Strong solutions for the nonhomogeneous Navier-Stokes equations in unbounded domains

Silva, Pablo Braz e; Rojas-Medar, Marko Antonio; Villamizar-Roa, Élder J.

doi:10.1002/mma.1178

Cited by 6 publications

(11 citation statements)

References 14 publications

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“…Moreover, there exists a positive constant ξ ∗ ≤ ξ such that, for any 0≤ θ < ξ ∗ , the bounds

\begin{matrix} \sup_{t \geq 0} e^{ξ^{*} t} ((), ∥ \nabla boldu (\cdot, t) ∥^{2} + ∥ \nabla boldw (\cdot, t) ∥^{2}) < + \infty, \\ \sup_{t \geq 0} e^{θt} ((), ∥ {boldu}_{t} (\cdot, t) ∥^{2} + ∥ {boldw}_{t} (\cdot, t) ∥^{2}) < + \infty, \\ \sup_{t \geq 0} e^{θt} ((), ∥ A boldu (\cdot, t) ∥^{2} + ∥ L boldw (\cdot, t) ∥^{2}) < + \infty, \\ \sup_{t \geq 0} \int_{0}^{t} e^{θs} ((), ∥ \nabla {boldu}_{t} (\cdot, s) ∥^{2} + ∥ \nabla {boldw}_{t} (\cdot, s) ∥^{2}) ds < + \infty, \\ \sup_{t \geq 0} ((), ∥ ρ_{t} (\cdot, t) ∥_{\infty} + ∥ \nabla ρ (\cdot, t) ∥_{\infty}) < + \infty, \end{matrix}

hold. A similar result was obtained in for the variable density Navier–Stokes equations (remark 18 in ).…”

Section: Preliminaries and Main Resultssupporting

confidence: 85%

“…Then v has second derivatives in L 2 (Ω), and

\begin{matrix} ∥ D^{2} boldv ∥ & \leq C_{\partial} (∥ P boldf ∥ + ∥ \nabla boldv ∥), \\ ∥ \nabla boldv ∥_{3} & \leq C_{\partial} (∥ P boldf ∥^{1 / 2} ∥ \nabla boldv ∥^{1 / 2} + ∥ \nabla boldv ∥), \end{matrix}

where C ∂ is a positive constant depending only on the regularity of ∂ Ω. It depends neither on the ‘size’ of ∂ Ω nor on the ‘size’ of Ω.Remark (, Remark 5) The assumption of C 3 ‐regularity of the domain Ω is used only to prove Lemma independently of potential theory. Via potential theory, Lemma is valid for domains with C 2 boundary.…”

Section: Preliminaries and Main Resultsmentioning

confidence: 98%

“…We use these bounds to study the case of unbounded domains through an expanding sequences of bounded domains. This method was used in for the Navier–Stokes equations with constant density and in for the variable density Navier–Stokes equations. Our first result is the following.…”

Section: Preliminaries and Main Resultsmentioning

confidence: 99%

“…Then, there exists a local semi‐strong solution ( ρ , u , w ) of – in Ω×(0, T ). The functions u , w , ρ , and p , defined on (0, T ∗ ), for some T ∗ ∈(0, T ], are such that

\begin{matrix} boldu \in L^{\infty} (0, T^{*}; boldV (normalΩ)), \\ boldw \in L^{\infty} ((), 0, T^{*}; {boldH}_{0}^{1} (normalΩ)), \\ {boldu}_{t}, {boldw}_{t}, D^{2} boldu, D^{2} boldw, \nabla p \in L^{2} ((), 0, T^{*}; {boldL}^{2} (normalΩ)), \\ ρ \in L^{\infty} (0, T^{*}; L^{\infty} (normalΩ)), \\ ρ_{t} \in L^{\infty} ((), 0, T^{*}; L_{loc}^{\infty} (normalΩ)), \\ \nabla ρ \in L^{\infty} ((), 0, T^{*}; {boldL}_{loc}^{\infty} (normalΩ)), \\ ∥ \nabla boldu (t) - \nabla {boldu}_{0} ∥ \to 0 and ∥ \nabla boldw (t) - \nabla {boldw}_{0} ∥ \to 0 as t \to 0^{+} . \end{matrix}

Remark (, Remark 2) If Ω is either bounded or

normalΩ = {double-struckR}^{3}

, the regularity for the density is ρ t ∈ L ∞ (0, T ∗ ; L ∞ (Ω)) and ∇ ρ ∈ L ∞ (0, T ∗ ; L ∞ (Ω)). This fact is due to known results on the regularity of the Stokes problem in such domains (Remark ).…”

Section: Preliminaries and Main Resultsmentioning

confidence: 99%

“…Here, we adapt the techniques used in by J. G. Heywood for the classical Navier–Stokes equations and in by P. Braz e Silva, M. A. Rojas‐Medar, and E. J. Villamizar‐Roa for the Navier–Stokes equations with variable density, together with arguments used in to show the existence of local semi‐strong solutions and existence of global strong solutions for systems – in L 2 (Ω) = ( L 2 (Ω)) 3 , in the case of unbounded three‐dimensional domains with boundary uniformly of class C 3 .…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Semi‐strong and strong solutions for variable density asymmetric fluids in unbounded domains

Silva

Cruz

Rojas-Medar

2016

Math Methods in App Sciences

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We establish the existence of local in time semi‐strong solutions and global in time strong solutions for the system of equations describing flows of viscous and incompressible asymmetric fluids with variable density in general three‐dimensional domains with boundary uniformly of class C3. Under suitable assumptions, uniqueness of local semi‐strong solutions is also proved. Copyright © 2016 John Wiley & Sons, Ltd.

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“…Moreover, there exists a positive constant ξ ∗ ≤ ξ such that, for any 0≤ θ < ξ ∗ , the bounds

\begin{matrix} \sup_{t \geq 0} e^{ξ^{*} t} ((), ∥ \nabla boldu (\cdot, t) ∥^{2} + ∥ \nabla boldw (\cdot, t) ∥^{2}) < + \infty, \\ \sup_{t \geq 0} e^{θt} ((), ∥ {boldu}_{t} (\cdot, t) ∥^{2} + ∥ {boldw}_{t} (\cdot, t) ∥^{2}) < + \infty, \\ \sup_{t \geq 0} e^{θt} ((), ∥ A boldu (\cdot, t) ∥^{2} + ∥ L boldw (\cdot, t) ∥^{2}) < + \infty, \\ \sup_{t \geq 0} \int_{0}^{t} e^{θs} ((), ∥ \nabla {boldu}_{t} (\cdot, s) ∥^{2} + ∥ \nabla {boldw}_{t} (\cdot, s) ∥^{2}) ds < + \infty, \\ \sup_{t \geq 0} ((), ∥ ρ_{t} (\cdot, t) ∥_{\infty} + ∥ \nabla ρ (\cdot, t) ∥_{\infty}) < + \infty, \end{matrix}

hold. A similar result was obtained in for the variable density Navier–Stokes equations (remark 18 in ).…”

Section: Preliminaries and Main Resultssupporting

confidence: 85%

“…Then v has second derivatives in L 2 (Ω), and

\begin{matrix} ∥ D^{2} boldv ∥ & \leq C_{\partial} (∥ P boldf ∥ + ∥ \nabla boldv ∥), \\ ∥ \nabla boldv ∥_{3} & \leq C_{\partial} (∥ P boldf ∥^{1 / 2} ∥ \nabla boldv ∥^{1 / 2} + ∥ \nabla boldv ∥), \end{matrix}

Section: Preliminaries and Main Resultsmentioning

confidence: 98%

Section: Preliminaries and Main Resultsmentioning

confidence: 99%

“…Then, there exists a local semi‐strong solution ( ρ , u , w ) of – in Ω×(0, T ). The functions u , w , ρ , and p , defined on (0, T ∗ ), for some T ∗ ∈(0, T ], are such that

\begin{matrix} boldu \in L^{\infty} (0, T^{*}; boldV (normalΩ)), \\ boldw \in L^{\infty} ((), 0, T^{*}; {boldH}_{0}^{1} (normalΩ)), \\ {boldu}_{t}, {boldw}_{t}, D^{2} boldu, D^{2} boldw, \nabla p \in L^{2} ((), 0, T^{*}; {boldL}^{2} (normalΩ)), \\ ρ \in L^{\infty} (0, T^{*}; L^{\infty} (normalΩ)), \\ ρ_{t} \in L^{\infty} ((), 0, T^{*}; L_{loc}^{\infty} (normalΩ)), \\ \nabla ρ \in L^{\infty} ((), 0, T^{*}; {boldL}_{loc}^{\infty} (normalΩ)), \\ ∥ \nabla boldu (t) - \nabla {boldu}_{0} ∥ \to 0 and ∥ \nabla boldw (t) - \nabla {boldw}_{0} ∥ \to 0 as t \to 0^{+} . \end{matrix}

Remark (, Remark 2) If Ω is either bounded or

normalΩ = {double-struckR}^{3}

Section: Preliminaries and Main Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Semi‐strong and strong solutions for variable density asymmetric fluids in unbounded domains

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Global unique solvability of nonhomogeneous asymmetric fluids: A Lagrangian approach

Silva

Cruz

Loayza

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Uniform regularity for the incompressible Navier-Stokes system with variable density and Navier boundary conditions

2018

Quart. Appl. Math.

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We investigate the uniform regularity for the nonhomogeneous incompressible Navier-Stokes system with Navier boundary conditions and the inviscid limit to the Euler system. It is shown that there exists a unique strong solution of the Navier-Stokes equations in an interval of time that is uniform with respect to the viscosity parameter. The uniform estimate in conormal Sobolev spaces is established. Based on the uniform estimate, we show the convergence of the viscous solutions to the inviscid ones in L ∞ ( [ 0 , T ] × Ω ) L^\infty ([0,T]\times \Omega ) . This improves the result obtained by Ferreira et al. [SIAM J. Math. Anal. Vol. 45, No. 4, (2013), pp. 2576-2595], where L ∞ ( [ 0 , T ] , L 2 ( Ω ) ) L^\infty ([0,T],L^2(\Omega )) convergence was proved.

show abstract

Strong solutions for the nonhomogeneous Navier-Stokes equations in unbounded domains

Abstract: We show the existence of strong solutions for the nonhomogeneous Navier-Stokes equations in three-dimensional domains with boundary uniformly of class C 3 . Under suitable assumptions, uniqueness is also proved.

Cited by 6 publications

References 14 publications

Semi‐strong and strong solutions for variable density asymmetric fluids in unbounded domains

Semi‐strong and strong solutions for variable density asymmetric fluids in unbounded domains

Global unique solvability of nonhomogeneous asymmetric fluids: A Lagrangian approach

Uniform regularity for the incompressible Navier-Stokes system with variable density and Navier boundary conditions

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