Zero Viscosity Limit for Analytic Solutions, of the Navier-Stokes Equation on a Half-Space.¶I. Existence for Euler and Prandtl Equations

Sammartino, M.; Caflisch, Russel E.

doi:10.1007/s002200050304

Cited by 391 publications

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“…In fact we show that the time of existence of a regular solution does not depend on the boundary layer solution whereas in [4] and [5] the size of the domain of analyticity was shrinking at each step of the asymptotic expansion.…”

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confidence: 80%

“…Our analysis will strictly follow Sammartino and Caflish ( [4] and [5]). In their papers the authors proved that the solution of the Navier-Stokes equations with analytic initial data can be decomposed in the form of an asymptotic series.…”

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confidence: 99%

“…In Section 3 we will state the Abstract Cauchy-Kovalevskaya Theorem (ACK) in the form proposed by Safonov [3]. In Section [4] we shall introduce the Navier-Stokes operator we will need to put the Navier-Stokes equations in a suitable form for the application of the ACK theorem. Through the NavierStokes operator the Navier-Stokes equations will be solved in Section 5 and the iterative procedure of the ACK theorem will be shown to converge to a unique solution.…”

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confidence: 99%

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Analytic solutions of the Navier-Stokes equations

Lombardo

2001

Rend. Circ. Mat. Palermo

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Analytic solutions of the Navier-Stokes equations

Lombardo

2001

Rend. Circ. Mat. Palermo

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“…The stationary Prandtl system has been widely studied [12]. Despite its importance in engeneering applications [7], [15], very few results are known for the existence of (global in time) solutions of the instationary system [12,13,14]. In both cases, the stationary and the instationary one, a possible way to tackle the problem consists in using the so-called Crocco transformation [12].…”

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The Linearized Crocco Equation

Buchot

Raymond

2006

J. math. fluid mech.

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Abstract. In this paper, we study the existence and uniqueness of a degenerate parabolic equation, with nonhomogeneous boundary conditions, coming from the linearization of the Crocco equation [12]. The Crocco equation is a nonlinear degenerate parabolic equation obtained from the Prandtl equations with the so-called Crocco transformation. The linearized Crocco equation plays a major role in stabilization problems of fluid flows described by the Prandtl equations [5]. To study the infinitesimal generator associated with the adjoint linearized Crocco equation -with homogeneous boundary conditions -we first study degenerate parabolic equations in which the x-variable plays the role of a time variable. This equation is doubly degenerate: the coefficient in front of ∂x vanishes on a part of the boundary, and the coefficient of the elliptic operator vanishes in another part of the boundary. This makes very delicate the proof of uniqueness of solution. To overcome this difficulty, a uniqueness result is first obtained for an equation in which the elliptic operator is symmetric, and it is next extended to the original equation by combining an iterative process and a fixed point argument (see Th. 4.9). This kind of argument is also used to prove estimates, which cannot be obtained in a classical way. Mathematics Subject Classification (2000). Primary 35K65 ; Secondary 35Q35, 76D10.

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“…(due to the compatibility conditions), and on the usual estimates on the heat operators given, e.g., in [3,6]. Notice the loss of regularity (one derivative) in the radial component due to the incompressibility condition.…”

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Asymptotic Analysis of the Linearized Navier–stokes Equation on an Exterior Circular Domain: Explicit Solution and the Zero Viscosity Limit

Lombardo

Caflisch

Sammartino

2001

Communications in Partial Differential Equations

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In this paper we study and derive explicit formulas for the linearized Navier-Stokes equations on an exterior circular domain in space dimension two. Through an explicit construction, the solution is decomposed into an inviscid solution, a boundary layer solution and a corrector. Bounds on these solutions are given, in the appropriate Sobolev spaces, in terms of the norms of the initial and boundary data. The correction term is shown to be of the same order of magnitude as the square root of the viscosity.

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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

Cited by 391 publications

References 10 publications

Analytic solutions of the Navier-Stokes equations

Analytic solutions of the Navier-Stokes equations

The Linearized Crocco Equation

Asymptotic Analysis of the Linearized Navier–stokes Equation on an Exterior Circular Domain: Explicit Solution and the Zero Viscosity Limit

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