Local‐in‐Time Existence and Uniqueness of Solutions to the Prandtl Equations by Energy Methods

By Nader Masmoudi, Tak Kwong Wong. 2012

267 citations recorded by the source corpus.

We prove local existence and uniqueness for the two‐dimensional Prandtl system in weighted Sobolev spaces under Oleinik's monotonicity assumption. In particular we do not use the Crocco transform or any change of variables. Our proof is based on a new nonlinear energy estimate for the Prandtl system. This new energy estimate is based on a cancellation property that is valid under the monotonicity assumption. To construct the solution, we use a regularization of the system that preserves this nonlinear structure. This new nonlinear structure may give some insight into the convergence properties from the Navier‐Stokes system to the Euler system when the viscosity goes to 0. © 2015 Wiley Periodicals, Inc.

Subfields: Applied Mathematics, Computational Mechanics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e7eb0b7bc9ec0e4ede05dd6bbd2106ae724e6e59

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In Applied Mathematics

In Computational Mechanics

In Mathematical Physics

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”
Almost Global Existence for the Prandtl Boundary Layer Equations (2015, methodology): “More recently in [32], Masmoudi and Wong use solely energy methods and a new change of variables to prove the local in time existence and uniqueness in weighted Sobolev spaces, under Oleinik’s monotonicity assumption.”
Well-posedness of the Prandtl equation without any structural assumption (2018, methodology): “Inspired by [27], we consider the regularized equation ∂tu+ (u∂x + v∂y)u+ (U e ǫ ∂x + V e ǫ ∂y)u+ (u∂x + v∂y)U e ǫ − ǫ∂(2) xu− ∂(2) yu = f e ǫ , (40) adding a tangential diffusion −ǫ∂2 xu.”
On the inviscid limit of the Navier-Stokes equations (2014, unknown): “This result goes back to Oleinik [Ole66], and we refer to [MW12a] for an elegant Sobolev energy-based proof.”
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.”
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”
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 of the hydrostatic Navier–Stokes equations (2018, background): “5) has local in time Sobolev solutions [34, 33].”
The Inviscid Limit of Navier–Stokes Equations for Analytic Data on the Half-Space (2017, unknown): “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).”
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.”