On the global existence of solutions to the Prandtl's system

By Zhouping Xin, Liqun Zhang. 2004

225 citations recorded by the source corpus.

Abstract In this paper we establish a global existence of weak solutions to the two-dimensional Prandtl's system for unsteady boundary layers in the class considered by Oleinik (J. Appl. Math. Mech. 30 (1966) 951) provided that the pressure is favourable. This generalizes the local well-posedness results due to Oleinik (1966; Mathematical Models in Boundary Layer Theory, Chapman & Hall, London, 1999). For the proof, we introduce a viscous splitting method so that the asymptotic behaviour of the solution near the boundary can be estimated more accurately by methods applicable to the degenerate parabolic equations.

Subfields: Computational Mechanics, Computational Theory and Mathematics, Control and Systems Engineering

View Paper: https://www.semanticscholar.org/paper/9a6f0c3a13c5ca4a099f316f08c913b7b7e77dc6

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In Computational Mechanics

In Computational Theory and Mathematics

In Control and Systems Engineering

Citations in the corpus

Local‐in‐Time Existence and Uniqueness of Solutions to the Prandtl Equations by Energy Methods (2012, background): “These solutions can be extended as global weak solutions if the pressure gradient is favorable (∂xp ≤ 0) [33, 34].”
Global existence and decay of solutions to Prandtl system with small analytic data. (2019, unknown): “[21] M. Paicu, P. Zhang and Z. Zhang, On the hydrostatic approximation of the Navier-Stokes equations in a thin strip, arXiv:1904.04438.”
Well-posedness of the Prandtl equation in Sobolev spaces (2012, methodology): “In addition to Oleinik’s monotonicity assumption on the velocity field, by imposing a so called favorable condition on the pressure, Xin and Zhang obtained the existence of global weak solutions to the Prandtl equation in [22].”
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.”
On the local existence of analytic solutions to the Prandtl boundary layer equations (2013, unknown): “[28, 29, 37]); or analyticity in thex variable, Sobolev regularity in they variable, and exponential decay of u(x, Y, t) − U(x, t) asY → ∞ (cf.”
Well-posedness of the hydrostatic Navier–Stokes equations (2018, background): “n Sobolev spaces [12, 15]. Local in time wellposedness can be achieved when the initial datum is real analytic [39, 25], and even under the milder condition of Gevrey regularity in x[14]. We refer to [10, 42, 13, 20, 26, 9] and references therein for more results on the Prandtl system such as singul”
Almost Global Existence for the Prandtl Boundary Layer Equations (2015, background): “Xin and Zhang prove in [40] that if the pressure gradient has a favorable sign, that is ∂x pE (t, x) 0 for all t > 0 and x ∈ R, and the initial condition uP 0 of Prandtl is monotone in y, the solutions are global.”
Well‐Posedness in Gevrey Function Space for 3D Prandtl Equations without Structural Assumption (2020, background): “Let us also mention the work [33] on the global existence of weak solutions under an additional favorable pressure condition.”
Vanishing Viscous Limits for 3D Navier–Stokes Equations with a Navier-Slip Boundary Condition (2012, background): “tions for boundary value problem of the 2-dimensional Prandtl’s equations [25]. In this case, the existence and uniqueness of global solutions to the Prandtl’s equations was established by Xin, Zhang [37](also see [36]). In [27], Sammartino and Caﬂisch obtained the local existence of the analytic so”