Analyticity of solutions to the primitive equations

Giga, Yoshikazu; Gries, Mathis; Hieber, Matthias; Hussein, Amru; Kashiwabara, Takahito

doi:10.1002/mana.201700401

Cited by 19 publications

(28 citation statements)

References 38 publications

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“…Our main result, Theorem 3.1, yields that for each time T there exists an ε > 0 such that the solution exists on (0, T ). d) The primitive equations are well-posed for all times T > 0 provided u 0 ∈ X γ , see [5]. e) Our method can be adjusted to the case with perturbed initial data.…”

Section: Resultsmentioning

confidence: 99%

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

et al. 2020

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Consider the anisotropic Navier-Stokes equations as well as the primitive equations. It is shown that the horizontal velocity of the solution to the anisotropic Navier-Stokes equations in a cylindrical domain of height ε with initial data u 0 = (v 0 , w 0) ∈ B 2−2/p q,p , 1/q + 1/p ≤ 1 if q ≥ 2 and 4/3q +2/3p ≤ 1 if q ≤ 2, converges as ε → 0 with convergence rate O(ε) to the horizontal velocity of the solution to the primitive equations with initial data v 0 with respect to the maximal-L p-L q-regularity norm. Since the difference of the corresponding vertical velocities remains bounded with respect to that norm, the convergence result yields a rigorous justification of the hydrostatic approximation in the primitive equations in this setting. It generalizes in particular a result by Li and Titi for the L 2-L 2-setting. The approach presented here does not rely on second order energy estimates but on maximal L p-L q-estimates for the heat equation.

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Section: Resultsmentioning

confidence: 99%

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

et al. 2020

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“…Taking advantage of the regularization of solutions for t > 0 one passes into the setting discussed in [11] and [9], and thus we obtain the following corollary. Our main result on the hydrostatic semigroup acting on X σ reads as follows.…”

Section: Resultsmentioning

confidence: 85%

“…It has been proven in [7] that the operator A p,σ possesses maximal L q -regularity. In [9] the authors applied this to develop a solution theory for initial data…”

Section: It Is Well Known Thatmentioning

confidence: 99%

“…There, the existence of a unique, global, strong solution to the primitive equations for initial data belonging to H 2/p,p was proved for the case of mixed Dirichlet-Neumann boundary conditions. This approach was transfered in [8,9] to the case of pure Neumann boundary conditions and global, strong well-posedness of the primitive equations was obtained for data a of the form a = a 1 + a 2 , where a 1 ∈ C(G; L 1 (−h, 0)) and a 2 ∈ L ∞ (G; L 1 (−h, 0)) with a 2 being small. These spaces are scaling invariant and represent the anisotropic character of the primitive equations.…”

Section: Introductionmentioning

confidence: 99%

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The hydrostatic Stokes semigroup and well-posedness of the primitive equations on spaces of bounded functions

Giga

Gries

Hieber

et al. 2020

Journal of Functional Analysis

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Consider the 3-d primitive equations in a layer domain Ω = G × (−h, 0), G = (0, 1) 2 , subject to mixed Dirichlet and Neumann boundary conditions at z = −h and z = 0, respectively, and the periodic lateral boundary condition. It is shown that this equation is globally, strongly well-posed for arbitrary large data of the form a = a 1 + a 2 , where a 1 ∈ C(G; L p (−h, 0)), a 2 ∈ L ∞ (G; L p (−h, 0)) for p > 3, and where a 1 is periodic in the horizontal variables and a 2 is sufficiently small. In particular, no differentiability condition on the data is assumed. The approach relies on L ∞ H L p z (Ω)-estimates for terms of the form t 1/2 ∂z e tA σ Pf L ∞for t > 0, where e tA σ denotes the hydrostatic Stokes semigroup. The difficulty in proving estimates of this form is that the hydrostatic Helmholtz projection P fails to be bounded with respect to the L ∞ -norm. The global strong well-posedness result is then obtained by an iteration scheme, splitting the data into a smooth and a rough part and by combining a reference solution for smooth data with an evolution equation for the rough part.2010 Mathematics Subject Classification. Primary: 35Q35; Secondary: 47D06, 76D03, 86A05.

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“…(e) Since the proof is based on the construction of strong solutions via maximal L 2 -regularity, one can prove using the so-called 'parameter-trick', cf. [14] and the references given therein, that solutions become real analytic in time and space for t > 0 provided the force terms has this property. For the periodic setting the solution is real analytic even on the boundary.…”

Section: Correspondingly For the Hyper-viscous Primitive Equations Onmentioning

confidence: 89%

Partial and full hyper-viscosity for Navier-Stokes and primitive equations

Hussein

2020

Journal of Differential Equations

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The 3-D primitive equations and incompressible Navier-Stokes equations with full hyperviscosity and only horizontal hyper-viscosity are considered on the torus, i.e., the diffusion term −∆ isHyper-viscosity is applied in many numerical schemes, and in particular horizontal hyper-viscosity appears in meteorological models. A classical result by Lions states that for the Navier-Stokes equations uniqueness of global weak solutions for initial data in L 2 holds if −∆ is replaced by (−∆) 5/4 . Here, for the primitive equations the corresponding result is proven for (−∆) 8/5 . For the case of horizontal hyperviscosity l = 2 is sufficient in both cases. Strong convergence for ε → 0 of hyper-viscous solutions to a weak solution of the Navier-Stokes and primitive equations, respectively, is proven as well. The approach presented here is based on the construction of strong solutions via an evolution equation approach for initial data in L 2 and weak-strong uniqueness.2010 Mathematics Subject Classification. Primary: 35Q35; Secondary: 35K25, 35K30, 35Q30, 76D03, 86A05.

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Analyticity of solutions to the primitive equations

Cited by 19 publications

References 38 publications

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

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

The hydrostatic Stokes semigroup and well-posedness of the primitive equations on spaces of bounded functions

Partial and full hyper-viscosity for Navier-Stokes and primitive equations

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