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Niveau: Supérieur, Doctorat, Bac+8

Lp-SOLUTIONS OF THE STEADY-STATE NAVIER–STOKES WITH ROUGH EXTERNAL FORCES CLAYTON BJORLAND, LORENZO BRANDOLESE, DRAGOS¸ IFTIMIE, AND MARIA E. SCHONBEK Abstract. In this paper we address the existence, the asymptotic behavior and sta- bility in Lp and Lp,∞, 32 < p ≤ ∞, for solutions to the steady state 3D Navier-Stokes equations with possibly very singular external forces. We show that under certain small- ness conditions of the forcing term there exists solutions to the stationary Navier-Stokes equations in Lp spaces, and we prove the stability of these solutions as fixed points of the non-stationary Navier–Stokes. The non-stationary solutions can be large. We also give non-existence results of stationary solutions in Lp, for 1 ≤ p ≤ 32 . 1. Introduction In this paper we consider the solutions to the three-dimensional steady state Navier– Stokes equations in the whole space R3, (1.1) { ? · (U ? U) +?P = ∆U + f ? · U = 0. Here U = (U1, U2, U3) is the velocity, P the pressure and f = (f1, f2, f3) a given time independent external force. Equation (1.1) will be complemented with a boundary condi- tion at infinity of the form U(x)? 0 in a weak sense: typically, we express this condition requiring that U belongs to some Lp spaces.

Lp-SOLUTIONS OF THE STEADY-STATE NAVIER–STOKES WITH ROUGH EXTERNAL FORCES CLAYTON BJORLAND, LORENZO BRANDOLESE, DRAGOS¸ IFTIMIE, AND MARIA E. SCHONBEK Abstract. In this paper we address the existence, the asymptotic behavior and sta- bility in Lp and Lp,∞, 32 < p ≤ ∞, for solutions to the steady state 3D Navier-Stokes equations with possibly very singular external forces. We show that under certain small- ness conditions of the forcing term there exists solutions to the stationary Navier-Stokes equations in Lp spaces, and we prove the stability of these solutions as fixed points of the non-stationary Navier–Stokes. The non-stationary solutions can be large. We also give non-existence results of stationary solutions in Lp, for 1 ≤ p ≤ 32 . 1. Introduction In this paper we consider the solutions to the three-dimensional steady state Navier– Stokes equations in the whole space R3, (1.1) { ? · (U ? U) +?P = ∆U + f ? · U = 0. Here U = (U1, U2, U3) is the velocity, P the pressure and f = (f1, f2, f3) a given time independent external force. Equation (1.1) will be complemented with a boundary condi- tion at infinity of the form U(x)? 0 in a weak sense: typically, we express this condition requiring that U belongs to some Lp spaces.

- stationary solution
- h˙sp can
- sf? ?
- sobolev spaces
- navier stokes equations
- space
- see also
- valued ?-measurable

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Published by | mijec |

Reads | 13 |

Language | English |

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