defect. The result turns out to be very subtle, since with β > 3 is not lower semicontinuous, as is shown in this paper. The key idea behind semicontinuity is a duality representation for J 3 and . The duality representation is also important for obtaining a lower bound by using J 3 + for the relaxation limit of the Ginzburg–Landau type energy for gradient fields. The lower bound obtained here agrees with the conjectured value of the relaxation limit.