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.
In this paper, we justify the hydrostatic approximation of the primitive equations in maximal $$L^p$$ L p -$$L^q$$ L q -settings in the three-dimensional layer domain $$\varOmega = \mathbb {T} ^2 \times (-1, 1)$$ Ω = T 2 × ( - 1 , 1 ) under the no-slip (Dirichlet) boundary condition in any time interval (0, T) for $$T>0$$ T > 0 . We show that the solution to the $$\epsilon $$ ϵ -scaled Navier–Stokes equations with Besov initial data $$u_0 \in B^{s}_{q,p}(\varOmega )$$ u 0 ∈ B q , p s ( Ω ) for $$s > 2 - 2/p + 1/ q$$ s > 2 - 2 / p + 1 / q converges to the solution to the primitive equations with the same initial data in $$\mathbb {E}_1 (T) = W^{1, p}(0, T ; L^q (\varOmega )) \cap L^p(0, T ; W^{2, q} (\varOmega )) $$ E 1 ( T ) = W 1 , p ( 0 , T ; L q ( Ω ) ) ∩ L p ( 0 , T ; W 2 , q ( Ω ) ) with order $$O(\epsilon )$$ O ( ϵ ) , where $$(p,q) \in (1,\infty )^2$$ ( p , q ) ∈ ( 1 , ∞ ) 2 satisfies $$ \frac{1}{p} \le \min ( 1 - 1/q, 3/2 - 2/q ) $$ 1 p ≤ min ( 1 - 1 / q , 3 / 2 - 2 / q ) and $$\epsilon $$ ϵ has the length scale. The global well-posedness of the scaled Navier–Stokes equations by $$\epsilon $$ ϵ in $$\mathbb {E}_1 (T)$$ E 1 ( T ) is also proved for sufficiently small $$\epsilon >0$$ ϵ > 0 . Note that $$T = \infty $$ T = ∞ is included.
In this paper we justify the hydrostatic approximation of the primitive equations in the maximal L p -L q -setting in the three-dimensional layer domain Ω = T 2 × (−1, 1) under the no-slip (Dirichlet) boundary condition in any time interval (0, T ) for T > 0. We show that the solution to the scaled Navier-Stokes equations with Besov initial data u 0 ∈ B s q,p (Ω) for s > 2 − 2/p + 1/q converges to the solution to the primitive equations with the same initial data in E 1 (T ) = W 1,p (0, T ; L q (Ω)) ∩ L p (0, T ; W 2,q (Ω)) with order O(ǫ) where (p, q) ∈ (1, ∞) 2 satisfies 1 p ≤ min(1 − 1/q, 3/2 − 2/q). The global well-posedness of the scaled Navier-Stokes equations in E 1 (T ) is also proved for sufficiently small ǫ > 0. Note that T = ∞ is included.
A Correction to this paper has been published: 10.1007/s00028-021-00674-6
In this paper we consider maximal regularity for the vector-valued quasi-steady linear elliptic problems. The equations are the elliptic equation in the domain and the evolution equations on its boundary. We prove the maximal Lp-Lq regularity for these problems and give examples that our results are applicable. The Lopatinskii-Shapiro and the asymptotic Lopatinskii-Shapiro conditions are important to get boundedness of solution operators.
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