scholarly journals Rigorous justification of the hydrostatic approximation for the primitive equations by scaled Navier–Stokes equations

Nonlinearity ◽  
2020 ◽  
Vol 33 (12) ◽  
pp. 6502-6516
Author(s):  
Ken Furukawa ◽  
Yoshikazu Giga ◽  
Matthias Hieber ◽  
Amru Hussein ◽  
Takahito Kashiwabara ◽  
...  
Keyword(s):  
Stokes Equations ◽  
Primitive Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Hydrostatic Approximation

scholarly journals The primitive equations in the scaling-invariant space $$L^{\infty }(L^1)$$

2021 ◽  
Author(s):  
Yoshikazu Giga ◽  
Mathis Gries ◽  
Matthias Hieber ◽  
Amru Hussein ◽  
Takahito Kashiwabara
Keyword(s):  
Initial Data ◽  
Stokes Equations ◽  
Primitive Equations ◽  
Navier Stokes ◽  
Invariant Space ◽  
Unique Strong Solution ◽  
Navier Stokes Equations ◽  
Anisotropic Character ◽  
Scaling Invariant ◽  
Well Posed

AbstractConsider the primitive equations on $$\mathbb {R}^2\times (z_0,z_1)$$ R 2 × ( z 0 , z 1 ) with initial data a of the form $$a=a_1+a_2$$ a = a 1 + a 2 , where $$a_1 \in \mathrm{BUC}_\sigma (\mathbb {R}^2;L^1(z_0,z_1))$$ a 1 ∈ BUC σ ( R 2 ; L 1 ( z 0 , z 1 ) ) and $$a_2 \in L^\infty _\sigma (\mathbb {R}^2;L^1(z_0,z_1))$$ a 2 ∈ L σ ∞ ( R 2 ; L 1 ( z 0 , z 1 ) ) . These spaces are scaling-invariant and represent the anisotropic character of these equations. It is shown that for $$a_1$$ a 1 arbitrary large and $$a_2$$ a 2 sufficiently small, this set of equations admits a unique strong solution which extends to a global one and is thus strongly globally well posed for these data provided a is periodic in the horizontal variables. The approach presented depends crucially on mapping properties of the hydrostatic Stokes semigroup in the $$L^\infty (L^1)$$ L ∞ ( L 1 ) -setting. It can be seen as the counterpart of the classical iteration schemes for the Navier–Stokes equations, now for the primitive equations in the $$L^\infty (L^1)$$ L ∞ ( L 1 ) -setting.

REGULARIZATION BY MONOTONE PERTURBATIONS OF THE HYDROSTATIC APPROXIMATION OF NAVIER–STOKES EQUATIONS

2004 ◽  
Vol 14 (12) ◽  
pp. 1819-1848 ◽  
Author(s):  
FRANCISCO ORTEGÓN GALLEGO
Keyword(s):  
Existence Of Solutions ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Energy Identity ◽  
Nonlinear Perturbations ◽  
Smooth Functions ◽  
Regularization Technique ◽  
Navier Stokes Equations ◽  
Hydrostatic Approximation ◽  
Functional Setting

Due to the lack of regularity of the solutions to the hydrostatic approximation of Navier–Stokes equations, an energy identity cannot be deduced. By including certain nonlinear perturbations to the hydrostatic approximation equations, the solutions to the perturbed problem are smooth enough so that they satisfy the corresponding energy identity. The perturbations considered in this paper are of the monotone class. Three kinds of problems are then studied. To do that, we introduce a functional setting and show in every case that the set of smooth functions with compact support is dense in the space where we search for solutions. When the perturbations are small enough in a certain sense, the solutions of the perturbed problem are close to those of the original one. As a result, this gives a new proof of the existence of solutions to the hydrostatic approximation of Navier–Stokes equations. Finally, this regularization technique has been applied to the analysis of a one-equation hydrostatic turbulence model.

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