Well-Posedness of the Prandtl Equations Without Any Structural Assumption

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].

Real Analytic Local Well‐Posedness for the Triple Deck

“…appearing in the weight above is bounded. It would be interesting to determine whether one can relax the growth rate as y 3 I of , for instance, by following the strategy of [13] or of [41]. We denote the Fourier transform a function f in the x-variable only, at frequency P R, as f h f .y; t /.…”

Global Existence and the Decay of Solutions to the Prandtl System with Small Analytic Data

“…Lately, Gérvard-Varet and Masmoudi [11] proved the well-posedness of (1.1) for a class of data with Gevrey regularity. This result was improved to be optimal in sense of [10] in [7] by Dietert and Gérvard-Varet. The question of the long time existence for Prandtl system with small analytic data was first addressed in [28] and an almost global existence result was provided in [15].…”

Well-posedness in Sobolev spaces of the two-dimensional MHD boundary layer equations without viscosity

“…• Without any structural assumption on initial data the well-posedness for 2D and 3D MHD boundary systems was established in Gevrey space by the first author and T. Yang [15] with Gevrey index up to 3/2, and it remains interesting to relax the Gevrey index therein to 2 inspired the previous works of [4,12] on the well-posedness for the Prandtl equations in Gevrey space with optimal index 2. • Under the structural assumption that the tangential magnetic field dominates, i.e., f = 0, the well-posedness in weighted Sobolev space was established by Liu-Xie-Yang [18] and Liu-Wang-Xie-Yang [16] without Oleinik's monotonicity assumption, where the two cases that with both viscosity and resistivity and with only viscosity are considered, respectively; see also the work of Gérard-Varet and Prestipino [8] for the stability analysis of the MHD boundary layer system with insulating boundary conditions (i.e.…”

“…The mathematical study on the Prandtl boundary layer has a long history, and there have been extensive works concerning its well/ill-posedness theories. So far the twodimensional (2D) Prandtl equation is well-explored in various function spaces, see, e.g., [1,2,4,5,6,7,9,10,11,13,14,22,23,24] and the references therein. Compared with the Prandtl equation the treatment is more complicated since we have a new difficulty caused by the additional loss of tangential derivative in the magnetic field.…”