Navier–Stokes Equations in a Curved Thin Domain, Part II: Global Existence of a Strong Solution

Miura, Tatsu-Hiko

doi:10.1007/s00021-020-00534-2

Cited by 6 publications

(30 citation statements)

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“…Problem settings. In this paper, as a continuation of [57,58], we consider the Navier-Stokes equations with Navier's slip boundary conditions…”

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confidence: 99%

“…with a constant c > 0 independent of ε. Using them and average operators in the thin direction we established in the second part [58] the global-in-time existence of a strong solution u ε to (1.1) for large data u ε 0 and f ε in the sense that…”

mentioning

confidence: 99%

“…with constants c, α, β > 0 independent of ε under additional conditions on u ε 0 and f ε (see Theorem 2.5). In this paper, based on the results of [57,58], we study the behavior as ε → 0 of the average in the thin direction of u ε given by M u ε (y, t) := 1 εg(y) εg1(y) εg0(y) u ε (y + rn(y), t) dr, (y, t) ∈ Γ × (0, ∞).…”

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confidence: 99%

“…Main ideas and outline of the proofs. To prove the main results of this paper (Theorems 2.6 and 2.7) we use the results of the first and second parts [57,58] of our study. So let us first explain main ideas used in [57,58] and then give the outline of the proofs of Theorems 2.6 and 2.…”

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confidence: 99%

“…In [58] we proved the global existence of a strong solution u ε to (1.1) and obtained the estimates (1.7) for u ε by a standard energy method. The main ingredient for the proof was the following estimate for the trilinear term:…”

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confidence: 99%

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Navier-Stokes equations in a curved thin domain, Part III: thin-film limit

Miura¹

2020

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We consider the Navier-Stokes equations with Navier's slip boundary conditions in a three-dimensional curved thin domain around a given closed surface. Under suitable assumptions we show that the average in the thin direction of a strong solution to the bulk Navier-Stokes equations converges weakly in appropriate function spaces on the limit surface as the thickness of the thin domain tends to zero. Moreover, we characterize the limit as a weak solution to limit equations, which are the damped and weighted Navier-Stokes equations on the limit surface. We also prove the strong convergence of the average of a strong solution to the bulk equations towards a weak solution to the limit equations by showing estimates for the difference between them. In some special case our limit equations agree with the Navier-Stokes equations on a Riemannian manifold in which the viscous term contains the Ricci curvature. This is the first result on a rigorous derivation of the surface Navier-Stokes equations on a general closed surface by the thin-film limit.

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“…Problem settings. In this paper, as a continuation of [57,58], we consider the Navier-Stokes equations with Navier's slip boundary conditions…”

mentioning

confidence: 99%

mentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

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confidence: 99%

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Navier-Stokes equations in a curved thin domain, Part III: thin-film limit

Miura¹

2020

Preprint

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Approximation of a Solution to the Stationary Navier–Stokes Equations in a Curved Thin Domain by a Solution to Thin-Film Limit Equations

Miura

2024

J. Math. Fluid Mech.

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Korn inequalities for incompatible tensor fields in three space dimensions with conformally invariant dislocation energy

2021

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Let $$\Omega \subset \mathbb {R}^3$$ Ω ⊂ R 3 be an open and bounded set with Lipschitz boundary and outward unit normal $$\nu $$ ν . For $$1<p<\infty $$ 1 < p < ∞ we establish an improved version of the generalized $$L^p$$ L p -Korn inequality for incompatible tensor fields P in the new Banach space $$\begin{aligned}&W^{1,\,p,\,r}_0({{\,\mathrm{dev}\,}}{{\,\mathrm{sym}\,}}{{\,\mathrm{Curl}\,}}; \Omega ,\mathbb {R}^{3\times 3}) \\&\quad = \{ P \in L^p(\Omega ; \mathbb {R}^{3 \times 3}) \mid {{\,\mathrm{dev}\,}}{{\,\mathrm{sym}\,}}{{\,\mathrm{Curl}\,}}P \in L^r(\Omega ; \mathbb {R}^{3 \times 3}),\ {{\,\mathrm{dev}\,}}{{\,\mathrm{sym}\,}}(P \times \nu ) = 0 \text { on }\partial \Omega \} \end{aligned}$$ W 0 1 , p , r ( dev sym Curl ; Ω , R 3 × 3 ) = { P ∈ L p ( Ω ; R 3 × 3 ) ∣ dev sym Curl P ∈ L r ( Ω ; R 3 × 3 ) , dev sym ( P × ν ) = 0 on ∂ Ω } where $$\begin{aligned} r \in [1, \infty ), \qquad \frac{1}{r} \le \frac{1}{p} + \frac{1}{3}, \qquad r >1 \quad \text {if }p = \frac{3}{2}. \end{aligned}$$ r ∈ [ 1 , ∞ ) , 1 r ≤ 1 p + 1 3 , r > 1 if p = 3 2 . Specifically, there exists a constant $$c=c(p,\Omega ,r)>0$$ c = c ( p , Ω , r ) > 0 such that the inequality $$\begin{aligned} \Vert P \Vert _{L^p(\Omega ,\mathbb {R}^{3\times 3})}\le c\,\left( \Vert {{\,\mathrm{sym}\,}}P \Vert _{L^p(\Omega ,\mathbb {R}^{3\times 3})} + \Vert {{\,\mathrm{dev}\,}}{{\,\mathrm{sym}\,}}{{\,\mathrm{Curl}\,}}P \Vert _{L^{r}(\Omega ,\mathbb {R}^{3\times 3})}\right) \end{aligned}$$ ‖ P ‖ L p ( Ω , R 3 × 3 ) ≤ c ‖ sym P ‖ L p ( Ω , R 3 × 3 ) + ‖ dev sym Curl P ‖ L r ( Ω , R 3 × 3 ) holds for all tensor fields $$P\in W^{1,\,p, \, r}_0({{\,\mathrm{dev}\,}}{{\,\mathrm{sym}\,}}{{\,\mathrm{Curl}\,}}; \Omega ,\mathbb {R}^{3\times 3})$$ P ∈ W 0 1 , p , r ( dev sym Curl ; Ω , R 3 × 3 ) . Here, $${{\,\mathrm{dev}\,}}X :=X -\frac{1}{3} {{\,\mathrm{tr}\,}}(X)\,{\mathbb {1}}$$ dev X : = X - 1 3 tr ( X ) 1 denotes the deviatoric (trace-free) part of a $$3 \times 3$$ 3 × 3 matrix X and the boundary condition is understood in a suitable weak sense. This estimate also holds true if the boundary condition is only satisfied on a relatively open, non-empty subset $$\Gamma \subset \partial \Omega $$ Γ ⊂ ∂ Ω . If no boundary conditions are imposed then the estimate holds after taking the quotient with the finite-dimensional space $$K_{S,dSC}$$ K S , d S C which is determined by the conditions $${{\,\mathrm{sym}\,}}P =0$$ sym P = 0 and $${{\,\mathrm{dev}\,}}{{\,\mathrm{sym}\,}}{{\,\mathrm{Curl}\,}}P = 0$$ dev sym Curl P = 0 . In that case one can replace $$\Vert {{\,\mathrm{dev}\,}}{{\,\mathrm{sym}\,}}{{\,\mathrm{Curl}\,}}P \Vert _{L^r(\Omega ,\mathbb {R}^{3\times 3})} $$ ‖ dev sym Curl P ‖ L r ( Ω , R 3 × 3 ) by $$\Vert {{\,\mathrm{dev}\,}}{{\,\mathrm{sym}\,}}{{\,\mathrm{Curl}\,}}P \Vert _{W^{-1,p}(\Omega ,\mathbb {R}^{3\times 3})}$$ ‖ dev sym Curl P ‖ W - 1 , p ( Ω , R 3 × 3 ) . The new $$L^p$$ L p -estimate implies a classical Korn’s inequality with weak boundary conditions by choosing $$P=\mathrm {D}u$$ P = D u and a deviatoric-symmetric generalization of Poincaré’s inequality by choosing $$P=A\in {{\,\mathrm{\mathfrak {so}}\,}}(3)$$ P = A ∈ so ( 3 ) . The proof relies on a representation of the third derivatives $$\mathrm {D}^3 P$$ D 3 P in terms of $$\mathrm {D}^2 {{\,\mathrm{dev}\,}}{{\

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Navier–Stokes Equations in a Curved Thin Domain, Part II: Global Existence of a Strong Solution

Cited by 6 publications

References 35 publications

Navier-Stokes equations in a curved thin domain, Part III: thin-film limit

Navier-Stokes equations in a curved thin domain, Part III: thin-film limit

Approximation of a Solution to the Stationary Navier–Stokes Equations in a Curved Thin Domain by a Solution to Thin-Film Limit Equations

Korn inequalities for incompatible tensor fields in three space dimensions with conformally invariant dislocation energy

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