Limiting Case of the Sobolev Inequality in BMO,¶with Application to the Euler Equations

Abstract. In this paper we study the analytic structure of a two-dimensional mass balance equation of an incompressible fluid in a porous medium given by Darcy's law. We obtain local existence and uniqueness by the particle-trajectory method and we present different global existence criterions. These analytical results with numerical simulations are used to indicate nonformation of singularities. Moreover, blow-up results are shown in a class of solutions with infinite energy.

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“…Now, we use the following inequality given in [14]: Let f ∈ W s,p with 1 < p < ∞ and s > 2/p, then, there exists a constant C=C(p,s) such that…”

Section: Singularities With Infinite Energymentioning

confidence: 99%

Analytical behavior of two-dimensional incompressible flow in porous media

Córdoba

Gancedo

Orive

2007

Journal of Mathematical Physics

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“…In a parallel development, a number of regularity criteria have been derived for the Euler equations [2,13,15,16,25]. However, the results are suboptimally applicable or even inapplicable to the Navier-Stokes case.…”

Section: Introductionmentioning

confidence: 99%

“…To make use of the remaining dissipation term in (25) we observe that ∇|u| q/2 = q|u| (q−2)/2 ∇|u|/2. So…”

Section: Introductionmentioning

confidence: 99%

Depletion of nonlinearity in the pressure force driving Navier–Stokes flows

Tran

2015

Nonlinearity

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Abstract. The dynamics of the velocity norms ||u|| L q , for q ≥ 3, in Navier-Stokes flows is studied. The pressure term that drives this dynamics has a high degree of nonlinear depletion, which owes its origin to a genuine negative correlation between |u| and |∇|u||, among other things. Under viscous effects, such depletion may give rise to mild growth of ||u|| L q . We explore the possibility of non-singular growth of ||u|| L q .

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“…This criterion is a straightforward extension of the celebrated BKM criterion (Beale, Kato & Majda 1984) for usual (non-magnetic) fluids governed by the Euler or Navier-Stokes equations. As in the BKM criterion, where the BMO norm of the vorticity ω BMO can replace ω ∞ (Kozono & Tanuichi 2000), we have the following slightly weaker version of (2.4):…”

mentioning

confidence: 99%

“…This theorem, together with the relation f BMO 2 f L ∞ , means that the BMO and L ∞ norms are almost equivalent, up to a logarithmic 'correction'. This fact enables Kozono & Tanuichi (2000) to replace ω ∞ by ω BMO in the BKM criterion, by essentially accommodating a logarithmic factor in the estimate of the growth rate of high-order derivatives of u.…”

mentioning

confidence: 99%

Two-dimensional magnetohydrodynamic turbulence in the limits of infinite and vanishing magnetic Prandtl number

Tran

Blackbourn

2013

J. Fluid Mech.

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We study both theoretically and numerically two-dimensional magnetohydrodynamic turbulence at infinite and zero magnetic Prandtl number $\mathit{Pm}$ (and the limits thereof), with an emphasis on solution regularity. For $\mathit{Pm}= 0$, both $\Vert \omega \Vert ^{2} $ and $\Vert j\Vert ^{2} $, where $\omega $ and $j$ are, respectively, the vorticity and current, are uniformly bounded. Furthermore, $\Vert \boldsymbol{\nabla} j\Vert ^{2} $ is integrable over $[0, \infty )$. The uniform boundedness of $\Vert \omega \Vert ^{2} $ implies that in the presence of vanishingly small viscosity $\nu $ (i.e. in the limit $\mathit{Pm}\rightarrow 0$), the kinetic energy dissipation rate $\nu \Vert \omega \Vert ^{2} $ vanishes for all times $t$, including $t= \infty $. Furthermore, for sufficiently small $\mathit{Pm}$, this rate decreases linearly with $\mathit{Pm}$. This linear behaviour of $\nu \Vert \omega \Vert ^{2} $ is investigated and confirmed by high-resolution simulations with $\mathit{Pm}$ in the range $[1/ 64, 1] $. Several criteria for solution regularity are established and numerically tested. As $\mathit{Pm}$ is decreased from unity, the ratio $\Vert \omega \Vert _{\infty } / \Vert \omega \Vert $ is observed to increase relatively slowly. This, together with the integrability of $\Vert \boldsymbol{\nabla} j\Vert ^{2} $, suggests global regularity for $\mathit{Pm}= 0$. When $\mathit{Pm}= \infty $, global regularity is secured when either $\Vert \boldsymbol{\nabla} \boldsymbol{u}\Vert _{\infty } / \Vert \omega \Vert $, where $\boldsymbol{u}$ is the fluid velocity, or $\Vert j\Vert _{\infty } / \Vert j\Vert $ is bounded. The former is plausible given the presence of viscous effects for this case. Numerical results over the range $\mathit{Pm}\in [1, 64] $ show that $\Vert \boldsymbol{\nabla} \boldsymbol{u}\Vert _{\infty } / \Vert \omega \Vert $ varies slightly (with similar behaviour for $\Vert j\Vert _{\infty } / \Vert j\Vert $), thereby lending strong support for the possibility $\Vert \boldsymbol{\nabla} \boldsymbol{u}\Vert _{\infty } / \Vert \omega \Vert \lt \infty $ in the limit $\mathit{Pm}\rightarrow \infty $. The peak of the magnetic energy dissipation rate $\mu \Vert j\Vert ^{2} $ is observed to decrease rapidly as $\mathit{Pm}$ is increased. This result suggests the possibility $\Vert j\Vert ^{2} \lt \infty $ in the limit $\mathit{Pm}\rightarrow \infty $. We discuss further evidence for the boundedness of the ratios $\Vert \omega \Vert _{\infty } / \Vert \omega \Vert $, $\Vert \boldsymbol{\nabla} \boldsymbol{u}\Vert _{\infty } / \Vert \omega \Vert $ and $\Vert j\Vert _{\infty } / \Vert j\Vert $ in conjunction with observation on the density of filamentary structures in the vorticity, velocity gradient and current fields.

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Limiting Case of the Sobolev Inequality in BMO,¶with Application to the Euler Equations

Cited by 187 publications

References 11 publications

Analytical behavior of two-dimensional incompressible flow in porous media

Analytical behavior of two-dimensional incompressible flow in porous media

Depletion of nonlinearity in the pressure force driving Navier–Stokes flows

Two-dimensional magnetohydrodynamic turbulence in the limits of infinite and vanishing magnetic Prandtl number

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