Second Derivatives Estimate of Suitable Solutions to the 3D Navier–Stokes Equations

Vasseur, Alexis; Yang, Jincheng

doi:10.1007/s00205-021-01661-4

Cited by 8 publications

(4 citation statements)

References 20 publications

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“…Note however, that this direct proof collapses due to the transport term. In dimension three, u can be controlled at best in L 10/3 t,x while the best control of ∇ω is in the Lorentz spaces L 4/3,q t,x for any q > 4/3 (see [VY21]). But this is far from enough to bound the transport term u∇ω in L 1 t,x .…”

Section: Introductionmentioning

confidence: 99%

“…But the bound is in negative power of ν and so is useless for the asymptotic limit. However, we can use blow-up techniques inspired by [Vas10] (see also [CV14,VY21]) which naturally deplete the strength of the transport term.…”

Section: Introductionmentioning

confidence: 99%

“…As mentioned before, one reason is the transport term u • ∇ω is indeed hard to control. Using blow-up techniques along the trajectories of the flow first introduced in [Vas10],it was proved in [VY21] that without boundary in Ω = R 3 , ∇ 2 ω ∈ L 1,q locally for q > 1 but miss the endpoint L 1 . The bounded domain is even more complicated because of the lack of convenient global control on the pressure.…”

Section: Introductionmentioning

confidence: 99%

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Boundary Vorticity Estimates for Navier-Stokes and Application to the Inviscid Limit

Vasseur¹,

Yang²

2021

Preprint

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Consider the steady solution to the incompressible Euler equation ū = Ae 1 in the periodic tunnel Ω = T d−1 × (0, 1) in dimension d = 2, 3. Consider now the family of solutions u ν to the associated Navier-Stokes equation with the no-slip condition on the flat boundaries, for small viscosities ν = A/Re, and initial values close in L 2 to Ae 1 . Under a conditional assumption on the energy dissipation close to the boundary, Kato showed in 1984 that u ν converges to Ae 1 when the viscosity converges to 0 and the initial value converges to Ae 1 . It is still unknown whether this inviscid limit is unconditionally true. The convex integration method predicts the possibility of a layer separation with energy at time T up to u ν (T ) − Ae 1 2 L 2 (Ω) ≈ A 3 T. In this work, we prove that at the double limit for the inviscid asymptotic, where both the Reynolds number Re converges to infinity and the initial value u ν (0) converges to Ae 1 in L 2 , the energy of layer separation cannot be more than u ν (T ) − Ae 1 2A 3 T (log Re) N . This result holds unconditionally for any limit of Leray-Hopf solutions of Navier-Stokes equation. Especially, it shows that, even if the limit is not unique, the shear flow pattern is observable up to time A −1 (log Re) −N . This provides a notion of stability despite the possible non-uniqueness of the limit predicted by the convex integration theory. The result relies on a new boundary vorticity estimate for the Navier-Stokes equation. This new estimate, inspired by previous work on higher regularity estimates for Navier-Stokes, provides a non-linear control rescalable through the inviscid limit.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Boundary Vorticity Estimates for Navier-Stokes and Application to the Inviscid Limit

Vasseur¹,

Yang²

2021

Preprint

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“…As an application, see our subsequent paper ( [VY20]) which improves second derivative estimates for three dimensional incompressible Navier-Stokes equations. In this paper, we only use the maximal function to give an alternative proof for the results of Choi and Vasseur in [CV14], as an example of using the maximal function to go from local to global.…”

Section: Introductionmentioning

confidence: 99%

Construction of Maximal Functions associated with Skewed Cylinders Generated by Incompressible Flows and Applications

Yang

2020

Preprint

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We construct a maximal function associated with a family of skewed cylinders. These cylinders, which are defined as tubular neighborhoods of trajectories of a mollified flow, appear in the study of fluid equations such as the Navier-Stokes equations and the Euler equations. We define a maximal function subordinate to these cylinders, and show it is of weak type (1, 1) and strong type (p, p) by a covering lemma. As an application, we give an alternative proof for the higher derivatives estimate of smooth solutions to the three-dimensional Navier-Stokes equations.

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Intermittency and Lower Dimensional Dissipation in Incompressible Fluids

De Rosa,

Isett

2024

Arch Rational Mech Anal

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In the context of incompressible fluids, the observation that turbulent singular structures fail to be space filling is known as “intermittency”, and it has strong experimental foundations. Consequently, as first pointed out by Landau, real turbulent flows do not satisfy the central assumptions of homogeneity and self-similarity in the K41 theory, and the K41 prediction of structure function exponents $$\zeta _p={p}/{3}$$ ζ p = p / 3 might be inaccurate. In this work we prove that, in the inviscid case, energy dissipation that is lower-dimensional in an appropriate sense implies deviations from the K41 prediction in every p-th order structure function for $$p>3$$ p > 3 . By exploiting a Lagrangian-type Minkowski dimension that is very reminiscent of the Taylor’s frozen turbulence hypothesis, our strongest upper bound on $$\zeta _p$$ ζ p coincides with the $$\beta $$ β -model proposed by Frisch, Sulem and Nelkin in the late 70s, adding some rigorous analytical foundations to the model. More generally, we explore the relationship between dimensionality assumptions on the dissipation support and restrictions on the p-th order absolute structure functions. This approach differs from the current mathematical works on intermittency by its focus on geometrical rather than purely analytical assumptions. The proof is based on a new local variant of the celebrated Constantin-E-Titi argument that features the use of a third order commutator estimate, the special double regularity of the pressure, and mollification along the flow of a vector field.

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Second Derivatives Estimate of Suitable Solutions to the 3D Navier–Stokes Equations

Cited by 8 publications

References 20 publications

Boundary Vorticity Estimates for Navier-Stokes and Application to the Inviscid Limit

Boundary Vorticity Estimates for Navier-Stokes and Application to the Inviscid Limit

Construction of Maximal Functions associated with Skewed Cylinders Generated by Incompressible Flows and Applications

Intermittency and Lower Dimensional Dissipation in Incompressible Fluids

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