On certain models in the PDE theory of fluid flows

Šverák, Vladimír

doi:10.5802/jedp.658

Cited by 5 publications

(12 citation statements)

References 42 publications

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“…Thus, Theorem 2 implies that the advection in (1.2) can prevent singularity formation in the CLM model or the gCLM model for such initial data. Theorem 3 resolves the conjecture made in [19,44] that (1.2) develops a finite time singularity from initial data ω 0 ∈ C α or ω 0 ∈ H s for any α ∈ (0, 1) and s < 3 2 in the case of S 1 . The case of R has been resolved in [9] with ω 0 ∈ C ∞ c .…”

Section: Introductionsupporting

confidence: 80%

On the regularity of the De Gregorio model for the 3D Euler equations

Chen¹

2021

Preprint

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We study the regularity of the De Gregorio (DG) model ωt + uωx = uxω on S 1 for initial data ω 0 with period π and in class X: ω 0 is odd and ω 0 ≤ 0 (or ω 0 ≥ 0) on [0, π/2]. These sign and symmetry properties are the same as those of the smooth initial data that lead to singularity formation of the De Gregorio model on R or the generalized Constantin-Lax-Majda (gCLM) model on R or S 1 with a positive parameter. Thus, to establish global regularity of the DG model for general smooth initial data, which is a conjecture on the DG model, an important step is to rule out potential finite time blowup from smooth initial data in X. We accomplish this by establishing a one-point blowup criterion and proving global well-posedness for C 1,α initial data with any α ∈ (0, 1). On the other hand, for any α ∈ (0, 1), we construct a finite time blowup solution from a class of initial data with ω 0 ∈ C α ∩ C ∞ (S 1 \{0}) ∩ X. Our results imply that singularities developed in the DG model and the gCLM model on S 1 can be prevented by stronger advection.

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Section: Introductionsupporting

confidence: 80%

On the regularity of the De Gregorio model for the 3D Euler equations

Chen¹

2021

Preprint

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“…There are various 1D models proposed in the literature. We refer to [14,22,33] for excellent surveys of other 1D models for the 3D Euler equations, surface quasi-geostrophic equation and other equations. Throughout this paper, we call (1.1) the generalized Constantin-Lax-Majda equation (gCLM).…”

Section: Introductionmentioning

confidence: 99%

“…al. [33] and [25] proved that the equilibrium A sin(2(x − x 0 )) of the De Gregorio equation on the circle is nonlinearly stable.…”

Section: Introductionmentioning

confidence: 99%

Singularity formation and global Well-posedness for the generalized Constantin–Lax–Majda equation with dissipation

Chen

2020

Nonlinearity

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We study a generalization due to De Gregorio and Wunsch et.al. of the Constantin-Lax-Majda equation (gCLM) on the real linewhere H is the Hilbert transform and Λ = (−∂xx) 1/2 . We use the method in [5] to prove finite time self-similar blowup for a close to 1 2 and γ = 2 by establishing nonlinear stability of an approximate self-similar profile. For a > −1, we discuss several classes of initial data and establish global well-posedness and an one-point blowup criterion for different initial data. For a ≤ −1, we prove global well-posedness for gCLM with critical and supercritical dissipation.

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“…Perelman first considered parabolic theory as a high-dimensional limit of elliptic theory in [47]. This general principle was discussed in the blog of Tao [52], modified in the coursenotes of Sverak [51], then developed and applied in [17]. In our setting, we follow the ideas from [17]; namely, we use classical probabilistic formulae, essentially going back to Wiener [54], with a slight modification used by Sverak in [51].…”

Section: Introductionmentioning

confidence: 99%

“…This general principle was discussed in the blog of Tao [52], modified in the coursenotes of Sverak [51], then developed and applied in [17]. In our setting, we follow the ideas from [17]; namely, we use classical probabilistic formulae, essentially going back to Wiener [54], with a slight modification used by Sverak in [51]. However, to account for the presence of variable-coefficients, we have modified (and complicated) the change of variables formula from [17].…”

Section: Introductionmentioning

confidence: 99%

Parabolic Theory as a High-Dimensional Limit of Elliptic Theory

Davey

2017

Arch Rational Mech Anal

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The aim of this article is to show how certain parabolic theorems follow from their elliptic counterparts. This technique is demonstrated through new proofs of five important theorems in parabolic unique continuation and the regularity theory of parabolic equations and geometric flows. Specifically, we give new proofs of an L 2 Carleman estimate for the heat operator, and the monotonicity formulas for the frequency function associated to the heat operator, the two-phase free boundary problem, the flow of harmonic maps, and the mean curvature flow. The proofs rely only on the underlying elliptic theorems and limiting procedures belonging essentially to probability theory. In particular, each parabolic theorem is proved by taking a highdimensional limit of the related elliptic result.

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On certain models in the PDE theory of fluid flows

Cited by 5 publications

References 42 publications

On the regularity of the De Gregorio model for the 3D Euler equations

On the regularity of the De Gregorio model for the 3D Euler equations

Singularity formation and global Well-posedness for the generalized Constantin–Lax–Majda equation with dissipation

Parabolic Theory as a High-Dimensional Limit of Elliptic Theory

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