767 research outputs found
Recent Advances Concerning Certain Class of Geophysical Flows
This paper is devoted to reviewing several recent developments concerning
certain class of geophysical models, including the primitive equations (PEs) of
atmospheric and oceanic dynamics and a tropical atmosphere model. The PEs for
large-scale oceanic and atmospheric dynamics are derived from the Navier-Stokes
equations coupled to the heat convection by adopting the Boussinesq and
hydrostatic approximations, while the tropical atmosphere model considered here
is a nonlinear interaction system between the barotropic mode and the first
baroclinic mode of the tropical atmosphere with moisture.
We are mainly concerned with the global well-posedness of strong solutions to
these systems, with full or partial viscosity, as well as certain singular
perturbation small parameter limits related to these systems, including the
small aspect ratio limit from the Navier-Stokes equations to the PEs, and a
small relaxation-parameter in the tropical atmosphere model. These limits
provide a rigorous justification to the hydrostatic balance in the PEs, and to
the relaxation limit of the tropical atmosphere model, respectively. Some
conditional uniqueness of weak solutions, and the global well-posedness of weak
solutions with certain class of discontinuous initial data, to the PEs are also
presented.Comment: arXiv admin note: text overlap with arXiv:1507.0523
Strong solutions to the 3D primitive equations with only horizontal dissipation: near initial data
In this paper, we consider the initial-boundary value problem of the
three-dimensional primitive equations for oceanic and atmospheric dynamics with
only horizontal viscosity and horizontal diffusivity. We establish the local,
in time, well-posedness of strong solutions, for any initial data , by using the local, in space, type energy estimate. We also
establish the global well-posedness of strong solutions for this system, with
any initial data , such that , for some , by using the logarithmic type anisotropic
Sobolev inequality and a logarithmic type Gronwall inequality. This paper
improves the previous results obtained in [Cao, C.; Li, J.; Titi, E.S.: Global
well-posedness of the 3D primitive equations with only horizontal viscosity and
diffusivity, Comm. Pure Appl.Math., Vol. 69 (2016), 1492-1531.], where the
initial data was assumed to have regularity
Relative entropy in multi-phase models of 1d elastodynamics: Convergence of a non-local to a local model
In this paper we study a local and a non-local regularization of the system
of nonlinear elastodynamics with a non-convex energy. We show that solutions of
the non-local model converge to those of the local model in a certain regime.
The arguments are based on the relative entropy framework and provide an
example how local and non-local regularizations may compensate for
non-convexity of the energy and enable the use of the relative entropy
stability theory -- even if the energy is not quasi- or poly-convex
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
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