Steady vortex patch solutions to the vortex-wave system
aa r X i v : . [ m a t h . A P ] M a y Steady Vortex Patch Solutions to the Vortex-Wave System
Daomin Cao, Guodong Wang
Abstract
The vortex-wave system describes the motion of a two-dimensional ideal fluid in which thevorticity includes continuously distributed vorticity, which is called the background vorticity,and a finite number of concentrated vortices. In this paper we restrict ourselves to the caseof a single point vortex in bounded domains. We prove the existence of steady vortex patchsolutions to this system with prescribed distribution for the background vorticity. Moreover, weshow that the supports of these solutions “shrink” to a minimum point of the Kirchhoff-Routhfunction as the strength parameter of the background vorticity goes to infinity.
Keywords:
Vortex-wave system, Vortex patch, Euler equation, Desingularization,Kirchhoff-Routh function, Maximization
1. Introduction
The vortex-wave system was firstly introduced by Marchioro and Pulvirenti in [15] to describethe motion of a planar ideal fluid in which the vorticity consists of continuously distributedvorticity(wave part) and k concentrated vortices(vortex part). In the whole plane the systemcan be written as follows: ∂ t ω + u · ∇ ω = 0 , dx i dt = J ∇ Γ ∗ ω ( x i , t ) + P j = i κ j J ∇ Γ( x i − x j ) , i = 1 , · · · , k, u = J ∇ Γ ∗ ω + P kj =1 κ j J ∇ Γ( · − x j ) , (1.1)where Γ( x ) = − π ln | x | is the fundamental solution of − ∆ in R , J ( x , x ) = ( x , − x ) denotesclockwise rotation through π , and Γ ∗ ω is the Newton potential of ω defined byΓ ∗ ω ( x, t ) = − π Z R ln | x − y | ω ( y, t ) dy. (1.2)Let us explain system (1.1) briefly. The first equation is a transport equation for the back-ground vorticity ω ( x, t ), which means that the background vorticity is transported by the velocity“generated” by itself(the term J ∇ Γ ∗ ω ), and k point vortices(the term P kj =1 κ j J ∇ Γ( · − x j )). Email addresses: [email protected] (Daomin Cao), [email protected] (Guodong Wang)
Preprint submitted to Elsevier October 15, 2018 teady Vortex Patch Solutions to the Vortex-Wave System 2The second equation means that the evolution of each vortex x i ( t ) is influenced by the velocity“generated” by the background vorticity(the term J ∇ Γ ∗ ω ( x i , t )) and the other k − P j = i κ j J ∇ Γ( x i − x j )). If κ i = 0, i = 1 , · · · , k , then the system reduces to the vorticity formof the Euler equation, which has been extensively studied, see[11, 14, 22] for example. If thebackground vorticity vanishes, then the system becomes the Kirchhoff-Routh equation, which isa model to describe the motion of k concentrated vortices, see [10, 13, 19] for example.The existence and uniqueness to the non-stationary vortex-wave system in the whole planehave been extensively studied over the past decades, see [1, 5, 7, 8, 15, 16] for example. However,as far as we know, little work has been done in steady solutions to this system. Our purposehere is to construct steady vortex patch solutions in the case of a single vortex. More precisely,we will prove that for any vortex patch rearrangement class N µ defined by N µ = { ω ∈ L ∞ ( D ) | ω = µI A , A ⊂ D, µ | A | = 1 } , where µ is the vorticity strength parameter, there exists a steady solution to the vortex-wavesystem, say ( ω µ , x µ ), satisfying ω µ ∈ N µ . Moreover, as the strength parameter µ goes to infinity,both supp ( ω µ ) and x µ ”shrink” to a minimum point of the Kirchhoff-Routh function.The basic idea to prove the existence of ( ω µ , x µ ) for fixed µ is to construct a family of steadyvortex patch solutions to the Euler equation, in which one part of the vorticity belongs to therearrangement class N µ while the other part “shrinks” to a point, then we show the limit isin fact a steady solution to the vortex-wave system. We will use the result of Burton [2] onmaximization of convex functionals on rearrangement class to obtain approximate solutions,while the proof of the convergence is based on the idea of Turkington [20].It is worth mentioning that our result is closely related to the desingularization of point vor-tices for the Euler equation, which has been studied by many authors, see [1, 4, 6, 12, 13, 19, 21]for example. Roughly speaking, desingularization of vortices for the Euler equation is to jus-tify the Kirchhoff-Routh equation by approximation of the classical Euler equation. There aremainly two kinds of desingularization in the literature: the first kind is to consider a family ofinitial vorticity, which is sufficiently concentrated in k small regions, then the evolved vortic-ity according the Euler equation is also concentrated in k small regions for all time, and thelimiting positions of these small regions can be approximated by the Kirchhoff-Routh equation,see [12, 13, 19] and the references therein; the second kind is to construct a sequence of steadysolutions to the Euler equation that “shrinks” to a critical point of the Kirchhoff-Routh func-tion(or equivalently, a stationary solution to the Kirchhoff-Routh equation), see [4, 18, 20, 21]for example.Analogously, it is natural to consider the desingularization for the vortex-wave system. In [1],the author considered the first kind of desingularization, i.e., given a sequence of initial vorticitywhich is the sum of a given background vorticity and a concentrated vorticity “blob”, it wasproved that the sequence of the evolved solutions according to the Euler equation converges tothe vortex-wave system in some sense. In contrast to [1], in this paper we are concerned withthe the second kind of desingularization, i.e., we construct a family of steady Euler solutions inwhich one part of the vorticity belongs to a given rearrangement class while the support of otherpart “shrinks” to a point, and the limit is exactly a steady solution to the vortex-wave system.teady Vortex Patch Solutions to the Vortex-Wave System 3We end this section by giving outline of this paper. In Section 2, we introduce the vortex-wave system in bounded domains and state our main results. Then we devote Section 3 to theconstruction of approximate solutions by solving a certain variational problem. In Section 4by comparing energy we show that the limit of approximate solutions is in fact a steady vortexpatch solution to the vortex-wave system. In Section 5 we consider the limit of the steady vortexsolutions obtained in Section 4 as the strength of the background vorticity goes to infinity.
2. Main Results
Let D ⊂ R be a bounded and simply-connected domain with smooth boundary. The Green’sfunction for − ∆ in D with zero Dirichlet boundary condition is written as G ( x, y ) = 12 π ln 1 | x − y | − h ( x, y ) , x, y ∈ D, (2.1)where h ( x, y ) is the regular part of G . Note that h ( · , · ) is bounded from below in D × D . TheKirchhoff-Routh function of D is defined to be H ( x ) = 12 h ( x, x ) , x ∈ D, (2.2)and lim x → ∂D H ( x ) = + ∞ , see [19], Lemma 2.2 for example. 2 H is also called Robin function.We shall use the following notations throughout this paper: J ( a, b ) = ( b, − a ) denotes clock-wise rotation through π for any vector ( a, b ) ∈ R , | A | denotes the two-dimensional Lebesguemeasure for any measurable set A ⊂ R , A denotes the closure of some set A ⊂ R in theEuclidean topology, and I A denotes the characteristic function of some planar set A , that is, I A ( x ) = 1 if x ∈ A , I A ( x ) = 0 elsewhere. supp ( g ) denotes the support of some function g , thatis, supp ( g ) = { x | g ( x ) = 0 } . (2.3) dist ( · , · ) denotes the distance between two sets, dist ( A, B ) = inf x ∈ A,y ∈ B | x − y | . (2.4)For a given measurable function g on D , the rearrangement class of g is defined by R ( g ) = { f is measurable | for any a ∈ R , |{ f > a }| = |{ g > a }|} . (2.5)For any ω ∈ L ∞ ( D ), we also define the stream function of ω by G ∗ ω ( x ) = Z D G ( x, y ) ω ( y ) dy. (2.6)Note that since ω ∈ L p ( D ) for any p ∈ [1 , + ∞ ], by L p estimate and Sobolev embedding G ∗ ω ∈ W ,p ( D ) ∩ C ,α ( D ) for any p ∈ [1 , + ∞ ) and α ∈ (0 , We begin with a discussion on the Euler equation describing an ideal fluid with unit densitymoving in D , ∂ t u + ( u · ∇ ) u = −∇ P, ∇ · u = 0 , u · n | ∂D = 0 , (2.7)where u = ( u , u ) is the velocity field, P is the pressure, and n is the outward unit normal.Here we impose the impermeability boundary condition.Define the vorticity ω = ∂ u − ∂ u . Since D is simply connected, u can be uniquelydetermined by ω , u = J ∇ G ∗ ω, (2.8)see [14], § ω . Using theidentity ∇| u | = ( u · ∇ ) u + ωJ u , the first equation of (2.7) becomes u t + ∇ ( 12 | u | + P ) − ωJ u = 0 , (2.9)Taking the curl on both sides we obtain the vorticity form of the Euler equation ω t + u · ∇ ω = 0 , (2.10)which means that the vorticity is transport by the velocity u , where u is ”generated” by ω , i.e., u = J ∇ G ∗ ω .When the vorticity is sufficiently concentrated at k points, equation (2.10) is approximatedby the following Kirchhoff-Routh equation: dx i dt = k X j =1 ,j = i a j J ∇ x i G ( x i , x j ) − a i J ∇ H ( x i ) , i = 1 , · · · , k, (2.11)where x i ( t ) represents the position of the i -th vortex, and a i is the corresponding vorticitystrength. Equation (2.11) means that each vortex interacts with the others via the term a j J ∇ x i G ( x i , x j ) and with the boundary via the term − a i J ∇ H ( x i ). The approximation from theEuler equation to the Kirchhoff-Routh equation has been extensively studied, see [4, 6, 12, 13, 19]and the references therein.Now we combine the Euler equation and the Kirchhoff-Routh equation together, that is, weassume that the vorticity consists of both continuously distributed vorticity denoted by ω ( x, t )and k concentrated vortices x i ( t ), i = 1 , · · · , k . Then it is reasonable that the evolution of ω ( x, t )and x i ( t ) obey the following system: ∂ t ω + u · ∇ ω = 0 , dx i dt = J ∇ G ∗ ω ( x i , t ) + P kj =1 ,j = i a j J ∇ x i G ( x i , x j ) − a i J ∇ H ( x i ) , u = J ∇ G ∗ ω + P kj =1 a j J ∇ G ( x j , · ) , (2.12)teady Vortex Patch Solutions to the Vortex-Wave System 5which we call the vortex-wave system in bounded domains.Let us explain (2.12) briefly. The first equation in (2.12) means that evolution of the back-ground vorticity ω ( x, t ) is influenced by the velocity field J ∇ G ∗ ω “generated” by itself and thevelocity field P kj =1 a j J ∇ G ( x j , · ) “generated” by the k point vortices with strength a i , and theevolution of each x i ( t ) is influenced by the velocity field J ∇ G ∗ ω ( x i , t ) “generated” by ω and thevelocity P kj =1 ,j = i a j J ∇ x i G ( x i , x j ) “generated” by the other k − − a i J ∇ H ( x ). If ω ≡
0, then (2.12) is exactly the Kirchhoff-Routh equation;if a i = 0, i = 1 , · · · , k , then (2.12) becomes the vorticity form of the Euler equation. In the rest of this paper we will restrict ourselves to the stationary vortex-wave system witha single point vortex(that is k = 1), and we assume that the point vortex has unit strength forsimplicity (namely a = 1).More precisely, we will consider the following system: ( J ∇ ( G ∗ ω + G ( x, · )) · ∇ ω = 0 , ∇ G ∗ ω ( x ) − ∇ H ( x ) = 0 . (2.13)Since we are going to deal with vortex patch solutions which are discontinuous, it is necessaryto introduce the weak formulation for the first equation in (2.13). To motivate the definition,let us assume that ω is a smooth solution, then for any φ ∈ C ∞ c ( D ), Z D φJ ∇ ( G ∗ ω + G ( x, · )) · ∇ ωdy = 0 . (2.14)Now we claim that Z D φJ ∇ ( G ∗ ω + G ( x, · )) · ∇ ωdy = − Z D ωJ ∇ ( G ∗ ω + G ( x, · )) · ∇ φdy. (2.15)In fact, by the divergence theorem Z D φJ ∇ G ∗ ω · ∇ ωdy = Z D φ div ( ωJ ∇ G ∗ ω ) dy = Z D div ( φωJ ∇ G ∗ ω ) dy − Z D ωJ ∇ G ∗ ω · ∇ φdy = Z ∂D φωJ ∇ G ∗ ω · n dS − Z D ωJ ∇ G ∗ ω · ∇ φdy = − Z D ωJ ∇ G ∗ ω · ∇ φdy, (2.16)where we use the fact that J ∇ G ∗ ω is a divergence-free vector field. To calculate the integral R D φJ ∇ G ( x, · ) · ∇ ωdy , the singularity of ∇ G need to to be dealt with. To this end defineΩ a = { y ∈ D | G ( x, y ) > a } and D a = D \ Ω a . By the implicit function theorem, Ω a is a simplyteady Vortex Patch Solutions to the Vortex-Wave System 6connected domain with smooth boundary if a > Z D a φJ ∇ G ( x, · ) · ∇ ωdy = Z D a φ div ( ωJ ∇ G ( x, · )) dy = Z D a div ( φωJ ∇ G ( x, · )) dy − Z D a ωJ ∇ G ( x, · ) · ∇ φdy = Z ∂D a φωJ ∇ G ( x, · ) · n dS − Z D a ωJ ∇ G ( x, · ) · ∇ φdy = − Z D a ωJ ∇ G ( x, · ) · ∇ φdy. (2.17)On the other hand, by Lebesgue’s dominated convergence theorem(notice that ∇ G ( x, · ) ∈ L ( D )) we have lim a → + ∞ Z D a φJ ∇ G ( x, · ) · ∇ ωdy = Z D φJ ∇ G ( x, · ) · ∇ ωdy, (2.18)and lim a → + ∞ Z D a ωJ ∇ G ( x, · ) · ∇ φdy = Z D ωJ ∇ G ( x, · ) · ∇ φdy. (2.19)Taking the limit we obtain Z D φJ ∇ G ( x, · ) · ∇ ωdy = − Z D ωJ ∇ G ( x, · ) · ∇ φdy. (2.20)Hence we have proved (2.15). In conclusion, if ω is a smooth solution to the system (2.13), thenit must satisfy Z D ωJ ∇ ( G ∗ ω + G ( x, · )) · ∇ φdy = 0 . (2.21)Notice that the integral in (2.21) makes sense for any ω ∈ L ∞ ( D ) since G ∗ ω ∈ C ( D ) and ∇ G ( x, · ) ∈ L ( D ), so we have the following definition: Definition 2.1. ( ω, x ) is called a weak solution to (2.13) if ω ∈ L ∞ ( D ) , x ∈ D and R D ωJ ∇ ( G ∗ ω + G ( x, · )) · ∇ φdy = 0 , ∀ φ ∈ C ∞ c ( D ) ∇ G ∗ ω ( x ) − ∇ H ( x ) = 0 . (2.22)In this paper we are mainly interested in the vortex patch solution of (2.13), i.e., the solution( ω, x ) such that ω is of the form ω = aI A , where a is a real number representing the strength of ω and A ⊂ D is a Lebesgue measurable set.The main result of this paper is as follows:teady Vortex Patch Solutions to the Vortex-Wave System 7 Theorem 2.2.
Let µ be a positive real number satisfying µ > | D | , and N µ be a rearrangementclass defined by N µ = { ω ∈ L ∞ ( D ) | ω = µI A , A is a measurable set in D, µ | A | = 1 } . (2.23) Then there exist ω µ ∈ N µ and x µ ∈ D such that ( ω µ , x µ ) is a weak solution to the stationaryvortex-wave system (2.13) , moreover, ω µ has the form ω µ = µI { G ∗ ω µ + G ( x µ , · ) >b µ } (2.24) for some b µ > .Remark . If µ = | D | , then there is only one element in N µ , that is ω ≡ µ . In this case ω isa smooth function and the first equation in (2.13) is satisfied for any x ∈ D , so we need onlyconsider the second equation. Notice that H | ∂D = + ∞ , so we can always choose x ∈ D suchthat x is a maximum point, thus a critical point, for the function G ∗ ω − H .As for the asymptotic behavior of ( ω µ , x µ ) as µ → + ∞ , we can prove that up to a subsequence“most part” of ω µ concentrates near a minimum point of H , say x ∗ , and at the same time x µ → x ∗ . Theorem 2.4.
Let ( ω µ , x µ ) be the weak solution to the stationary vortex-wave system (2.13) obtained in Theorem 2.2, then up to a subsequence we have x µ → x ∗ as µ → + ∞ , where x ∗ isa minimum point of H . Moreover, there exists r µ , r µ → as µ → + ∞ , such that lim µ → + ∞ Z B rµ ( x ∗ ) ω µ ( x ) dx = 1 . (2.25) Remark . Recalling that R D ω µ ( x ) dx = 1, it is easy to check that ω µ → δ ( x ∗ ) as µ → + ∞ inthe distributional sense, where δ ( x ∗ ) is the Dirac measure located at x ∗ . More precisely,lim µ → + ∞ Z D ω µ ( x ) φ ( x ) dx = φ ( x ∗ ) , ∀ φ ∈ C ∞ c ( D ) . (2.26)In fact, (cid:12)(cid:12)(cid:12)(cid:12)Z D ω µ ( x ) φ ( x ) dx − φ ( x ∗ ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z D ( φ ( x ) − φ ( x ∗ )) ω µ ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z B rµ ( x ∗ ) ( φ ( x ) − φ ( x ∗ )) ω µ ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z D \ B rµ ( x ∗ ) ( φ ( x ) − φ ( x ∗ )) ω µ ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ sup x ∈ B rµ ( x ∗ ) | φ ( x ) − φ ( x ∗ ) | + 2 sup x ∈ D | φ ( x ) | Z D \ B rµ ( x ∗ ) ω µ ( x ) dx (2.27)which goes to 0 as µ → + ∞ , where we use (2.25) and the continuity of φ at x ∗ . Remark . When D is convex, H is a strictly convex function (see [3]), so there is only oneminimum point for H . In this case the phrase “up to a subsequence” in Theorem 2.4 can beenremoved.teady Vortex Patch Solutions to the Vortex-Wave System 8
3. Variational Problem
Throughout this section we assume that µ is a fixed positive real number. We will constructa family of steady vortex patch solutions to the Euler equation and analyze their properties.Let λ be a positive number. Define M λ = { ω ∈ L ∞ ( D ) | ω = ω + ω , ω ∈ N µ , ω = λI B , λ | B | = 1 , supp ( ω ) ∩ B = ∅ } . (3.1)Recall that N µ is defined in Theorem 2.2. For sufficiently large λ , since µ > | D | , we know that M λ is not empty. Moreover, it is easy to check that M λ is a rearrangement class of any elementin it if λ > µ , that is, for any ω ∈ M λ we have M λ = R ( ω ). In the following we always assume λ to be sufficiently large.Now define the energy functional on M λ by E ( ω ) = 12 Z D Z D G ( x, y ) ω ( x ) ω ( y ) dxdy, ω ∈ M λ , (3.2)which represents the kinetic energy of an ideal fluid in D with vorticity ω .Existence of a maximizer for E relative to M λ is an easy consequence of Corollary 3.4 in [2].Therein by choosing L = − ∆, E = Ψ, F = M λ and K as the Green’s operator, we have: Proposition 3.1.
There exists a maximizer for E relative to M λ ; moreover, if ω λ is a maxi-mizer, then ω λ = f ( G ∗ ω λ ) a.e. in D for some increasing function f : R → R .Remark . ω λ is in fact a steady weak solution to the Euler equation, we refer the interestedreader to [20] for a simple proof.Let ω λ ∈ M λ be a maximizer, then we can write ω λ = ω λ + ω λ , where ω λ ∈ N µ , ω λ = λI B λ , λ | B λ | = 1 , and supp ( ω λ ) ∩ B λ = ∅ . For convenience we shall write ψ λ = G ∗ ω λ and ψ λi = G ∗ ω λi , i = 1 , . Lemma 3.3. ω λ = λI { ψ λ >c λ } for some c λ > .Proof. Since |{ ω λ = λ }| > ω λ = f ( ψ λ ) a.e. in D , it follows that { t ∈ R | f ( t ) = λ } isnot empty, then we can define c λ = inf { t ∈ R | f ( t ) = λ } . By the fact that f is an increasingfunction and ψ λ > D (by strong maximum principle), we have c λ > c λ , ω λ = f ( ψ λ ) ≡ λ a.e. on { x ∈ D | ψ λ ( x ) > c λ } , and ω λ < λ a.e. on { x ∈ D | ψ λ ( x ) < c λ } . On the set { x ∈ D | ψ λ ( x ) = λ } , we have ∇ ψ λ ≡ ω λ = − ∆ ψ λ ≡ { x ∈ D | ω λ ( x ) = λ } = { x ∈ D | ψ λ ( x ) > c λ } , then bychoosing λ > a we have B λ = { x ∈ D | ψ λ ( x ) > c λ } , which is the desired result.Now we begin to analyze the asymptotic behavior of ω λ as λ → + ∞ . In this and the nextsection we shall use C to denote various constants not depending on λ . Lemma 3.4. E ( ω λ ) ≥ − π ln ε − C , where ε satisfies λπε = 1 . teady Vortex Patch Solutions to the Vortex-Wave System 9 Proof.
We take the test function as follows: for any fixed x ∈ D , define ¯ ω λ = ¯ ω λ + ¯ ω λ , where¯ ω λ = λI B ε ( x ) , ¯ ω λ ∈ N µ and ¯ ω λ = 0 a.e. in B ε ( x ). It’s easy to check that ¯ ω λ ∈ M λ , so wehave E ( ω λ ) ≥ E (¯ ω λ ). By simple calculation, E (¯ ω λ ) = 12 Z D Z D G ( x, y )¯ ω λ ( x )¯ ω λ ( y ) dxdy = 12 Z D Z D G ( x, y )(¯ ω λ ( x ) + ¯ ω λ ( x ))(¯ ω λ ( y ) + ¯ ω λ ( y )) dxdy = E (¯ ω λ ) + E (¯ ω λ ) + 12 Z D Z D G ( x, y )¯ ω λ ( x )¯ ω λ ( y ) dxdy + 12 Z D Z D G ( x, y )¯ ω λ ( x )¯ ω λ ( y ) dxdy = E (¯ ω λ ) + E (¯ ω λ ) + Z D Z D G ( x, y )¯ ω λ ( x )¯ ω λ ( y ) dxdy, (3.3)where we use the symmetry of the Green’s function, that is, G ( x, y ) = G ( y, x ) for any x, y ∈ D .Since G ∈ L ( D × D ), we have the following estimate for E (¯ ω λ ): | E (¯ ω λ ) | = (cid:12)(cid:12)(cid:12)(cid:12) Z D Z D G ( x, y )¯ ω λ ( x )¯ ω λ ( y ) dxdy (cid:12)(cid:12)(cid:12)(cid:12) ≤ µ (cid:12)(cid:12)(cid:12)(cid:12)Z D Z D G ( x, y ) dxdy (cid:12)(cid:12)(cid:12)(cid:12) ≤ C. (3.4)For the term R D R D G ( x, y )¯ ω λ ( x )¯ ω λ ( y ) dxdy in (3.3), by L p estimate we have (cid:12)(cid:12)(cid:12)(cid:12)Z D Z D G ( x, y )¯ ω λ ( x )¯ ω λ ( y ) dxdy (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z D G ∗ ¯ ω λ ( y )¯ ω λ ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12) ≤ C Z D ¯ ω λ ( y ) dy = C. (3.5)It remains to estimate the lower bound of E (¯ ω λ ), E (¯ ω λ ) = 12 Z D Z D G ( x, y )¯ ω λ ( x )¯ ω λ ( y ) dxdy = − π Z D Z D ln | x − y | ¯ ω λ ( x )¯ ω λ ( y ) dxdy − Z D Z D h ( x, y )¯ ω λ ( x )¯ ω λ ( y ) dxdy = − λ π Z B ε ( x ) Z B ε ( x ) ln | x − y | dxdy − Z D Z D h ( x, y )¯ ω λ ( x )¯ ω λ ( y ) dxdy, (3.6)Since | x − y | ≤ ε for x, y ∈ B ε ( x ), we have − λ π Z B ε ( x ) Z B ε ( x ) ln | x − y | dxdy ≥ − λ π Z B ε ( x ) Z B ε ( x ) ln | ε | dxdy = − π ln ε − π ln 2 . On the other hand, by the continuity of h ( x, y ) in D × D , the integral R B ε ( x ) R B ε ( x ) h ( x, y ) dxdy converges to h ( x , x ), thus is uniformly bounded, as λ → + ∞ , so E (¯ ω λ ) ≥ − π ln ε − C. (3.7)teady Vortex Patch Solutions to the Vortex-Wave System 10Using (3.3),(3.4),(3.5) and (3.7) we complete the proof.Now we define T ( ω λ ) = R D ω λ ( x )( ψ λ − c λ )( x ) dx , which represents the kinetic energy of thefluid on B λ . To simplify presentation we write ζ λ = ψ λ − c λ . By the fact that ζ λ = 0 on ∂B λ ,so T ( ω λ ) = 12 Z B λ ω λ ( x ) ζ λ ( x ) dx = 12 Z B λ |∇ ζ λ ( x ) | dx. (3.8)We have the following uniform estimate for T : Lemma 3.5. T ( ω λ ) ≤ C. Proof.
Firstly by H¨older’s inequality, we have T ( ω λ ) = 12 λ Z B λ ζ λ ( x ) dx ≤ λ | B λ | { Z B λ | ζ λ ( x ) | dx } . By the Sobolev embedding W , ( D ) ֒ → L ( D ), we have (cid:26)Z B λ | ζ λ ( x ) | dx (cid:27) = (cid:26)Z D | ( ζ λ ) + ( x ) | dx (cid:27) ≤ C Z D |∇ ( ζ λ ) + ( x ) | dx, where ( ζ λ ) + ( x ) = max { , ζ λ ( x ) } . It follows that T ( ω λ ) ≤ Cλ | B λ | Z D |∇ ( ζ λ ) + ( x ) | dx = Cλ | B λ | Z B λ |∇ ζ λ ( x ) | dx ≤ Cλ | B λ | (cid:26)Z B λ |∇ ζ λ ( x ) | dx (cid:27) . Notice that λ | B λ | = R D ω λ ( x ) dx = 1, we obtain T ( ω λ ) ≤ C (cid:26)Z B λ |∇ ζ λ ( x ) | dx (cid:27) . (3.9)By comparing (3.9) with (3.8) we get the desired result. Lemma 3.6.
There exists R > such that diam ( supp ( ω λ )) ≤ R ε .Proof. Firstly we estimate the lower bound for c λ . By the definition of T ( ω λ ), E ( ω λ ) = T ( ω λ ) + 12 Z D ω λ ( x ) ψ λ ( x ) dx + c λ , (3.10)It is easy to check that R D ω λ ( x ) ψ λ ( x ) dx has a uniform upper bounded. In fact, Z D ω λ ( x ) ψ λ ( x ) dx = Z D ω λ ( x ) G ∗ ( ω λ + ω λ )( x ) dx = Z D Z D G ( x, y ) ω λ ( x ) ω λ ( y ) dxdy + Z D Z D G ( x, y ) ω λ ( x ) ω λ ( y ) dxdy ≤ µ Z D Z D | G ( x, y ) | dxdy + | G ∗ ω λ | L ∞ ( D ) ≤ C. (3.11)teady Vortex Patch Solutions to the Vortex-Wave System 11Now (3.10) together with Lemma 3.4 and Lemma 3.5 gives c λ ≥ − π ln ε − C. (3.12)Now for any x ∈ supp ( ω λ ), we have ψ λ ( x ) ≥ c λ , that is, Z D G ( x, y ) w λ ( y ) dy ≥ − π ln ε − C. (3.13)Since h ( x, y ) is bounded from below on D × D , we have Z D ln 1 | x − y | ω λ ( y ) dy + ln ε ≥ − C, (3.14)or equivalently, Z D ln 1 | x − y | ω λ ( y ) dy + Z D ln 1 | x − y | ω λ ( y ) dy + ln ε ≥ − C, (3.15)Notice that (cid:12)(cid:12)(cid:12)(cid:12)Z D ln 1 | x − y | ω λ ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12) ≤ µ sup x ∈ D (cid:12)(cid:12)(cid:12)(cid:12)Z D ln | x − y | dy (cid:12)(cid:12)(cid:12)(cid:12) ≤ C, (3.16)so we get Z D ln ε | x − y | ω λ ( y ) dy ≥ − C. (3.17)Now let R > Z B Rε ( x ) ln ε | x − y | ω λ ( y ) dy + Z D \ B Rε ( x ) ln ε | x − y | ω λ ( y ) dy ≥ − C. (3.18)By the rearrangement inequality the first integral in (3.18) can be estimated as follows: Z B Rε ( x ) ln ε | x − y | ω λ ( y ) dy ≤ λ Z B ε ( x ) ln ε | x − y | dy = λ Z B ε (0) ln ε | y | dy = 12 . (3.19)By comparing (3.18) with (3.19) we obtain Z D \ B Rε ( x ) ln ε | x − y | ω λ ( y ) dy ≥ − C. We observe now that Z D \ B Rε ( x ) ln ε | x − y | ω λ ( y ) dy ≤ Z D \ B Rε ( x ) ln 1 R ω λ ( y ) dy, (3.20)therefore Z D \ B Rε ( x ) ω λ ( y ) dy ≤ C ln R , (3.21)teady Vortex Patch Solutions to the Vortex-Wave System 12which means Z B Rε ( x ) ω λ ( y ) dy ≥ − C ln R . (3.22)Choosing R large such that 1 − C ln R > , we have Z B Rε ( x ) ω λ ( y ) dy > . (3.23)Since x ∈ supp ( ω λ ) is arbitrary and R D ω λ ( y ) dy = 1, we get the desired result by choosing R = 2 R .Up to now we have established a family of functions ω λ and ω λ , moreover, we show that diam ( supp ( ω λ )) → λ → + ∞ . Now we are in a position to consider the limits of ω λ and ω λ . To this end, define the center of ω λ by x λ = Z D xω λ ( x ) dx. (3.24)Up to a subsequence, we can assume that as λ → + ∞ , there exists x µ ∈ D such that x λ → x µ . On the other hand, since { ω λ } is bounded in L ∞ ( D )(recall that µ is fixed in this section),up to a subsequence we assume that as λ → + ∞ ω λ → ω µ weakly star in L ∞ ( D )for some ω µ ∈ N µ , where N µ denotes the weak star closure of N µ in L ∞ ( D ). By standardelliptic equation theory we also have as λ → + ∞ G ∗ ω λ → G ∗ ω µ in C ,α ( D ) . We end this section by showing the following lemma which will be frequently used in thenext section.
Lemma 3.7.
We have(1), | G ∗ ω λ | L ∞ ( D ) ≤ C , for some C > not depending on λ .(2), E ( ω λ ) = E ( ω µ ) + o (1) ,(3), R D G ∗ ω λ ( x ) ω λ ( x ) dx = G ∗ ω µ ( x µ ) + o (1) ,where o (1) denotes quantities such that o (1) → as λ → + ∞ . teady Vortex Patch Solutions to the Vortex-Wave System 13 Proof.
To prove (1), it suffices to notice that ω λ is bounded in L ∞ ( D ), then the result followsfrom L p estimate and Sobolev embedding.Now we turn to the proof of (2). By simple calculation, (cid:12)(cid:12)(cid:12) E ( ω λ ) − E ( ω µ ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) Z D ω λ G ∗ ω λ dx − Z D ω µ G ∗ ω µ dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)Z D ω λ ( G ∗ ω λ − G ∗ ω µ ) dx (cid:12)(cid:12)(cid:12)(cid:12) + 12 (cid:12)(cid:12)(cid:12)(cid:12)Z D G ∗ ω µ ( ω λ − ω µ ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ µ (cid:12)(cid:12)(cid:12) G ∗ ω λ − G ∗ ω µ (cid:12)(cid:12)(cid:12) L ∞ ( D ) + o (1) , (3.25)which goes to 0 as λ → + ∞ . To prove (2), noting that diam ( supp ( ω λ )) → x λ → x µ , then we can choose r λ , r λ → λ → + ∞ , such that supp ( ω λ ) ⊂ B r λ ( x µ ). By the continuity of G ∗ ω µ and the fact G ∗ ω λ → G ∗ ω µ in L ∞ ( D ), it follows that (cid:12)(cid:12)(cid:12)(cid:12)Z D G ∗ ω λ ( x ) ω λ ( x ) dx − G ∗ ω µ ( x µ ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z D ( G ∗ ω λ ( x ) − G ∗ ω µ ( x µ )) ω λ ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z B rλ ( x µ ) ( G ∗ ω λ ( x ) − G ∗ ω µ ( x µ )) ω λ ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ sup x ∈ B rλ ( x µ ) | G ∗ ω λ ( x ) − G ∗ ω µ ( x µ ) |≤ sup x ∈ B rλ ( x µ ) | G ∗ ω λ ( x ) − G ∗ ω µ ( x ) | + sup x ∈ B rλ ( x µ ) | G ∗ ω µ ( x ) − G ∗ ω µ ( x µ ) |→ . (3.26)
4. Proof of Theorem 2.2
In this section we will give proof of Theorem 2.2. Before doing this we need to establishseveral preliminary lemmas first. We will show that the weakly star limit ω µ ∈ N µ of ω λ actually belongs to N µ , x µ ∈ ¯ D actually in D and ( ω µ , x µ ) is a weak solution to the stationaryvortex-wave system (2.13). Lemma 4.1.
Let ω ∈ L ∞ ( D ) , x ∈ D , then ( ω, x ) is a weak solution of (2.13) if the followingtwo conditions are satisfied(1). For any y ∈ D , G ∗ ω ( y ) − H ( y ) ≤ G ∗ ω ( x ) − H ( x ) .(2). For any v ∈ R ( ω ) , E ( v ) + G ∗ v ( x ) ≤ E ( ω ) + G ∗ ω ( x ) . (4.1)teady Vortex Patch Solutions to the Vortex-Wave System 14 Proof.
Condition (1) in Lemma 4.1 implies that x is a maximum point for the function G ∗ ω − H in D , so ∇ G ∗ ω ( x ) − ∇ H ( x ) = 0.In the following, for the sake of convenience set F ( v, y ) = E ( v ) + G ∗ v ( y ) , v ∈ R ( ω ) , y ∈ D. (4.2)For any given φ ∈ C ∞ ( D ), define a family of C transformations Φ t ( x ) : D ֒ → D for t ∈ ( −∞ , + ∞ ) by the following ordinary differential equation: ( d Φ t ( x ) dt = J ∇ φ (Φ t ( x )) , t ∈ R , Φ ( x ) = x, (4.3)where J denotes clockwise rotation through π as before. Note that (4.3) is solvable for all t since J ∇ φ is a smooth vector field with compact support in D . It’s easy to see that J ∇ φ is divergence-free, so by Liouville theorem(see [14], Appendix 1.1) Φ t ( x ) is area-preserving, orequivalently for any measurable set A ⊂ D | Φ t ( A ) | = | A | . (4.4)Now define a family of test functions ω ( t ) ( x ) , ω (Φ − t ( x )) . (4.5)Since Φ t is area-preserving, we have ω ( t ) ∈ R ( ω ), then condition (1) in Lemma 4.1 implies that F ( ω ( t ) , x ) attains its maximum at t = 0, so ddt F ( ω ( t ) , x ) | t =0 = 0. Expanding F ( ω ( t ) , x ) at t = 0gives F ( ω ( t ) , x ) = 12 Z D Z D G ( y, z ) ω (Φ − t ( y )) ω (Φ − t ( z )) dydz + Z D G ( x, y ) ω (Φ − t ( y )) dy = 12 Z D Z D G (Φ t ( y ) , Φ t ( z )) ω ( y ) ω ( z ) dydz + Z D G ( x, Φ t ( y )) ω ( y ) dy = E ( ω ) + t Z D ω ( y ) ∇ ( G ∗ ω ( y ) + G ( x, y )) · J ∇ φ ( y ) dy + o ( t ) , as t →
0. So we have Z D ω ( y ) ∇ ( G ∗ ω ( y ) + G ( x, y )) · J ∇ φ ( y ) dy = 0 , ∀ φ ∈ C ∞ c ( D ) , which completes the proof.To apply Lemma 4.1, we need more information about ( ω µ , x µ ). Lemma 4.2. x µ ∈ D . teady Vortex Patch Solutions to the Vortex-Wave System 15 Proof.
By Lemma 3.7 and the symmetry of the Green’s function, E ( ω λ ) = 12 Z D Z D G ( x, y )( ω λ + ω λ )( x )( ω λ + ω λ )( y ) dxdy = E ( ω λ ) + E ( ω λ ) + Z D Z D G ( x, y ) ω λ ( x ) ω λ ( y ) dxdy = E ( ω µ ) − π Z D Z D ln | x − y | ω λ ( x ) ω λ ( y ) dxdy + G ∗ ω µ ( x µ ) − H ( x µ ) + o (1) . (4.6)By rearrangement inequality(see [9], § − π Z D Z D ln | x − y | ω λ ( x ) ω λ ( y ) dxdy ≤ sup x ∈ D − λ π Z D ln | x − y | ω λ ( y ) dy ≤ − λ π Z B ε (0) ln | y | dy ≤ − π ln ε + C. (4.7)So we have E ( ω λ ) ≤ E ( ω µ ) − π ln ε + C + G ∗ ω µ ( x µ ) − H ( x µ ) + o (1) , (4.8)that is E ( ω λ ) + 14 π ln ε ≤ E ( ω µ ) + G ∗ ω µ ( x µ ) − H ( x µ ) + o (1) . (4.9)If x µ ∈ ∂D , then H ( x µ ) = + ∞ , which means that E ( ω λ ) + π ln ε → −∞ as λ → + ∞ , which isa contradiction to Lemma 3.4. Lemma 4.3. sup v ∈N µ ( E ( v ) + G ∗ v ( x µ )) = sup v ∈N µ ( E ( v ) + G ∗ v ( x µ )) .Proof. Firstly it is obvious that sup v ∈N µ ( E ( v ) + G ∗ v ( x µ )) ≤ sup v ∈N µ ( E ( v ) + G ∗ v ( x µ )).On the other hand, for any ω ∈ N µ we can choose a sequence { ω n } ⊂ N µ such that ω n → ω weakly star in L ∞ ( D ), then E ( ω n ) + G ∗ ω n ( x µ ) → E ( ω ) + G ∗ ω ( x µ ) , which means that sup v ∈N µ ( E ( v ) + G ∗ v ( x µ )) ≥ E ( ω ) + G ∗ ω ( x µ ). Since ω ∈ N µ is arbitrary,we have sup v ∈N µ ( E ( v ) + G ∗ v ( x µ )) ≥ sup v ∈N µ ( E ( v ) + G ∗ v ( x µ )) , which completes the proof. Lemma 4.4. E ( ω µ ) + G ∗ ω µ ( x µ ) = sup ω ∈N µ ( E ( ω ) + G ∗ ω ( x µ )) . teady Vortex Patch Solutions to the Vortex-Wave System 16 Proof.
Recall that ω λ = λI B λ . By choosing v λ = v λ + ω λ , such that v λ ∈ N µ , v λ ≡ B λ , it is obvious that v λ ∈ M λ . As a consequence we have E ( ω λ ) ≥ E ( v λ ), that is, E ( ω λ ) + E ( ω λ ) + Z D G ∗ ω λ ( x ) ω λ ( x ) dx ≥ E ( v λ ) + E ( ω λ ) + Z D G ∗ v λ ( x ) ω λ ( x ) dx, (4.10)which gives E ( ω λ ) + Z D G ∗ ω λ ( x ) ω λ ( x ) dx ≥ E ( v λ ) + Z D G ∗ v λ ( x ) ω λ ( x ) dx. (4.11)By Lemma 3.7 it follows E ( ω µ ) + G ∗ ω µ ( x µ ) ≥ E ( v λ ) + G ∗ v λ ( x µ ) + o (1) . (4.12)Since diam ( supp ( ω λ )) → E is a continuous functional on N µ , v λ can be any element in N µ as λ → + ∞ , that is E ( ω µ ) + G ∗ ω µ ( x µ ) ≥ E ( v ) + G ∗ v ( x µ ) , ∀ v ∈ N µ , (4.13)which, combined with Lemma 4.3 leads to the desired result. Lemma 4.5. ω µ ∈ N µ and ω µ = µI { G ∗ ω µ + G ( x µ , · ) >b µ } for some b µ > .Proof. Define F = { ω ∈ L ∞ ( D ) | ≤ ω ≤ µ, R D ω ( x ) dx = 1 } , then for F we have the followingtwo claims.Claim 1: N µ ⊂ F .Proof of Claim 1: By the definition of N µ it suffices to show that F is closed in the weakstar topology in L ∞ ( D ). Let ω n ∈ F , ω n → ω ∗ weakly star in L ∞ ( D ), that is,lim n → + ∞ Z D ω n ( x ) φ ( x ) dx = Z D ω ∗ ( x ) φ ( x ) dx, ∀ φ ∈ L ( D ) , (4.14)it suffices to show that ω ∗ ∈ F .Firstly by choosing φ ( x ) ≡ n → + ∞ Z D ω n ( x ) dx = Z D ω ∗ ( x ) dx = 1 . Now we prove 0 ≤ ω ∗ ≤ µ by contradiction. Suppose that |{ ω ∗ > µ }| >
0, then there exists ε > |{ ω ∗ ≥ µ + ε }| >
0. Denote A = { ω ∗ ≥ µ + ε } , then for φ = I A we have0 = lim n → + ∞ Z D ( ω ∗ − ω n )( x ) φ ( x ) dx = lim n → + ∞ Z A ω ∗ ( x ) − ω n ( x ) dx. On the other hand lim n → + ∞ Z A ( ω ∗ − ω n )( x ) dx ≥ ε | A | > , teady Vortex Patch Solutions to the Vortex-Wave System 17which is a contradiction. So we have ω ∗ ≤ µ a.e. on D .Lastly, a similar argument suggests ω ∗ ≥ D , which completes the proof of Claim 1.Claim 2: There exists ˜ ω ∈ F such that E (˜ ω ) + G ∗ ˜ ω ( x µ ) = sup ω ∈F E ( ω ) + G ∗ ω ( x µ ),moreover, any maximizer ˜ ω has the form ˜ ω = µλ { G ∗ ˜ ω + G ( x µ , · ) >b µ } for some b µ > ω ∈F E ( ω ) + G ∗ ω ( x µ ) < + ∞ . In fact, for any ω ∈ F , E ( ω ) + G ∗ ω ( x µ ) = 12 Z D Z D G ( x, y ) ω ( x ) ω ( y ) dxdy + Z D G ( x µ , y ) ω ( y ) dy ≤ µ Z D Z D | G ( x, y ) | dxdy + µ Z D G | ( x µ , y ) | dy ≤ C, (4.15)where C is a positive number not depending on ω (may depending on µ ). Now we choose ω n ∈ F such that ω n → ˜ ω and E ( ω n ) + G ∗ ω n ( x µ ) → sup ω ∈F E ( ω ) + G ∗ ω ( x µ ). An argument similarto the one used in Lemma 3.7 gives E (˜ ω ) + G ∗ ˜ ω ( x µ ) = sup ω ∈F ( E ( ω ) + G ∗ ω ( x µ )) . (4.16)Now we prove that ˜ ω is a vortex patch with the form ˜ ω = µλ { G ∗ ˜ ω + G ( x µ , · ) >b µ } for some b µ > ω ( s ) ( x ) = ˜ ω + s [ z ( x ) − z ( x )], s >
0, where z , z satisfies z , z ∈ L ∞ ( D ) , z , z ≥ , R D z dx = R D z dx,z = 0 in D \ { ˜ ω ≤ µ − δ } ,z = 0 in D \ { ˜ ω ≥ δ } , (4.17)here δ is any positive number. Note that for fixed z , z and δ , ω ( s ) ∈ F provided s is sufficientlysmall(depending on δ, z , z ). So we have dds [ E ( ω ( s ) ) + G ∗ ω ( s ) ( x µ )] (cid:12)(cid:12)(cid:12) s =0 + ≤ , (4.18)which gives sup { ˜ ω<µ } ( G ∗ ˜ ω + G ( x µ , · )) ≤ inf { ˜ ω> } ( G ∗ ˜ ω + G ( x µ , · )) . (4.19)Now it is obvious that there exists r > ω ≡ µ a.e. in B r ( x µ )(otherwise the left handside of (4.19) equals + ∞ ). Moreover, we can choose r sufficiently small such thatinf { ˜ ω> } ( G ∗ ˜ ω + G ( x µ , · )) = inf { ˜ ω> }∩ D r ( G ∗ ˜ ω + G ( x µ , · )) , (4.20)where D r = D \ B r ( x µ ). Then we havesup { ˜ ω<µ }∩ D r ( G ∗ ˜ ω + G ( x µ , · )) ≤ inf { ˜ ω> }∩ D r ( G ∗ ˜ ω + G ( x µ , · )) . (4.21)teady Vortex Patch Solutions to the Vortex-Wave System 18Since D r is connected (for sufficiently small r ) and { ˜ ω < µ } ∩ D r ∪ { ˜ ω > } ∩ D r = D r , we have { ˜ ω < µ } ∩ D r ∩ { ˜ ω > } ∩ D r = ∅ , then by the continuity of G ∗ ˜ ω + G ( x µ , · ) on D r ,sup { ˜ ω<µ }∩ D r ( G ∗ ˜ ω + G ( x µ , · )) = inf { ˜ ω> }∩ D r ( G ∗ ˜ ω + G ( x µ , · )) . (4.22)Now define b µ = sup { ˜ ω<µ }∩ D r ( G ∗ ˜ ω + G ( x µ , · )) = inf { ˜ ω> }∩ D r ( G ∗ ˜ ω + G ( x µ , · )) , (4.23)by maximum principle it is easy to see that µ >
0, and it is also obvious that ( ˜ ω = 0 a.e. in { G ∗ ˜ ω + G ( x µ , · ) < b µ } ∩ D r , ˜ ω = µ a.e. in { G ∗ ˜ ω + G ( x µ , · ) > b µ } ∩ D r . (4.24)On { G ∗ ˜ ω + G ( x µ , · ) = b µ } ∩ D r , we have ∇ ( G ∗ ˜ ω + G ( x µ , · )) = 0 a.e. , which gives ˜ ω = − ∆( G ∗ ˜ ω ) = − ∆( G ∗ ˜ ω + G ( x µ , · )) = 0. Now it remains to show that G ∗ ˜ ω + G ( x µ , · ) > b µ on B r ( x µ ) . This is an easy consequence of the maximum principle. In fact, by (4.20) b µ = inf { ˜ ω> }∩ D r ( G ∗ ˜ ω + G ( x µ , · )) , = inf { ˜ ω> } ( G ∗ ˜ ω + G ( x µ , · )) , ≤ inf B r ( x µ ) ( G ∗ ˜ ω + G ( x µ , · )) ≤ inf ∂B r ( x µ ) ( G ∗ ˜ ω + G ( x µ , · )) , (4.25)then by strong maximum principle we have G ∗ ˜ ω + G ( x µ , · ) > b µ on B r ( x µ ).In conclusion, we have proved that ˜ ω has the form ˜ ω = µI { G ∗ ˜ ω + G ( x µ , · ) >b µ } for some b µ > ω ∈N µ ( E ( ω ) + G ∗ ω ( x µ )) = sup ω ∈F ( E ( ω ) + G ∗ ω ( x µ )) , (4.26)therefore we obtain E ( ω µ ) + G ∗ ω µ ( x µ ) = sup ω ∈F ( E ( ω ) + G ∗ ω ( x µ )) . Using Claim 2 again we get the desired result.
Remark . Lemma 4.5 is essential to this paper. We remark that Corollary 3.4 in [2] can notbe applied here anymore since ∇ G is not a locally integrable function. The proof we give hereis based on the idea of Turkington in [20] with some modifications.Now we are ready to prove Theorem 2.2.teady Vortex Patch Solutions to the Vortex-Wave System 19 Proof of Theorem 2.2.
Firstly by Lemma 4.4 and Lemma 4.5, ω µ satisfies (2) in Lemma 4.1 andhas the form ω µ = µI { G ∗ ω µ + G ( x µ , · ) >b µ } for some b µ >
0. It suffices to show that x µ satisfies (1)in Lemma 4.1.Fix x ∈ D and define v λ = v λ + v λ , where v λ = λI B ε ( x ) , v λ ∈ N µ and v λ = 0 a.e. on B ε ( x ). It is easy to check that v λ ∈ M λ , so we have E ( ω λ ) ≥ E ( v λ ), that is, E ( ω λ ) + E ( ω λ ) + Z D G ∗ ω λ ( x ) ω λ ( x ) dx ≥ E ( v λ ) + E ( v λ ) + Z D G ∗ v λ ( x ) v λ ( x ) dx, (4.27)then by Lemma 3.7 for λ sufficiently large we have E ( ω µ ) − π Z D Z D ln | x − y | ω λ ( x ) ω λ ( y ) dxdy − H ( x µ ) + G ∗ ω µ ( x µ ) + o (1) ≥ E ( v λ ) − π Z D Z D ln | x − y | v λ ( x ) v λ ( y ) dxdy − H ( x ) + G ∗ v λ ( x ) . (4.28)On the other hand, by Riesz’s rearrangement inequality(see [9], § − π Z D Z D ln | x − y | ω λ ( x ) ω λ ( y ) dxdy ≤ − π Z D Z D ln | x − y | v λ ( x ) v λ ( y ) dxdy. (4.29)So we have E ( ω µ ) − H ( x µ ) + G ∗ ω µ ( x µ ) + o (1) ≥ E ( v λ ) − H ( x ) + G ∗ v λ ( x ) . (4.30)Again, since E is a continuous functional on N µ and | B ε ( x ) | → v λ can be any element in N µ as λ → + ∞ , that is, E ( ω µ ) − H ( x µ ) + G ∗ ω µ ( x µ ) ≥ E ( v ) − H ( x ) + G ∗ v ( x ) , ∀ v ∈ N µ . (4.31)Especially we can choose v = ω µ , then it follows − H ( x µ ) + G ∗ ω µ ( x µ ) ≥ − H ( x ) + G ∗ ω µ ( x ) , ∀ x ∈ D, (4.32)which means that x µ satisfies (1) in Lemma 4.1. Therefore we complete the proof.
5. Proof of Theorem 2.4
Up to now we have constructed ( ω µ , x µ ) as a steady vortex patch solution to the vortex-wavesystem for fixed µ . Now we consider the asymptotic behavior of ( ω µ , x µ ) when µ → + ∞ . Ashas been stated in Theorem 2.4, we will show that both the support of ω µ and x µ converge toa minimum point of H , which is a stationary solution to the Kirchhoff-Routh equation.In this section we shall use C to denote various positive numbers independent of µ . Theorem2.4 is an easy consequence of the following several lemmas.teady Vortex Patch Solutions to the Vortex-Wave System 20 Lemma 5.1.
For any ω ∈ N µ , x ∈ D , we have E ( ω ) + G ∗ ω ( x ) − H ( x ) ≤ E ( ω µ ) + G ∗ ω µ ( x µ ) − H ( x µ ) . Proof.
For fixed x ∈ D , define a family of test functions v λ = v λ + v λ , v λ = λI B ε ( x ) , v λ ∈ N µ and v λ = 0 a.e. on B ε ( x ). It is easy to check v λ ∈ M λ , then by definition E ( v λ ) ≤ E ( ω λ ), thatis, E ( v λ ) + E ( v λ ) + Z D G ∗ v λ ( y ) v λ ( y ) dy ≤ E ( ω λ ) + E ( ω λ ) + Z D G ∗ ω λ ( y ) ω λ ( y ) dy, (5.1)again by Lemma 3.7 E ( v λ ) − π Z D Z D ln | y − z | v λ ( y ) v λ ( z ) dydz − H ( x ) + G ∗ v λ ( x ) ≤ E ( ω µ ) − π Z D Z D ln | y − z | ω λ ( y ) ω λ ( z ) dydz − H ( x µ ) + G ∗ ω µ ( x µ ) + o (1) , (5.2)where o (1) → λ → + ∞ . Using Riesz’s rearrangement inequality, from (5.2) we have E ( v λ ) − H ( x ) + G ∗ v λ ( x ) ≤ E ( ω µ ) − H ( x µ ) + G ∗ ω µ ( x µ ) + o (1) . (5.3)As λ → + ∞ , v λ can be any element in N µ , so we obtain E ( ω ) − H ( x ) + G ∗ ω ( x ) ≤ E ( ω µ ) − H ( x µ ) + G ∗ ω µ ( x µ ) , for all ( ω, x ) ∈ ( N µ , D ) . (5.4) Remark . One can also maximize E ( ω )+ G ∗ ω ( x ) − H ( x ) for ( ω, x ) ∈ ( N µ , D ) to obtain steadysolution to the vortex-wave system, but it is much more interesting to construct solutions fromthe Euler equation, because the vortex-wave itself is an approximation of the Euler equationwhen a part of the vorticity is sufficiently concentrated.In the following s will be the positive number defined by µπs = 1. Lemma 5.3.
There exists δ > , not depending on µ , such that dist ( x µ , ∂D ) > δ .Proof. Fix x ∈ D and define ¯ ω µ = µI B s ( x ) , then ¯ ω µ ∈ N µ , by Lemma 5.1 E (¯ ω µ ) + G ∗ ¯ ω µ ( x ) − H ( x ) ≤ E ( ω µ ) + G ∗ ω µ ( x µ ) − H ( x µ ) . (5.5)Using Riesz’s rearrangement inequality we get − H ( x ) − H ( x ) − H ( x ) + o (1) ≤ − Z D Z D h ( x, y ) ω µ ( x ) ω µ ( y ) dxdy − Z D h ( x µ , y ) ω µ ( y ) dy − H ( x µ ) , (5.6)since h is bounded from below in D × D , we have H ( x µ ) ≤ C, (5.7)then we get the desired result by the fact lim x → ∂D H ( x ) = + ∞ .teady Vortex Patch Solutions to the Vortex-Wave System 21 Lemma 5.4. G ∗ ω µ ( x µ ) ≥ − π ln s − C .Proof. Since dist ( x µ , ∂D ) > δ , we can define ¯ ω µ = µI B s ( x µ ) ∈ N µ , then by Lemma 4.4 E (¯ ω µ ) + G ∗ ¯ ω µ ( x µ ) ≤ E ( ω µ ) + G ∗ ω µ ( x µ ) . (5.8)Again by Riesz’s rearrangement inequality we have − Z D Z D h ( x, y )¯ ω µ ( x )¯ ω µ ( y ) dxdy − π Z D ln | x µ − y | ¯ ω µ ( y ) dy − Z D h ( x µ , y )¯ ω µ ( y ) dy ≤ − Z D Z D h ( x, y ) ω µ ( x ) ω µ ( y ) dxdy + G ∗ ω µ ( x µ ) , (5.9)since h is bounded from below in D × D and x µ is away from ∂D , we get G ∗ ω µ ( x µ ) ≥ − C − H ( x µ ) − π Z D ln | x µ − y | ¯ ω µ ( y ) dy − H ( x µ ) ≥ − µ π Z B s (0) ln | y | dy − C ≥ − π ln s − C, (5.10)where we use R B s (0) ln | y | dy = πs (ln s − ). Lemma 5.5.
There exists ρ µ satisfying ρ µ → and R B ρµ ( x µ ) ω µ ( x ) dx → as µ → + ∞ .Proof. By Lemma 5.4, − π Z D ln | x µ − y | ω µ ( y ) dy − Z D h ( x µ , y ) ω µ ( y ) dy ≥ − π ln s − C, (5.11)since h is bounded from below in D × D , we get Z D ln s | x µ − y | ω µ ( y ) dy ≥ − C. (5.12)Now choose R > Z B Rs ( x a ) ln s | x µ − y | ω µ ( y ) dy + Z D \ B Rs ( x µ ) ln s | x µ − y | ω µ ( y ) dy ≥ − C. (5.13)Observe that Z B Rs ( x µ ) ln s | x µ − y | ω µ ( y ) dy ≤ µ Z B s ( x µ ) ln s | x µ − y | dy = 12 , (5.14)so we get Z D \ B Rs ( x µ ) ln sRs ω µ ( y ) dy ≥ Z D \ B Rs ( x µ ) ln s | x µ − y | ω µ ( y ) dy ≥ − C, (5.15)teady Vortex Patch Solutions to the Vortex-Wave System 22which gives Z D \ B Rs ( x µ ) ω µ ( y ) dy ≤ C ln R , (5.16)but R D ω µ ( x ) dx = 1, we have 1 ≥ Z B Rs ( x µ ) ω µ ( y ) dy ≥ − C ln R , (5.17)then the lemma is proved by choosing R = s − and ρ µ = s .Since x µ is bounded and away from ∂D , we assume that x µ → x ∗ ∈ D (up to a subsequence)as µ → + ∞ . An argument similar to the one in Remark 2.5 shows that ω µ → δ ( x ∗ ) in thedistributional sense. Lemma 5.6. H ( x ∗ ) = min x ∈ D H ( x ) . Proof.
Since H = + ∞ on ∂D , there exists x such that H ( x ) = min x ∈ D H ( x ). It suffices toshow H ( x ) ≥ H ( x ∗ ). Define ¯ ω µ = µI B s ( x ) ∈ N µ , then by Lemma 5.1 E (¯ ω µ ) + G ∗ ¯ ω µ ( x ) − H ( x ) ≤ E ( ω µ ) + G ∗ ω µ ( x µ ) − H ( x µ ) , (5.18)that is, − π Z D Z D ln | x − y | ¯ ω µ ( x )¯ ω µ ( y ) dxdy − Z D Z D h ( x, y )¯ ω µ ( x )¯ ω µ ( y ) dxdy − π Z D ln | x − y | ¯ ω µ ( y ) dy − Z D h ( x , y )¯ ω µ ( y ) dy − H ( x ) ≤ − π Z D Z D ln | x − y | ω µ ( x ) ω µ ( y ) dxdy − Z D Z D h ( x, y ) ω µ ( x ) ω µ ( y ) dxdy − π Z D ln | x µ − y | ω µ ( y ) dy − Z D h ( x µ , y ) ω µ ( y ) dy − H ( x µ ) (5.19)Taking the limit in (5.19), by rearrangement inequality we obtain H ( x ) ≥ H ( x ∗ ) , (5.20)which completes the proof. Proof of Theorem 2.4.
By choosing r µ = ρ µ + | x µ − x ∗ | , Theorem 2.4 is an easy consequence ofLemma 5.5 and Lemma 5.6. Remark . There may be a better convergence for ω µ , that is, the support of ω µ shrinks to x ∗ as µ → + ∞ , but we have not yet proved this. The main difficulty to estimate the size of supp ( ω µ ) is that the mutual interaction energy between the background vorticity and the pointvortex is very large, and energy estimate does not provide enough information anymore.teady Vortex Patch Solutions to the Vortex-Wave System 23 Acknowledgements:
D. Cao was supported by NNSF of China (grant No. 11331010) andChinese Academy of Sciences by grant QYZDJ-SSW-SYS021. G. Wang was supported by NNSFof China (grant No.11771469).
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