Global existence, regularity, and uniqueness of infinite energy solutions to the Navier-Stokes equations

Bradshaw, Zachary; Tsai, Tai‐Peng

doi:10.1080/03605302.2020.1761386

Cited by 24 publications

(26 citation statements)

References 39 publications

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“…1.4 for a definition. 8 Under these assumptions and by interpolation, we infer that u is a 'smooth solution with sufficient decay' in the sense of the definition in Sect. 1.4.…”

Section: Introductionmentioning

confidence: 89%

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Quantitative Regularity for the Navier–Stokes Equations Via Spatial Concentration

Barker

Prange

2021

Commun. Math. Phys.

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This paper is concerned with quantitative estimates for the Navier–Stokes equations. First we investigate the relation of quantitative bounds to the behavior of critical norms near a potential singularity with Type I bound $$\Vert u\Vert _{L^{\infty }_{t}L^{3,\infty }_{x}}\le M$$ ‖ u ‖ L t ∞ L x 3 , ∞ ≤ M . Namely, we show that if $$T^*$$ T ∗ is a first blow-up time and $$(0,T^*)$$ ( 0 , T ∗ ) is a singular point then $$\begin{aligned} \Vert u(\cdot ,t)\Vert _{L^{3}(B_{0}(R))}\ge C(M)\log \Big (\frac{1}{T^*-t}\Big ),\,\,\,\,\,\,R=O((T^*-t)^{\frac{1}{2}-}). \end{aligned}$$ ‖ u ( · , t ) ‖ L 3 ( B 0 ( R ) ) ≥ C ( M ) log ( 1 T ∗ - t ) , R = O ( ( T ∗ - t ) 1 2 - ) . We demonstrate that this potential blow-up rate is optimal for a certain class of potential non-zero backward discretely self-similar solutions. Second, we quantify the result of Seregin (Commun Math Phys 312(3):833–845, 2012), which says that if u is a smooth finite-energy solution to the Navier–Stokes equations on $${\mathbb {R}}^3\times (0,1)$$ R 3 × ( 0 , 1 ) with $$\begin{aligned} \sup _{n}\Vert u(\cdot ,t_{(n)})\Vert _{L^{3}({\mathbb {R}}^3)}<\infty \,\,\,\text {and}\,\,\,t_{(n)}\uparrow 1, \end{aligned}$$ sup n ‖ u ( · , t ( n ) ) ‖ L 3 ( R 3 ) < ∞ and t ( n ) ↑ 1 , then u does not blow-up at $$t=1$$ t = 1 . To prove our results we develop a new strategy for proving quantitative bounds for the Navier–Stokes equations. This hinges on local-in-space smoothing results (near the initial time) established by Jia and Šverák (2014), together with quantitative arguments using Carleman inequalities given by Tao (2019). Moreover, the technology developed here enables us in particular to give a quantitative bound for the number of singular points in a Type I blow-up scenario.

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“…1.4 for a definition. 8 Under these assumptions and by interpolation, we infer that u is a 'smooth solution with sufficient decay' in the sense of the definition in Sect. 1.4.…”

Section: Introductionmentioning

confidence: 89%

“…Furthermore, C weak ∈ [1, ∞) is a universal constant from the embedding L 3,∞ (R 3 ) ⊂ L 2 uloc (R 3 ). See, for example, Lemma 6.2 in Bradshaw and Tsai's paper [8]. Assume that…”

Section: Backward Propagation Of Concentrationmentioning

confidence: 99%

Quantitative Regularity for the Navier–Stokes Equations Via Spatial Concentration

Barker

Prange

2021

Commun. Math. Phys.

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“…Various reformulations of local Leray solutions in L 2 uloc have been provided, such as Kikuchi and Seregin in 2007 [15] or Bradshaw and Tsai in 2019 [4]. The formulas proposed for the pressure, however, are actually equivalent, as they all imply that u is solution to the (MNS) problem.…”

Section: Solutions In Lmentioning

confidence: 99%

Characterisation of the pressure term in the incompressible Navier–Stokes equations on the whole space

Fernández-Dalgo¹,

Lemarié–Rieusset²

2021

DCDS-S

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“…Self-similar (SS) solutions, i.e., solutions invariant with respect to the scaling of (1.1) for all scaling factors λ > 0, are noteworthy candidates for the non-uniqueness of Leray-Hopf weak solutions, as demonstrated by [JS1,GuS]. On the other hand, discretely selfsimilar (DSS) solutions, i.e., solutions that satisfy the scaling invariance possibly only for some λ > 1, are candidates for the failure of eventual regularity [BT1,BT4]. For small data, the existence and uniqueness of such solutions follow easily from the classical well-posedness results; see [KT] and the references therein.…”

Section: Figure 1 the Cover Of Rmentioning

confidence: 99%

“…For global existence, these works assume some type of decay of the initial data as |x| → ∞, either pointwise decay of a locally determined quantity, the decay of the L 2 norm confined to balls of unit radius, or the decay the oscillation computed over balls of unit radius. In [LR2,BT4,BKT,FL2,KwT], existence results are given in weighted spaces, which allow for lack of decay in some directions. The papers [BKT,FL2] additionally allow for growth in some directions.…”

mentioning

confidence: 99%

Global weak solutions of the Navier-Stokes equations for intermittent initial data in half-space

Bradshaw¹,

Kukavica²,

Ożański³

2020

Preprint

Self Cite

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We prove existence of global-in-time weak solutions of the incompressible Navier-Stokes equations in the halfspace R 3 + with initial data in a weighted space that allow non-uniformly locally square integrable functions that grow at spatial infinity in an intermittent sense. The space for initial data is built on cubes whose sides R are proportional to the distance to the origin and the square integral of the data is allowed to grow as a power of R. The existence is obtained via a new a priori estimate and stability result in the weighted space, as well as new pressure estimates. Also, we prove eventual regularity of such weak solutions, up to the boundary, for (x, t) satisfying t > c 1 |x| 2 + c 2 , where c 1 , c 2 > 0, for a large class of initial data u 0 , with c 1 arbitrarily small. As an application of the existence theorem, we construct global discretely self-similar solutions, thus extending the theory on the half-space to the same generality as the whole space.

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Global existence, regularity, and uniqueness of infinite energy solutions to the Navier-Stokes equations

Cited by 24 publications

References 39 publications

Quantitative Regularity for the Navier–Stokes Equations Via Spatial Concentration

Quantitative Regularity for the Navier–Stokes Equations Via Spatial Concentration

Characterisation of the pressure term in the incompressible Navier–Stokes equations on the whole space

Global weak solutions of the Navier-Stokes equations for intermittent initial data in half-space

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