Gevrey-Class-3 Regularity of the Linearised Hyperbolic Prandtl System on a Strip

De Anna, Francesco; Kortum, Joshua; Scrobogna, Stefano

doi:10.1007/s00021-023-00821-8

Cited by 3 publications

(1 citation statement)

References 34 publications

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“…Gevrey spaces have recently attracted attention in the mathematical community, particularly in relation to wellposedness issues in models involving boundary layers (cf. [13][14][15][16][17][18][19]). Roughly speaking, the initial data belongs to the Gevrey-class 3/2 if the corresponding Fourier coefficients decay as e −σ |k| 2/3 , when the frequencies diverge.…”

Section: Gevrey-type Solutions Of the Linearized Modelmentioning

confidence: 99%

Existence of solutions for stress-rate type strain-limiting viscoelasticity in Gevrey spaces

Bachmann,

De Anna,

Schlömerkemper

et al. 2023

Phil. Trans. R. Soc. A.

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In this work, we deal with a one-dimensional stress-rate type model for the response of viscoelastic materials, in relation to the strain-limiting theory. The model is based on a constitutive relation of stress-rate type. Unlike classical models in elasticity, the unknown of the model under consideration is uniquely the stress, avoiding the use of the deformation. Here, we treat the case of periodic boundary conditions for a linearized model. We determine an optimal function space that ensures the local existence of solutions to the linearized model around certain steady states. This optimal space is known as the Gevrey-class 3 / 2 , which characterizes the regularity properties of the solutions. The exponent 3 / 2 in the Gevrey-class reflects the specific dispersion properties of the equation itself. This article is part of the theme issue ‘Foundational issues, analysis and geometry in continuum mechanics’.

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Section: Gevrey-type Solutions Of the Linearized Modelmentioning

confidence: 99%

Existence of solutions for stress-rate type strain-limiting viscoelasticity in Gevrey spaces

Bachmann,

De Anna,

Schlömerkemper

et al. 2023

Phil. Trans. R. Soc. A.

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Global‐in‐time well‐posedness of solutions for the 2D hyperbolic Prandtl equations in an analytic framework

Dong

2024

Math Methods in App Sciences

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Quantitative aspects on the ill-posedness of the Prandtl and hyperbolic Prandtl equations

De Anna,

Kortum,

Scrobogna

2024

Z. Angew. Math. Phys.

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We address the Prandtl equations and a physically meaningful extension known as hyperbolic Prandtl equations. For the extension, we show that the linearised model around a non-monotonic shear flow is ill-posed in any Sobolev spaces. Indeed, shortly in time, we generate solutions that experience a dispersion relation of order $$\root 3 \of {k}$$ k 3 in the frequencies of the tangential direction, akin the pioneering result of Gérard-Varet, D., Dormy, E.: On the ill-posedness of the Prandtl equation. J. Am. Math. Soc., 23(2), 591–609 (2010) for Prandtl (where the dispersion was of order $$\sqrt{k}$$ k ). We emphasise, however, that this growth rate does not imply (a-priori) ill-posedness in Gevrey-class m, with $$m>3$$ m > 3 . We relate these aspects to the original Prandtl equations in Gevrey-class m, with $$m>2$$ m > 2 : We show that the result in Gérard-Varet, D., Dormy, E.: On the ill-posedness of the Prandtl equation. J. Am. Math. Soc., 23(2), 591–609 (2010) determines a dispersion relation of order $$\sqrt{k}$$ k for a short time proportional to $$\ln (\sqrt{k})/\sqrt{k}$$ ln ( k ) / k . Therefore, the ill-posedness in Gérard-Varet, D., Dormy, E.: On the ill-posedness of the Prandtl equation. J. Am. Math. Soc., 23(2), 591–609 (2010) in its generality is momentarily constrained to Sobolev spaces rather than extending to the Gevrey classes.

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Gevrey-Class-3 Regularity of the Linearised Hyperbolic Prandtl System on a Strip

Cited by 3 publications

References 34 publications

Existence of solutions for stress-rate type strain-limiting viscoelasticity in Gevrey spaces

Existence of solutions for stress-rate type strain-limiting viscoelasticity in Gevrey spaces

Global‐in‐time well‐posedness of solutions for the 2D hyperbolic Prandtl equations in an analytic framework

Quantitative aspects on the ill-posedness of the Prandtl and hyperbolic Prandtl equations

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