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GLOBAL EXISTENCE AND UNIQUENESS OF WEAK SOLUTIONS

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arXiv:0802.2131v1 [math.AP] 15 Feb 2008GLOBAL EXISTENCE AND UNIQUENESS OF WEAKSOLUTIONS OF 3-D EULER EQUATIONS WITH HELICALSYMMETRY IN THE ABSENCE OF VORTICITY STRETCHINGBORIS ETTINGER AND EDRISS S. TITIAbstract. We prove uniqueness and existence of the weak solutions of Eulerequations with helical symmetry, with initial vorticity in L∞under ”no vortic-ity stretching” geometric constraint. Our article follows the argument of theseminal work of Yudovich. We adjust the argument to resolve the difficultieswhich are specific to the helical symmetry.1. IntroductionIdeal incompres sible homogeneous fluid of density ρ0and confined in three-dimensional domain D ⊆ R3is governed by the Euler equatio ns:∂u∂t+ (u ·∇)u = −1ρ0∇p + F,(1.1a)∇ · u = 0,(1.1b)supplemented with the no-normal flow boundary conditions(1.2) u · n = 0, on ∂D, where n is the normal vector to ∂D,and initial velocity u0(x). u : D×[0, T ) → R3is the velocity field, p : D×[0, T ) → Ris the pressure, determined by the incompressibility condition and F : D ×[0, T ) →R3is the given external body forcing term. We will consider constant densityρ0= 1.In this article, we will investigate the solutions of equations (1.1), which areinvariant under a helical symmetry group Gκ. The group Gκis a one-parametergroup of isometries of R3(1.3) Gκ= {Sρ: R3→ R3|ρ ∈ R}.The transformation Sρ(S stands for ”s c rew motion”) is defined by:(1.4) Sρxyz=x cos ρ + y sin ρ−x sin ρ + y cos ρz + κρ,where κ is a fixed nonzero constant le ngth s c ale. In fact, Sρis a sup e rposition ofa s imultaneous rotation around the ˆz-axis with a translation along the ˆz-axis. Thesymmetry lines (orbits of Gκ) are concentric helices. We call the solutions, andmore generally functions, which are invariant under Gκ- ”helical”. Since the Eulerequations in R3are invariant under isometries, then (under mild assumptions ofDate: February 14, 2008.2000 Mathematics Subject Classification. 76B03,35Q35,35D05,76B47.Key words and phrases. Inviscid helical flows, three-dimensional Euler equations.12 HELICAL EULERuniqueness) solving the Euler equa tio ns in a domain which is invariant under helicalsymmetry with a helical initial condition and a helical body forcing will give riseto a solution, which is helical for the whole interval of time of it’s existence.Observe that S2πis a translation by 2πκ in the ˆz directio n. Therefore, helicalsymmetry imposes a per iodic boundary conditions in the ˆz direction. We will alsoassume that the physical domain D ⊆ R3is bounded in the ˆx and ˆy directions, thusimpo sing a no-normal flow. We say that D is bounded in an (infinite) cylinder alo ngthe ˆz-axis, which has a finite radius. Since D is invariant under Gκ, it’s boundary∂D can be thought as being the unionSρ∈RSρC, where C is a closed planar curve.The difficulty to establish the g lobal regularity for the solutions of the 3D Eulerequations can be bes t appreciated, when one examines the evolution of the vorticityfield Ω = ∇ ∧ u, where ∇∧ is the curl (rotor) operator. Taking the curl of bothsides of equation (1.1a) we get:(1.5)∂Ω∂t+ (u ·∇)Ω + (Ω · ∇)u = ∇ ∧ F.The last term o f the left-hand side, (Ω · ∇)u, is called the vorticity stretchingterm. This term is the main obstacle to a chieve global in time regularity of thethree-dimensional Euler Equations, (see [3],[6],[17] for the latest discussions of thisquestion). The difficulty remains after imposing the helical symmetry on the solu-tion, since helical flows can undergo nontrivial vorticity stretching.We will therefore include an additional requirement. We will demand that thevelocity field u = (ux, uy, uz)T, where ux, uy, uzare components of the vector fieldin the basic directions, obeys the following constraint:(1.6) yux− xuy+ κuz= 0.This condition is an orthogonality of the velocity field to the symmetry lines of thegroup Gκ. This co ndition together with the assumption of the helical symmetrylead to vanishing of the vorticity stretching term. We will prove in Sectio n 2 thatunder these conditions, the vorticity Ω is direc ted along the symmetry lines andit’s magnitude, up to a no rmalization is transported by the flow. This control o fthe L∞norm of the vorticity is the key to o ur argument of global existence anduniqueness. It is consistent with the celebrated result of Beal-Kato-Majda [4], andthe works of Yudovich [23] and Ukhovskii and Yudovich [19] for the 2D case andaxi-symmetric flow (without swirl), re spec tively.Our article is inspired by the seminal work of Yudovich [23], who proved existenceand uniqueness for a certain class of weak solutions in two space dimensions. Weadapt his ideas: the stream function weak formulation of the problem and theelliptic regula rity to the case of helical flows. We follow closely his article, espe c iallyin Section 4, where we prove uniqueness. A different route would have been toextend the ideas of Bardos [2], adding viscosity with artificial boundary conditionsand passing to the limit of viscosity going to zero. In such case, the equationwith the added viscosity is not the Navier-Stokes e quation, because of the differentboundary conditions and the limit also has a non-physical boundary conditions.In his work, Yudovich [23] proved global existence and uniqueness of the solutionsof the two-dimensional Euler equations, whose initial vorticity belongs to the spaceof essentially bounded functions, L∞. Uniqueness of the solutions was extendedfor a wider classes of functions in the work Yudovich himself [24] to the classof functions, which are not bounded but whose Lpnorms grow ”slowly enough”.HELICAL EULER 3Uniqueness and global existence was also proven by Vishik [20] for vorticity inthe Besov-like spaces. Global exis tence of solutions of the two-dimensional E ulerequations was proven for initial vorticity Ω0∈ Lpby Majda and DiPerna [9] andalso for solutions, whose initial vorticity is a positive Radon measure by Delort [8].No uniqueness is known in these cas e s.The helical flows fall within a class of ”two-and-a-half” dimensional flows, namelyflows in a three-dimensional domains with a certain co ntinuous spatial symmetry.The most heavily investigated within this class are the axi-symmetric flows, whichare invariant under a rotation around a certain axis of symmetry. The viscousaxi-symmetric flows1were analyzed by


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