144 research outputs found
Global Strong Well-posedness of the Three Dimensional Primitive equations in -spaces
In this article, an -approach to the primitive equations is developed.
In particular, it is shown that the three dimensional primitive equations admit
a unique, global strong solution for all initial data provided . To this end, the hydrostatic
Stokes operator defined on , the subspace of associated with
the hydrostatic Helmholtz projection, is introduced and investigated. Choosing
large, one obtains global well-posedness of the primitive equations for
strong solutions for initial data having less differentiability properties
than , hereby generalizing in particular a result by Cao and Titi (Ann.
Math. 166 (2007), pp. 245-267) to the case of non-smooth initial data.Comment: 26 page
Dynamics of the Ericksen-Leslie Equations with General Leslie Stress I: The Incompressible Isotropic Case
The Ericksen-Leslie model for nematic liquid crystals in a bounded domain
with general Leslie and isotropic Ericksen stress is studied in the case of a
non-isothermal and incompressible fluid. This system is shown to be locally
well-posed in the -setting, and a dynamic theory is developed. The
equilibria are identified and shown to be normally stable. In particular, a
local solution extends to a unique, global strong solution provided the initial
data are close to an equilibrium or the solution is eventually bounded in the
topology of the natural state manifold. In this case, the solution converges
exponentially to an equilibrium, in the topology of the state manifold. The
above results are proven {\em without} any structural assumptions on the Leslie
coefficients and in particular {\em without} assuming Parodi's relation
Stokes Resolvent Estimates in Spaces of Bounded Functions
The Stokes equation on a domain is well understood in
the -setting for a large class of domains including bounded and exterior
domains with smooth boundaries provided . The situation is very
different for the case since in this case the Helmholtz projection
does not act as a bounded operator anymore. Nevertheless it was recently proved
by the first and the second author of this paper by a contradiction argument
that the Stokes operator generates an analytic semigroup on spaces of bounded
functions for a large class of domains. This paper presents a new approach as
well as new a priori -type estimates to the Stokes equation. They
imply in particular that the Stokes operator generates a -analytic
semigroup of angle on , or a non--analytic
semigroup on for a large class of domains. The
approach presented is inspired by the so called Masuda-Stewart technique for
elliptic operators. It is shown furthermore that the method presented applies
also to different type of boundary conditions as, e.g., Robin boundary
conditions.Comment: 22 pages, to appear in Ann. Sci. \'Ec. Norm. Sup\'er. (4
Heat-kernels and maximal Lp − Lq−estimates: the non-autonomous case
International audienceIn this paper, we establish maximal Lp−Lq estimates for non autonomous parabolic equations of the type u′(t) + A(t)u(t) = f(t), u(0) = 0 under suitable conditions on the kernels of the semigroups generated by the operators −A(t), t ∈ [0; T]. We apply this result on semilinear problems of the form u′(t) + A(t)u(t) = f(t; u(t)), u(0) = 0
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