Tosio Kato

Remarks on the breakdown of smooth solutions for the 3-D Euler equations

The authors prove that the maximum norm of the vorticity controls the breakdown of smooth solutions of the 3-D Euler equations. In other words, if a solution of the Euler equations is initially smooth and loses its regularity at some later time, then the maximum vorticity necessarily grows without bound as the critical time approaches; equivalently, if the vorticity remains bounded, a smooth solution persists.

Nonstationary flows of viscous and ideal fluids in R3

Abstract The Cauchy problem for the nonstationary Navier-Stokes equation in R3 is considered. It is shown that the solution exists on a time interval independent of the viscosity v and tends as v → 0 to the solution of the limiting equation, provided that the initial velocity field and the external force field are sufficiently smooth and small at infinity (in the sense that they belong to the Sobolev space over R3 of order 3). Such a result is not altogether new but the proof, which depends on the theory of nonlinear evolution equation in Hilbert space, is simpler than the existing one due to Swann.

Wave Operators and Similarity for Some Non-selfadjoint Operators

The purpose of the present paper is to develop a new method for establishing the similarity of a perturbed operator T(ϰ), formally given by T + ϰV, to the unperturbed operator T. It is basically a “small perturbation” theory, since the parameter ϰ is assumed to be sufficiently small. Otherwise the setting of the problem is rather general; T or V need not be symmetric or bounded, although they are assumed to act in a separable Hilbert space r and ϰ need not be real. The basic assumptions are that the spectrum of T is a subset of the real axis, that V can be written formally as V = B* A, where A is T-smooth and B is T*-smooth (see Definition 1.2), and that A(T − ζ)−1 B* is uniformly bounded for nonreal ζ. Here A and B are (in general unbounded) operators from r to another Hilbert space r′ (r′ = r is permitted). Strictly speaking, we are dealing with a certain extension T(ϰ) of T + ϰB* A, which is uniquely determined by T, A, and B; T(ϰ) = T + ϰB* A is true if A and B are bounded1).

On the Korteweg-de Vries equation

Existence, uniqueness, and continuous dependence on the initial data are proved for the local (in time) solution of the (generalized) Korteweg-de Vries equation on the real line, with the initial function ϕ in the Sobolev space of order s>3/2 and the solution u(t) staying in the same space, s=∞ being included For the proper KdV equation, existence of global solutions follows if s≥2. The proof is based on the theory of abstract quasilinear evolution equations developed elsewhere.

On nonlinear Schrödinger equations, II.HS-solutions and unconditional well-posedness

AbstractWe consider the nonlinear Schrödinger equation (NLS) (see below) with a general “potential”F(u), for which there are in general no conservation laws. The main assumption onF(u) is a growth rateO(|u|k) for large |u|, in addition to some smoothness depending on the problem considered. A uniqueness theorem is proved with minimal smoothness assumption onF andu, which is useful in eliminating the “auxiliary conditions” in many cases. A new local existence theorem forHS-solutions is proved using an auxiliary space of Lebesgue type (rather than Besov type); here the main assumption is thatk≤1+4/(m−2s) ifsm/2). Moreover, a general existence theorem is proved for globalHS-solutions with small initial data, under the main additional condition thatF(u)=O(|u|1+4/m) for small |u|; in particularF(u) need not be (quasi-) homogeneous or in the critical case. The results are valid for alls≥0 ifm≤6; there are some restrictions ifm≥7 and ifF(u) isnot a polynomial inu and

Strong solutions of the Navier-Stokes equation in Morrey spaces

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Remarks on Zero Viscosity Limit for Nonstationary Navier- Stokes Flows with Boundary

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Nonlinear Evolution Equations and Analyticity. I

Hilbert space is a special case of Banach space, but it deserves separate consideration because of its importance in applications. In Hilbert spaces the general results deduced in previous chapters are strengthened and, at the same time, new problems arise.