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

We show the critical Sobolev inequalities in the Besov spaces with the logarithmic form such as Brezis-Gallouet-Wainger and Beale-Kato-Majda. As an application of those inequalities, the regularity problem under the critical condition to the Navier-Stokes equations, the Euler equations in 
${mathbb R}^n$
 and the gradient flow to the harmonic map to the sphere are discussed. Namely the Serrin-Ohyama type regularity criteria are improved in the terms of the Besov spaces.

The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations

We obtain a family of nonlinear maximum principles for linear dissipative nonlocal operators, that are general, robust, and versatile. We use these nonlinear bounds to provide transparent proofs of global regularity for critical SQG and critical d-dimensional Burgers equations. In addition we give applications of the nonlinear maximum principle to the global regularity of a slightly dissipative anti-symmetric perturbation of 2D incompressible Euler equations and generalized fractional dissipative 2D Boussinesq equations.

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Nonlinear maximum principles for dissipative linear nonlocal operators and applications

The equations of motion of an incompressible, Newtonian fluid — usually called Navier-Stokes equations — have been written almost one hundred eighty years ago. In fact, they were proposed in 1822 by the French engineer C.M.L.H. Navier upon the basis of a suitable molecular model. It is interesting to observe, however, that the law of interaction between the molecules postulated by Navier were shortly recognized to be totally inconsistent from the physical point of view for several materials and, in particular, for liquids. It was only more than twenty years later that the same equations were rederived by the twenty-six year old G. H. Stokes 1845 in a quite general way, by means of the theory of continua.

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An Introduction to the Navier-Stokes Initial-Boundary Value Problem

We prove that the BMO norm of the velocity and the vorticity controls the blow-up phenomena of smooth solutions to the Navier-Stokes equations Our result is applied to the criterion on uniqueness and regularity of weak solutions in the marginal class

Bilinear estimates in BMO and the Navier-Stokes equations

We shall prove a logarithmic Sobolev inequality by means of the BMO-norm in the critical exponents. As an application, we shall establish a blow-up criterion of solutions to the Euler equations.

Limiting case of the Sobolev inequality in BMO, with application to the Euler equations

We prove a local existence theorem for the Navier-Stokes equations with the initial data in B0∞,∞ containing functions which do not decay at infinity. Then we establish an extension criterion on our local solutions in terms of the vorticity in the homogeneous Besov space B·0∞,∞.

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Navier-stokes equations in the besov space near l∞ and bmo

We shall prove the global existence theorem for the 2 dimensional Euler equations in Open image in new window with the initial vorticity in bmo containing functions which do not decay at infinity and have logarithmic singularities.

Yasushi Taniuchi

Papers

The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations

Bilinear estimates in BMO and the Navier-Stokes equations

Limiting case of the Sobolev inequality in BMO, with application to the Euler equations

Navier-stokes equations in the besov space near l∞ and bmo

Uniformly Local L p Estimate for 2-D Vorticity Equation and Its Application to Euler Equations with Initial Vorticity in bmo