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Well-posedness of the Prandtl equation without any structural assumption
Helge Dietert,David Gérard-Varet +1 more
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In this paper, the authors show the local in time well-posedness of the Prandtl equation for data with Gevrey $2$ regularity in $x$ and H^1$ regularities in $y$.Abstract:
We show the local in time well-posedness of the Prandtl equation for data with Gevrey $2$ regularity in $x$ and $H^1$ regularity in $y$. The main novelty of our result is that we do not make any assumption on the structure of the initial data: no monotonicity or hypothesis on the critical points. Moreover, our general result is optimal in terms of regularity, in view of the ill-posedness result of [9].read more
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The Inviscid Limit for the Navier–Stokes Equations with Data Analytic Only Near the Boundary
TL;DR: In this paper, the inviscid limit for the Navier-Stokes equations in a half space, with initial datum that is analytic only close to the boundary of the domain, and that has Sobolev regularity in the complement, was studied.
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On the effect of rotation on the life-span of analytic solutions to the $3D$ inviscid primitive equations
TL;DR: Ghoul, Tej-Eddine, Ibrahim, Slim, Lin, Quyuan; Titi, Edriss S as mentioned in this paper studied the effect of the rotation on the life-span of solutions to the inviscid primitive equations (PEs).
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Well-posedness in Gevrey function space for 3D Prandtl equations without Structural Assumption
TL;DR: In this paper, the authors established the well-posedness in Gevrey function space with optimal class of regularity 2 for the three dimensional Prandtl system without any structural assumption.
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Magnetic effects on the solvability of 2D MHD boundary layer equations without resistivity in Sobolev spaces.
TL;DR: In this article, the magnetic effect on the Sobolev solvability of boundary layer equations for the 2D incompressible viscous MHD system without resistivity was investigated. And the authors showed that if the tangential magnetic field shear layer is degenerate at one point, then the linearized MHD boundary layer system around the shear surface profile is ill-posed in the SINR settings provided that the initial velocity shear flow is non-degenerately critical at the same point.
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Finite-time blowup and ill-posedness in Sobolev spaces of the inviscid primitive equations with rotation
TL;DR: In this article, the inviscid PEs with rotation are shown to be ill-posed in Sobolev spaces in the sense that their perturbation around a certain steady state background flow is both linearly and nonlinearly illposed.
References
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Gevrey class regularity for the solutions of the Navier-Stokes equations
Ciprian Foias,Roger Temam +1 more
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Zero Viscosity Limit for Analytic Solutions, of the Navier-Stokes Equation on a Half-Space.I. Existence for Euler and Prandtl Equations
TL;DR: In this article, the authors prove short time existence theorems for the Euler and Prandtl equations with analytic initial data in either two or three spatial dimensions, using abstract Cauchy-Kowalewski theorem.
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Zero Viscosity Limit for Analytic Solutions of the Navier-Stokes Equation on a Half-Space. II. Construction of the Navier-Stokes Solution
TL;DR: In this article, the Navier-Stokes solution is constructed through a composite asymptotic expansion involving the solutions of the Euler and Prandtl equations, plus an error term.
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On the nonlinear instability of Euler and Prandtl equations
Abstract: In this paper we give examples of nonlinearly unstable solutions of Euler equations in the whole space ℝ2, the half space ℝ × ℝ+, the periodic strip ℝ × , the strip ℝ × [−1,1], and the periodic torus 2, with an application to vortex sheets. Using the same methods, we prove an instability result for Prandtl-type boundary layers that appear in ℝ × ℝ+ and × ℝ+. © 2000 John Wiley & Sons, Inc.
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