Incompressible 3D Navier–Stokes Equations as a Limit of a Nonlinear Parabolic System

Rusin, Walter

doi:10.1007/s00021-011-0074-x

Cited by 8 publications

(21 citation statements)

References 12 publications

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“…A weak solution to problem is a vector field

u^{ε} \in L^{\infty} false(0, \infty false), L^{2} false({double-struckR}^{n} false) false) \cap L^{2} false((0, \infty), {\dot{H}}^{1} (R^{n}) false)

such that for every

φ \in D (R^{n}, false[0, \infty false))

div φ = 0

, we have

\int_{0}^{\infty} \int (\nabla u^{ε} \cdot \nabla φ + (u^{ε} \cdot \nabla u^{ε}) \cdot φ + \frac{1}{2} u^{ε} \cdot φ div u^{ε} - u^{ε} \cdot \partial_{t} φ) d x d t = \int u_{0} \cdot φ false(\cdot, 0 false) d x

u_{0} \in L^{2} false({double-struckR}^{n} false)

, and

div u_{0} = 0

, then the existence (and the uniqueness for

n = 2

) of a weak solution to problem was established in . As for the classical Navier–Stokes equations such solution satisfies the energy inequality

\int false| u^{ε} {false(x, t false) |}^{2} d x + 2 \int_{s}^{t} \int (| \nabla)

…”

Section: Statement Of the Main Resultsmentioning

confidence: 99%

“…The associated linear problem to equation is

({, \begin{matrix} \partial_{t} u^{ε} - Δ u^{ε} - \frac{1}{ε} \nabla div false(u^{ε} false) = g \\ u false(x, 0 false) = u_{0} false(x false) . \end{matrix})

The integral formulation associated with this linear problem reads

u^{ε} false(x, t false) = M_{ε} false(t false) u_{0} false(x false) + \int_{0}^{t} M_{ε} false(t - s false) g false(s false) d s,

where

M_{ε} false(t false) u_{0} false(x false)

is given by the convolution integral

M_{ε} false(t false) u_{0} false(x false) = \int M_{ε} false(x - y, t false) u_{0} false(y false) d y .

The properties of the kernel

M_{ε} false(x, t false)

have been studied in detail by Rusin in . Its symbol is

{({\overset{M}{true}}_{ε} false(ξ, t false))}_{k, l} = e^{- {t | ξ |}^{2}} (δ_{k, l} - \frac{ξ_{k} ξ_{l}}{{| ξ |}^{2}} (() 1 - e^{- t}))

…”

Section: Global Strong Solutions Uniformly Integrable In Timementioning

confidence: 99%

“…We complement here the analysis in of the kernel

M_{ε} false(t false)

by observing that, for

t, ε > 0

, the kernel

M_{ε} false(\cdot, t false)

belongs to the Schwartz class. Indeed, we can write

{({\overset{M}{true}}_{ε} false(ξ, t false))}_{k, l} = e^{- {t | ξ |}^{2}} (δ_{k, l} + \frac{t}{ε} ξ_{k} ξ_{l} {H false(- t false| ξ false|}^{2} / ε)),

where

H false(r false) = false(e^{- r} - 1 false) / r = \sum_{k = 0}^{\infty} {(- r)}^{k} / false(k + 1 false)!

.…”

Section: Global Strong Solutions Uniformly Integrable In Timementioning

confidence: 99%

“…Motivated by recent studies by Niche, Schonbek and Rusin , we consider the following system, proposed by Temam as a useful model for the effective approximation of solutions to the Navier–Stokes equations:

({, \begin{matrix} \partial_{t} u^{ε} - Δ u^{ε} + u^{ε} \cdot \nabla u^{ε} + \frac{1}{2} u^{ε} div false(u^{ε} false) - \frac{1}{ε} \nabla div false(u^{ε} false) = 0, x \in R^{n}, t > 0 \\ u false(x, 0 false) = u_{0} false(x false), div false(u_{0} false) = 0 . \end{matrix})

Here

ε > 0

is a parameter measuring how much the vector field

u^{ε}

is far from being incompressible. Note that

div (u^{ε})

, in general, will not be equal to zero for

t > 0

.…”

Section: Introductionmentioning

confidence: 99%

“…The construction of global weak solutions to the system and the convergence problem as

ε \to 0

of these solutions to the corresponding Leray solutions of the classical Navier–Stokes system are successfully addressed in and more recently in .…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

On a non-solenoidal approximation to the incompressible Navier-Stokes equations

Brandolese

2017

J. London Math. Soc.

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We establish an asymptotic profile that sharply describes the behavior as t→∞ for solutions to a non‐solenoidal approximation of the incompressible Navier–Stokes equations introduced by Temam. The solutions of Temam's model are known to converge to the corresponding solutions of the classical Navier–Stokes, for example, in L loc 3false(double-struckR+×double-struckR3false), provided ε→0, where ε>0 is the physical parameter related to the artificial compressibility term. However, we show that such model is no longer a good approximation of Navier–Stokes for large times: indeed, its solutions can decay much slower as t→∞ than the corresponding solutions of Navier–Stokes.

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“…A weak solution to problem is a vector field

u^{ε} \in L^{\infty} false(0, \infty false), L^{2} false({double-struckR}^{n} false) false) \cap L^{2} false((0, \infty), {\dot{H}}^{1} (R^{n}) false)

such that for every

φ \in D (R^{n}, false[0, \infty false))

div φ = 0

, we have

\int_{0}^{\infty} \int (\nabla u^{ε} \cdot \nabla φ + (u^{ε} \cdot \nabla u^{ε}) \cdot φ + \frac{1}{2} u^{ε} \cdot φ div u^{ε} - u^{ε} \cdot \partial_{t} φ) d x d t = \int u_{0} \cdot φ false(\cdot, 0 false) d x

u_{0} \in L^{2} false({double-struckR}^{n} false)

, and

div u_{0} = 0

, then the existence (and the uniqueness for

n = 2

) of a weak solution to problem was established in . As for the classical Navier–Stokes equations such solution satisfies the energy inequality

\int false| u^{ε} {false(x, t false) |}^{2} d x + 2 \int_{s}^{t} \int (| \nabla)

…”

Section: Statement Of the Main Resultsmentioning

confidence: 99%

“…The associated linear problem to equation is

({, \begin{matrix} \partial_{t} u^{ε} - Δ u^{ε} - \frac{1}{ε} \nabla div false(u^{ε} false) = g \\ u false(x, 0 false) = u_{0} false(x false) . \end{matrix})

The integral formulation associated with this linear problem reads

u^{ε} false(x, t false) = M_{ε} false(t false) u_{0} false(x false) + \int_{0}^{t} M_{ε} false(t - s false) g false(s false) d s,

where

M_{ε} false(t false) u_{0} false(x false)

is given by the convolution integral

M_{ε} false(t false) u_{0} false(x false) = \int M_{ε} false(x - y, t false) u_{0} false(y false) d y .

The properties of the kernel

M_{ε} false(x, t false)

have been studied in detail by Rusin in . Its symbol is

{({\overset{M}{true}}_{ε} false(ξ, t false))}_{k, l} = e^{- {t | ξ |}^{2}} (δ_{k, l} - \frac{ξ_{k} ξ_{l}}{{| ξ |}^{2}} (() 1 - e^{- t}))

…”

Section: Global Strong Solutions Uniformly Integrable In Timementioning

confidence: 99%

“…We complement here the analysis in of the kernel

M_{ε} false(t false)

by observing that, for

t, ε > 0

, the kernel

M_{ε} false(\cdot, t false)

belongs to the Schwartz class. Indeed, we can write

{({\overset{M}{true}}_{ε} false(ξ, t false))}_{k, l} = e^{- {t | ξ |}^{2}} (δ_{k, l} + \frac{t}{ε} ξ_{k} ξ_{l} {H false(- t false| ξ false|}^{2} / ε)),

where

H false(r false) = false(e^{- r} - 1 false) / r = \sum_{k = 0}^{\infty} {(- r)}^{k} / false(k + 1 false)!

.…”

Section: Global Strong Solutions Uniformly Integrable In Timementioning

confidence: 99%

({, \begin{matrix} \partial_{t} u^{ε} - Δ u^{ε} + u^{ε} \cdot \nabla u^{ε} + \frac{1}{2} u^{ε} div false(u^{ε} false) - \frac{1}{ε} \nabla div false(u^{ε} false) = 0, x \in R^{n}, t > 0 \\ u false(x, 0 false) = u_{0} false(x false), div false(u_{0} false) = 0 . \end{matrix})

Here

ε > 0

is a parameter measuring how much the vector field

u^{ε}

is far from being incompressible. Note that

div (u^{ε})

, in general, will not be equal to zero for

t > 0

.…”

Section: Introductionmentioning

confidence: 99%

“…The construction of global weak solutions to the system and the convergence problem as

ε \to 0

of these solutions to the corresponding Leray solutions of the classical Navier–Stokes system are successfully addressed in and more recently in .…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On a non-solenoidal approximation to the incompressible Navier-Stokes equations

Brandolese

2017

J. London Math. Soc.

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Comparison of decay of solutions to two compressible approximations to Navier-Stokes equations

Niche

Schonbek

2016

Bull Braz Math Soc, New Series

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In this article, we use the decay character of initial data to compare the energy decay rates of solutions to different compressible approximations to the NavierStokes equations. We show that the system having a nonlinear damping term has slower decay than its counterpart with an advection-like term. Moreover, me characterize a set of initial data for which the decay of the first system is driven by the difference between the full solution and the solution to the linear part, while for the second system the linear part provides the decay rate.

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On a Toy-Model Related to the Navier–Stokes Equations

Hounkpe

2022

J Math Sci

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A parabolic toy-model for the incompressible Navier-Stokes system is considered. This model shares a lot of similar features with the incompressible model, including the energy inequality, the scaling symmetry, and it is also supercritical in 3D. A goal is to establish some regularity results for this toy-model in order to get, if possible, better insight to the standard Navier-Stokes system. A Caffarelli-Kohn-Nirenberg type result for the model is also proved in a direct manner. Finally, the absence of divergence-free constraint allows us to study this model in the radially symmetric setting for which full regularity is established. Bibliography: 16 titles.

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Incompressible 3D Navier–Stokes Equations as a Limit of a Nonlinear Parabolic System

Abstract: In this paper, we consider the Cauchy problem for a nonlinear parabolic system u t − Δu + u · ∇uWe analyze the convergence of its solutions to a solution of the incompressible Navier-Stokes system as → 0.Mathematics Subject Classification (2010). Primary 35Q30; Secondary 76D05.

Cited by 8 publications

References 12 publications

On a non-solenoidal approximation to the incompressible Navier-Stokes equations

On a non-solenoidal approximation to the incompressible Navier-Stokes equations

Comparison of decay of solutions to two compressible approximations to Navier-Stokes equations

On a Toy-Model Related to the Navier–Stokes Equations

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