The L ∞-Stokes semigroup in exterior domains

The Stokes semigroup on a bounded domain is an analytic semigroup on spaces of bounded functions as was recently shown by the authors based on an a priori L ^∞-estimate for solutions to the linear Stokes equations. In this paper, we extend our approach to exterior domains and prove that the Stokes semigroup is uniquely extendable to an analytic semigroup on spaces of bounded functions.     

Viscosity solutions with shocks

A solution of single nonlinear first order equations may develop jump discontinuities even if initial data is smooth. Typical examples include a crude model equation describing some bunching phenomena observed in epitaxial growth of crystals as well as conservation laws where jump discontinuities are called shocks. Conventional theory of viscosity solutions does not apply. We introduce a notion of proper (viscosity) solutions to track whole evolutions for such equations in multi‐dimensional spaces. We establish several versions of comparison principles. We also study the vanishing viscosity method to construct a unique global proper solution at least when the evolution is monotone in time or the initial data is monotone in some sense under additional technical assumptions. In fact, we prove that the graph of approximate solutions converges to that of a proper solution in the Hausdorff distance topology. Such a convergence is also established for conservation laws with monotone data. In particular, local uniform convergence outside shocks is proved. © 2001 John Wiley & Sons, Inc.     

Analyticity of the Stokes semigroup in spaces of bounded functions

The analyticity of the Stokes semigroup with the Dirichlet boundary condition is established in spaces of bounded functions when the domain occupied with fluid is bounded or more generally admissible which admits a special estimate for the Helmholtz decomposition. The proof is based on a blow-up argument. This is the first proof of the analyticity in spaces of bounded functions which was left open more than thirty years.     

A Comparison Principle for Singular Diffusion Equations with Spatially Inhomogeneous Driving Force for Graphs

We introduce the notions of viscosity super- and subsolutions suitable for singular diffusion equations of non-divergence type with a general spatially inhomogeneous driving term. In particular, the viscosity super- and subsolutions support facets and allow a possible facet bending. We prove a comparison principle by a modified doubling variables technique. Finally, we present examples of viscosity solutions. Our results apply to a general crystalline curvature flow with a spatially inhomogeneous driving term for a graph-like curve.     

Large time behavior of the vorticity of two-dimensional viscous flow and its application to vortex formation

We consider the Cauchy problem for the two-dimensional vorticity equation. We show that the solution behaves like a constant multiple of the Gauss kernel having the same total vorticity as time tends to infinity. No particular structure of initial data ₀=(x, 0) is assumed except the restriction that the Reynolds numberR=|₀|dx/v is small, wherev is the kinematic viscosity. Applying a time-dependent scale transformation, we show a stability of Burgers' vortex, which physically implies formation of a concentrated vortex.Partly supported by Grant-in-Aid for Scientific Research No. B60460042, the Japan Ministry of Education, Science and Culture     

An Approach to Rotating Boundary Layers Based on Vector Radon Measures

In this paper we develop a new approach to rotating boundary layers via Fourier transformed finite vector Radon measures. As an application we consider the Ekman boundary layer. By our methods we can derive very explicit bounds for existence intervals and solutions of the linearized and the nonlinear Ekman system. For example, we can prove these bounds to be uniform with respect to the angular velocity of rotation which has proved to be relevant for several aspects (see introduction). Another advantage of our approach is that we obtain well-posedness in classes containing nondecaying vector fields such as almost periodic functions. These outcomes give respect to the nature of boundary layer problems and cannot be obtained by approaches in standard function spaces such as Lebesgue, Bessel-potential, Hölder or Besov spaces.