44 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
On a strong solution of the non-stationary Navier-Stokes equations under slip or leak boundary conditions of friction type
Strong solutions of the non-stationary Navier-Stokes equations under
non-linearized slip or leak boundary conditions are investigated. We show that
the problems are formulated by a variational inequality of parabolic type, to
which uniqueness is established. Using Galerkin's method and deriving a priori
estimates, we prove global and local existence for 2D and 3D slip problems
respectively. For leak problems, under no-leak assumption at we prove
local existence in 2D and 3D cases. Compatibility conditions for initial states
play a significant role in the estimates.Comment: 20 page
Penalty method with Crouzeix-Raviart approximation for the Stokes equations under slip boundary condition
The Stokes equations subject to non-homogeneous slip boundary conditions are
considered in a smooth domain . We
propose a finite element scheme based on the nonconforming P1/P0 approximation
(Crouzeix-Raviart approximation) combined with a penalty formulation and with
reduced-order numerical integration in order to address the essential boundary
condition on . Because the
original domain must be approximated by a polygonal (or polyhedral)
domain before applying the finite element method, we need to take
into account the errors owing to the discrepancy , that
is, the issues of domain perturbation. In particular, the approximation of
by makes it non-trivial whether we
have a discrete counterpart of a lifting theorem, i.e., right-continuous
inverse of the normal trace operator ; . In this paper
we indeed prove such a discrete lifting theorem, taking advantage of the
nonconforming approximation, and consequently we establish the error estimates
and for the velocity in
the - and -norms respectively, where if and
if . This improves the previous result [T. Kashiwabara et
al., Numer. Math. 134 (2016), pp. 705--740] obtained for the conforming
approximation in the sense that there appears no reciprocal of the penalty
parameter in the estimates.Comment: 21 page
Strong Well-Posedness for a Class of Dynamic Outflow Boundary Conditions for Incompressible Newtonian Flows
Based on energy considerations, we derive a class of dynamic outflow boundary
conditions for the incompressible Navier-Stokes equations, containing the
well-known convective boundary condition but incorporating also the stress at
the outlet. As a key building block for the analysis of such problems, we
consider the Stokes equations with such dynamic outflow boundary conditions in
a halfspace and prove the existence of a strong solution in the appropriate
Sobolev-Slobodeckij-setting with (in time and space) as the base space
for the momentum balance. For non-vanishing stress contribution in the boundary
condition, the problem is actually shown to have -maximal regularity under
the natural compatibility conditions. Aiming at an existence theory for
problems in weakly singular domains, where different boundary conditions apply
on different parts of the boundary such that these surfaces meet orthogonally,
we also consider the prototype domain of a wedge with opening angle
and different combinations of boundary conditions: Navier-Slip
with Dirichlet and Navier-Slip with the dynamic outflow boundary condition.
Again, maximal regularity of the problem is obtained in the appropriate
functional analytic setting and with the natural compatibility conditions.Comment: 31 pages, 1 figur
Analyticity of solutions to the primitive equations
This article presents the maximal regularity approach to the primitive
equations. It is proved that the primitive equations on cylindrical
domains admit a unique, global strong solution for initial data lying in the
critical solonoidal Besov space for with
. This solution regularize instantaneously and becomes even
real analytic for .Comment: 19 page