Existence Theorem for the Flow of an Ideal Incompressible Fluid in Two Dimensions

Schaeffer, A. C.

doi:10.2307/1989741

Cited by 5 publications

(6 citation statements)

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“…Then one obtains easily that u is bounded in C1,a (see Wolibner [14], Schaeffer [13] and Kato [11] for details), but the bounds involve constants, like those in (2.14), rapidly growing with /. Nevertheless, for smooth data we have m E WXco(Q).…”

Section: Application To the Euler Equation In Two Dimensionsmentioning

confidence: 97%

The Vortex Method with Finite Elements

Bardos¹,

Bercovier²,

Pironneau³

1981

Mathematics of Computation

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Abstract. This work shows that the method of charcteristics is well suited for the numerical solution of first order hyperbolic partial differential equations whose coefficients are approximated by functions piecewise constant on a finite element triangulation of the domain of integration. We apply this method to the numerical solution of Euler's equation and prove convergence when the time step and the mesh size tend to zero. The proof is based upon the results of regularity given by Kato and Wolibner and on L°° estimates for the solution of the Dirichlet problem given by Nitsche. The method obtained belongs to the family of vortex methods usually studied in a finite difference context.

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Section: Application To the Euler Equation In Two Dimensionsmentioning

confidence: 97%

The Vortex Method with Finite Elements

Bardos¹,

Bercovier²,

Pironneau³

1981

Mathematics of Computation

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“…Other classical results in this direction are due to Gyunter (see [7] for a list of references). Further classical, fundamental contributions are those of Wolibner [15], Hölder [6], Leray [9], and Shaeffer [12]. In particular, these authors show the existence of a global solution in Hölder spaces.…”

Section: Remarkmentioning

confidence: 99%

Developable Surfaces as Generators of the ?Isobaric Solutions? to the Euler Equations

Veiga

2004

J. math. fluid mech.

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The simplest solutions to the Euler equations (1.1) for which the pressure vanishes identically are those representing the motion of lines parallel to a fixed direction r moving in the same direction (each line with an independent, given, constant velocity). Are there many other solutions to this problem? If yes, is there a simple characterization of all the initial data (volume Ω occupied by the fluid at time t = 0 and initial velocity u 0 (x), x ∈ Ω) that gives rise to the general solutions? In this paper we show that the answer to both questions is positive. We prove, in particular, that there is a natural correspondence between solutions in R 2 of this problem and (Cartesian pieces of) developable surfaces in R 3 . See Theorem 3. Mathematics Subject Classification (2000). 35Q35, 35L60, 35B99.

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“…In this section we will assume that the domain increases fast enough compared with the size of the data and we will show that with this hypothesis the method of Wolibner [8] can be adapted Wolibner has noticed that from the estimate curl (o e L^iGj) one cannot deduce an estimate on the sup norm of V^w. Therefore the first step is to prove, using an analysis of the pair dispersion, that o is bounded in some Holder space C Oa .…”

Section: Construction Of a Smooth Solutionmentioning

confidence: 99%

“…Now the relation (28) is a direct conse-quence of (31), To prove the relation (29) we will adapt the method of Wolibner [8].…”

Section: In the Relation (28) The Constant K Dénotes The Sup Norm Of mentioning

confidence: 99%

“…a 2 ) dénotes the components of the outward unitary normal to 30(0 in ^2-It is known (cf. Wolibner [8] or Ladysenskaia and Uraltceva [6]) that there is no uniform bound of the gradient of u in term of the uniform norm of V A U and of the norme of u. a in C 1 (dQ(i)). On the other hand assuming (for sake of simplicity) that w.oc is bounded in C 2 (dQ(t)) one can prove (Wolibner [8], Kato [2]) the following estimate :…”

Section: In the Relation (28) The Constant K Dénotes The Sup Norm Of mentioning

confidence: 99%

See 1 more Smart Citation

On the existence and the regularity of an initial boundary problem of vorticity equation

Bardos¹,

Yu²

1983

RAIRO. Anal. numér.

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Existence Theorem for the Flow of an Ideal Incompressible Fluid in Two Dimensions

Cited by 5 publications

References 0 publications

The Vortex Method with Finite Elements

The Vortex Method with Finite Elements

Developable Surfaces as Generators of the ?Isobaric Solutions? to the Euler Equations

On the existence and the regularity of an initial boundary problem of vorticity equation

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