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research
Strong solutions to the 3D primitive equations with only horizontal dissipation: near
H
1
H^1
H
1
initial data
Authors
Chongsheng Cao
Jinkai Li
Edriss S. Titi
Publication date
21 July 2016
Publisher
View
on
arXiv
Abstract
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
(
v
0
,
T
0
)
β
H
1
(v_0, T_0)\in H^1
(
v
0
β
,
T
0
β
)
β
H
1
, 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
(
v
0
,
T
0
)
β
H
1
β©
L
β
(v_0, T_0)\in H^1\cap L^\infty
(
v
0
β
,
T
0
β
)
β
H
1
β©
L
β
, such that
β
z
v
0
β
L
m
\partial_zv_0\in L^m
β
z
β
v
0
β
β
L
m
, for some
m
β
(
2
,
β
)
m\in(2,\infty)
m
β
(
2
,
β
)
, 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
(
v
0
,
T
0
)
(v_0, T_0)
(
v
0
β
,
T
0
β
)
was assumed to have
H
2
H^2
H
2
regularity
Similar works
Full text
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eScholarship - University of California
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Last time updated on 25/12/2021