Lp SOLUTIONS OF THE STEADY STATE NAVIER–STOKES WITH ROUGH EXTERNAL FORCES

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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.

  • stationary solution

  • h˙sp can

  • sf? ?

  • sobolev spaces

  • navier stokes equations

  • space

  • see also

  • valued ?-measurable


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Lp-SOLUTIONS OF THE STEADY-STATE NAVIER–STOKES WITH ROUGH EXTERNAL FORCES
CLAYTONBJORLAND,LORENZOBRANDOLESE,DRAGO¸SIFTIMIE, AND MARIA E. SCHONBEK
Abstract.In this paper we address the existence, the asymptotic behavior and sta-bility inLpandLp,,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 inLpspaces, 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 inLp, for 1p23.
1.Introduction
In this paper we consider the solutions to the three-dimensional steady state Navier– Stokes equations in the whole spaceR3, (1.1)r ∙(UU) +rP= ΔU+f r ∙U= 0.
HereU= (U1 U2 U3) is the velocity,Pthe pressure andf= (f1 f2 f3) a given time independent external force. Equation (1.1) will be complemented with a boundary condi-tion at infinity of the formU(x) typically, we express this condition0 in a weak sense: requiring thatUbelongs to someLp problems will be addressed.spaces. Three We will first establish the existence of solutionsULp, with32< p≤ ∞, to equations (1.1) for (small) functionsfas possible, and non-existence results in the rangeas general 1p23. Next we will study the asymptotic properties as|x| → ∞for a relevant subclass of the solutions obtained. The third problem at hand is the stability of the solutions in the sense of solutions to (1.1) being “fixed point” inLpto the non-stationary incompressible Navier-Stokes equations
(1.2tu+u∙ ru+rp= Δu+f )r ∙u= 0 u(0) =u0 whereu,pthe time dependent velocity and pressure of the flow.are  will show that We thatsmallstationary solutionsUof (1.1) will attract all global non-stationary solutionsu
2000Mathematics Subject Classification.Primary 76D05; Secondary 35B40. Key words and phrases.Steady Navier-Stokes. The work of L. Brandolese, C. Bjorland, D. Iftimie and M. Schonbek were partially supported by FBF GrantSC-08-34. The work of M. Schonbek was also partially supported by NSF Grant DMS-0600692. 1
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BJORLAND, BRANDOLESE, IFTIMIE, AND SCHONBEK
to (1.2) verifying mild regularity conditions, and emanating from possiblylargedatau0. This will be achieved by first proving that a wide class of global solutions of (1.2) must become small inL3,after some time, and then applying the stability theory of small solutions inL3,developed, e.g., in [8, 18, 28].as  addition, for small solutions, we In will extend the results on the stability in the existing literature by giving necessary and sufficient conditions to have thatu(t)UinLpast→ ∞. The existence and stability of stationary solutions is well understood in the case of bounded domains. See for example [9]. For related results in exterior domains we refer the reader to [10, 11, 12, 15]. A wider list of references regarding connected literature can be found in [3]. For example, the existence and the stability of stationary solutions inLp withpn, wherenis the dimension of the space, is obtained in [22], under the condition that the Reynolds number is sufficiently small, and in [18], [28] under the assumption that the external force is small in a Lorentz space. Similar results in the whole domainRn , always forpn, have been obtained also in [17], [7], [8]. On the other hand, not so much can be found in the literature about the existence and stability of stationary solutions inRnwithp < n. This problem have been studied recently in the casen= 3 andp= 2 in [3]. In this paper we extend the results of [3] to the range23< p≤ ∞, and improve such results also in the casep= 2 by considering a more general class of forcing functions. The methods in this paper differ completely from the ones used in [3]. In the former paper the construction of solutions with finite energy was based on a well known formal observation: if Φ is the fundamental solution for the heat equation thenR0Φ(t)dtis the fundamental solution for Poisson’s equation. Using that idea it was possible to make a time dependent PDE similar to the Navier-Stokes equation withfthat can be formally integrated in time to find aas initial data with a solution solution of (1.1). As we shall see, the conditions onfin the present paper which yield thatULpare, essentially, necessary and sufficient. This will be made possible by a systematic use of suitable function spaces. One could also complement the system (1.1) with different type of boundary condition at infinity. For example, conditions of the formU(x)Uas|x| → ∞, whereUR3 andU6 However the properties of stationary solutions satisfying= 0 are also of interest. such condition are already quite well understood. We refer to the treatise of Galdi [13] for a comprehensive study of this question. On the other hand, the understanding of the problem in the caseU= 0, is still to a more primitive level. For example, the construction of solutions obeying to the natural energy equality (ob-tained multiplying the equation (1.1) byUand formally integrating by parts), without putting any smallness assumption onf The, is still an open problem. main difficulty, for example when Ω =R3ehPno(wravatbaliytilonsiinsuaeqncoi´earlaattheusule),istha prioriestimate on the Dirichlet integral krUkL2≤ kfkH1 ˙ ensures only thatUH˙1L6: but to give a sense to the integral in the formal equality Zr ∙(UU)U dx= 0 one would need,e.g., thatUbelongs also toL4. More generally, one motivation for developing theLptheory (especially forlow values ofp) of stationary solutions is that this provides additional information on the asymptotic