Asymptotic Study of the 2D-DQGE Solutions

We study the regularity of the solutions of the surface quasi-geostrophic equation with subcritical exponent1/2<α≤1. We prove that if the initial data is small enough in the critical spaceH˙2-2α(R2), then the regularity of the solution is of exponential growth type with respect to time and itsH˙2-2α(R2)norm decays exponentially fast. It becomes then infinitely differentiable with respect to time and has value in all homogeneous Sobolev spacesH˙s(R2)fors≥2-2α. Moreover, we give some general properties of the global solutions.

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Almost critical local well-posedness for the space-time monopole equation in Lorenz gauge

Communications in Contemporary Mathematics ◽

10.1142/s0219199714500436 ◽

2015 ◽

Vol 17 (03) ◽

pp. 1450043 ◽

Cited By ~ 3

Author(s):

Achenef Tesfahun

Keyword(s):

Initial Data ◽

Critical Exponent ◽

Sobolev Spaces ◽

Lebesgue Space ◽

Space Time ◽

Well Posedness ◽

Critical Space ◽

Lorenz Gauge

Recently, Candy and Bournaveas proved local well-posedness of the space-time monopole equation in Lorenz gauge for initial data in Hs with [Formula: see text]. The equation is L2-critical, and hence a [Formula: see text] derivative gap is left between their result and the scaling prediction. In this paper, we consider initial data in the Fourier–Lebesgue space [Formula: see text] for 1 < p ≤ 2 which coincides with Hs when p = 2 but scales like lower regularity Sobolev spaces for 1 < p < 2. In particular, we will see that as p → 1+, the critical exponent [Formula: see text], in which case [Formula: see text] is the critical space. We shall prove almost optimal local well-posedness to the space-time monopole equation in Lorenz gauge with initial data in the aforementioned spaces that correspond to p close to 1.

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Viscous Flow Around a Rigid Body Performing a Time-periodic Motion

Journal of Mathematical Fluid Mechanics ◽

10.1007/s00021-021-00556-4 ◽

2021 ◽

Vol 23 (1) ◽

Author(s):

Thomas Eiter ◽

Mads Kyed

Keyword(s):

Rigid Body ◽

Sobolev Spaces ◽

Periodic Motion ◽

Viscous Incompressible Fluid ◽

Body Motion ◽

Translational Velocity ◽

Ill Posed ◽

Homogeneous Sobolev Spaces ◽

The Mean ◽

Time Periodic

AbstractThe equations governing the flow of a viscous incompressible fluid around a rigid body that performs a prescribed time-periodic motion with constant axes of translation and rotation are investigated. Under the assumption that the period and the angular velocity of the prescribed rigid-body motion are compatible, and that the mean translational velocity is non-zero, existence of a time-periodic solution is established. The proof is based on an appropriate linearization, which is examined within a setting of absolutely convergent Fourier series. Since the corresponding resolvent problem is ill-posed in classical Sobolev spaces, a linear theory is developed in a framework of homogeneous Sobolev spaces.

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Bilinear forms on non-homogeneous Sobolev spaces

Forum Mathematicum ◽

10.1515/forum-2019-0311 ◽

2020 ◽

Vol 32 (4) ◽

pp. 995-1026

Author(s):

Carme Cascante ◽

Joaquín M. Ortega

Keyword(s):

Sobolev Spaces ◽

Bilinear Form ◽

Positive Measure ◽

Bilinear Forms ◽

Homogeneous Sobolev Spaces

AbstractIn this paper, we show that if {b\in L^{2}(\mathbb{R}^{n})}, then the bilinear form defined on the product of the non-homogeneous Sobolev spaces {H_{s}^{2}(\mathbb{R}^{n})\times H_{s}^{2}(\mathbb{R}^{n})}, {0<s<1}, by(f,g)\in H_{s}^{2}(\mathbb{R}^{n})\times H_{s}^{2}(\mathbb{R}^{n})\to\int_{% \mathbb{R}^{n}}(\mathrm{Id}-\Delta)^{\frac{s}{2}}(fg)(\mathbf{x})b(\mathbf{x})% \mathop{}\!d\mathbf{x}is continuous if and only if the positive measure {\lvert b(\mathbf{x})\rvert^{2}\mathop{}\!d\mathbf{x}} is a trace measure for {H_{s}^{2}(\mathbb{R}^{n})}.

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