Well-posedness of the Prandtl equation in Sobolev spaces

By Radjesvarane Alexandre, Ya-Guang Wang, Chao-Jiang Xu, Tong Yang. 2012

230 citations recorded by the source corpus.

We develop a new approach to study the well-posedness theory of the Prandtl equation in Sobolev spaces by using a direct energy method under a monotonicity condition on the tangential velocity field instead of using the Crocco transformation. Precisely, we firstly investigate the linearized Prandtl equation in some weighted Sobolev spaces when the tangential velocity of the background state is monotonic in the normal variable. Then to cope with the loss of regularity of the perturbation with respect to the background state due to the degeneracy of the equation, we apply the Nash-Moser-Hormander iteration to obtain a well-posedness theory of classical solutions to the nonlinear Prandtl equation when the initial data is a small perturbation of a monotonic shear flow.

Subfields: Applied Mathematics, Computational Mechanics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/45752d570c8054b8ab793f5f924d2021b8560236

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Citations in the corpus

Gevrey stability of Prandtl expansions for 2-dimensional Navier–Stokes flows (2016, methodology): “e proof is based on a change of variables and unknowns called Crocco transform. Recently, the wellposedness theory for Y-increasing data was revisited in the Eulerian form (1.0.3) and Sobolev setting [1, 19]. • Without themonotonicityassumption, the situationis muchlessfavorable. Existenceof local i”
Well-posedness of the Prandtl equation without any structural assumption (2018, methodology): “More recently, the local wellposedness result of Oleinik was revisited using the standard Eulerian form of the equation, see [1, 27, 21] for the local theory in Sobolev spaces.”
Local‐in‐Time Existence and Uniqueness of Solutions to the Prandtl Equations by Energy Methods (2012, background): “Finally, let us mention that the new preprint [1] also considers the existence for the Prandtl system in physical space.”
On the Inviscid Limit Problem of the Vorticity Equations for Viscous Incompressible Flows in the Half‐Plane (2012, methodology): “t; x1; Y2/dY2: Under the assumptions on the monotonicity of the data the solvability of the Prandtl equations is proved by [41, 46, 57] using the Crocco transformation, and recently also by [2, 39] whose proofs are based on a direct energy method.”
Well‐Posedness in Gevrey Function Space for 3D Prandtl Equations without Structural Assumption (2020, methodology): “The cancellations in the convection terms were observed in recent years by two research groups independently [1,28] to overcome the difficulty of the loss of derivatives in the system.”
Global existence and decay of solutions to Prandtl system with small analytic data. (2019, unknown): “Recently, by ingenious use of the cancelation property of the bad terms containing the tangential derivative, the authors of [1] and [17] succeeded in proving the existence of local smooth solution to (1.”
The Inviscid Limit for the Navier–Stokes Equations with Data Analytic Only Near the Boundary (2019, methodology): “We refer the reader to [1, 10, 17, 28, 35, 36, 37, 39, 47, 51, 52] for the well-posedness theory for the Prandtl equations, to [14, 26, 18, 38] for the identification of ill-posed regimes, and to [19, 20, 21, 22, 23] for recent works which show the invalidity of the Prandtl expansion at the level of”
Well-posedness for the Prandtl system without analyticity or monotonicity (2013, methodology): “The novelty in [20] is to obtain such well-posedness result without using the Crocco transform, namely performing Sobolev estimates in the original Eulerian formulation (see also [1] where estimates on the linearized problem were done).”
The Inviscid Limit of Navier–Stokes Equations for Analytic Data on the Half-Space (2017, background): “ers of √ ν. Since then, the Prandtl boundary layers have been intensively studied in the mathematical literature. Notably, solutions to the Prandtl equations have been constructed for monotonic data ([31, 32, 1, 29]) or data with Gevrey or analytic regularity ([35, 11, 22], among others). In the cas”