Existence and asymptotic behavior of large axisymmetric solutions for steady Navier-Stokes system in a pipe
aa r X i v : . [ m a t h . A P ] J a n EXISTENCE AND ASYMPTOTIC BEHAVIOR OF LARGEAXISYMMETRIC SOLUTIONS FOR STEADY NAVIER-STOKESSYSTEM IN A PIPE
YUN WANG AND CHUNJING XIE
Abstract.
In this paper, the existence and uniqueness of strong axisymmetric solutionswith large flux for the steady Navier-Stokes system in a pipe are established even whenthe external force is also suitably large in L . Furthermore, the exponential convergencerate at far fields for the arbitrary steady solutions with finite H distance to the Hagen-Poiseuille flows is established as long as the external forces converge exponentially at farfields. The key point to get the existence of these large solutions is the refined estimate forthe derivatives in the axial direction of the stream function and the swirl velocity, whichexploits the good effect of the convection term. An important observation for the asymptoticbehavior of general solutions is that the solutions are actually small at far fields when theyhave finite H distance to the Hagen-Poiseuille flows. This makes the estimate for thelinearized problem play a crucial role in studying the convergence of general solutions at farfields. Introduction and Main Results
An important physical problem in fluid mechanics is to study the flows in nozzles. Givenan infinitely long nozzle ˜Ω, a natural problem is to investigate the well-posedness theory forthe steady Navier-Stokes system(1) ( ( u · ∇ ) u − ∆ u + ∇ p = 0 , in ˜Ωdiv u = 0 , in ˜Ω , supplemented with the no slip conditions, i.e.,(2) u = 0 on ∂ ˜Ω , where u = ( u x , u y , u z ) denotes the velocity field of the flows. If ˜Ω is a straight cylinder ofthe form Σ × R where Σ is a smooth two dimensional domain, then there exists a solution Mathematics Subject Classification.
Key words and phrases.
Hagen-Poiseuille flows, steady Navier-Stokes system, large solution, asymptoticbehavior, pipe.Updated on January 14, 2020. u = (0 , , u z ( x, y )) satisfying (1) and (2). The solution is called the Poiseuille flow and isuniquely determined by the flow flux Φ defined by(3) Φ = Z Σ u z ( x, y ) dS. In particular, if the straight cylinder is a pipe, i.e., Σ is the unit disk B (0), then theassociated Poiseuille flows ¯ U = ¯ U ( r ) e z have the explicit form as follows,(4) ¯ U ( r ) = 2Φ π (1 − r ) with r = p x + y , which are also called Hagen-Poiseuille flows.Given an infinitely long nozzle ˜Ω tending to a straight cylinder Ω at far fields, the problemon the well-posedness theory for (1)-(2) together with the condition that the velocity fieldconverges to the Posieuille flows in Ω, is called Leray’s problem nowadays and it was firstaddressed by Leray ( [14]) in 1933. The first significant contribution to the solvability ofLeray’s problem is due to Amick [3,4], who reduced the proof of existence to the resolution ofa well-known variational problem related to the stability of the Poiseuille flow in a straightcylinder. However, Amick left out the investigation of uniqueness and existence of thesolutions with large flux. A rich and detailed analysis of the problem is due to Ladyzhenskayaand Solonnikov [13]. However, the asymptotic far field behavior of the solutions obtainedin [13] is not very clear. Therefore, in order to get a complete resolution for Leray’s problem,the key issue is to study the asymptotic behavior for the solutions of the steady Navier-Stokes equations in infinitely long nozzles. The asymptotic behavior for steady Navier-Stokes system in a nozzle was studied extensively in the literature, see [2, 10, 11, 13, 21] andreferences therein. A classical and straightforward way to prove the asymptotic behavior forsteady solutions of Navier-Stokes system is to derive a differential inequality for the localizedenergy [10]. This approach was later refined in [2, 11, 13, 21] and the book by Galdi [6], etc.The asymptotic behavior obtained via this method is also only for the solutions of steadyNavier-Stokes system with small fluxes. A significant open problem posed in [6, p. 19] is theglobal well-posedness for Leray’s problem in a general nozzle when the flux Φ is large.Another approach to prove the convergence to Poiseuille flows of steady solutions forNavier-Stokes system is the blowup technique. In fact, with the aid of the compactnessobtained in [13], global well-posedness for the Leray’s problem in a general nozzle tending toa straight cylinder could be established even when the flux Φ is large, provided that we canprove global uniqueness or some Liouville type theorem for Poiseuille flows in the straightcylinder. In order to study the global uniqueness of Poiseuille flows in a straight cylinder, animportant step is to prove the uniqueness in a bounded set in a suitable metric space. Theuniqueness of Hagen-Poiseuille flows in a uniformly small neighborhood (independent of the XISTENCE AND ASYMPTOTIC BEHAVIOR OF LARGE AXISYMMETRIC SOLUTIONS 3 size of the flux) was obtained in [24]. More precisely, suppose that Ω = B (0) × R is a pipeand F = ( F x , F y , F z ) is external force, does the problem(5) ( ( u · ∇ ) u − ∆ u + ∇ p = F , in Ω , div u = 0 , in Ω , supplemented with no-slip boundary condition(6) u = 0 on ∂ Ωand the flux constraint(7) Z B (0) u z ( x, y, z ) dxdy = Φhave a unique solution in the neighborhood of the Hagen-Poiseuille flows when the externalforce is small? The uniform nonlinear structural stability of Hagen-Poiseuille flows wasestablished in [24] in the axisymmetric setting. It was proved in [24] that the problem(5)-(7) has a unique axisymmetric solution u satisfying(8) k u − ¯ U k H (Ω) ≤ C k F k L (Ω) and(9) k u − ¯ U k H (Ω) ≤ C (1 + Φ ) k F k L (Ω) . when the L − norm of F is smaller than a uniform constant independent of the flux Φ.The main goal of this paper contains two parts. The first one is to show the existenceand uniqueness of strong solutions for the problem (5)-(7), when F is large in the casethat the flux is large. The second goal in this paper is to investigate the convergence rateof steady solutions of Navier-Stokes system in a pipe which have finite H distance to theHagen-Poiseuille flows, even when the flows have large fluxes.Our first main result is stated as follows. Theorem 1.1.
Assume that F ∈ L (Ω) and F = F ( r, z ) is axisymmetric. There exists aconstant Φ ≥ , such that if (10) Φ ≥ Φ and k F k L (Ω) ≤ Φ , then the problem (5) – (7) admits a unique axisymmetric solution u satisfying (11) k u − ¯ U k H (Ω) ≤ C Φ and k u r k L (Ω) ≤ Φ − , where C is a constant independent of Φ and F . Moreover, the solution u satisfies that (12) k u − ¯ U k H (Ω) ≤ C Φ . YUN WANG AND CHUNJING XIE
We have the following remarks on Theorem 1.1.
Remark 1.1.
The crucial point of Theorem 1.1 is that the external force F can be very largewhen the flux of the flow is large. Remark 1.2. If Φ is sufficiently large, F = 0 and k u − ¯ U k H (Ω) ≤ Φ , then u ≡ ¯ U .It means that ¯ U is the unique solution in a bounded set with large radius Φ . This can beregarded as a step to get Liouville type theorem for steady Navier-Stokes system in a pipe. In case that F has additional structure at far fields, we have the following asymptoticbehavior of solutions of Navier-Stokes system in a pipe. Theorem 1.2.
Assume that F ∈ H (Ω) and F = F ( r, z ) is axisymmetric. There exists aconstant α depending only on Φ , such that if F = F ( r, z ) satisfies (13) k e α | z | F k L (Ω) < + ∞ , with some α ∈ (0 , α ) , and u is an axisymmetric solution to the problem (5) – (7) , satisfying (14) k u − ¯ U k H (Ω) < + ∞ , then one has (15) k e αz ( u − ¯ U ) k H (Ω ∩{ z ≥ } ) + k e − αz ( u − ¯ U ) k H (Ω ∩{ z ≤ } ) < + ∞ . There are a few remarks in order.
Remark 1.3.
The key point of Theorem 1.2 is that there is neither smallness assumptionon the flux Φ nor the smallness on the deviation of u with ¯ U . Remark 1.4.
It follows from (9) and (12) that the solutions obtained in [24] and Theorem1.1 satisfy the condition (14) . Hence if F in [24] and Theorem 1.1 also satisfies (13) , thenthe associated solutions must converge to Hagen-Poiseuille flows exponentially fast. Remark 1.5. If F decays to zero with an algebraic rate, i.e., F satisfies (16) k z k F k L (Ω) < + ∞ , with some k ∈ N , then under the condition (14) , using the same idea of the proof for Theorem1.2 yields that the axisymmetric solution u of the problem (5) – (7) converges to the Hagen-Poiseuille flows with the same algebraic rates, i.e., u satisfies (17) k z k ( u − ¯ U ) k H (Ω) < + ∞ . Remark 1.6.
The same asymptotic behavior also holds for the axisymmetric flows in semi-infinite pipes.
XISTENCE AND ASYMPTOTIC BEHAVIOR OF LARGE AXISYMMETRIC SOLUTIONS 5
The structure of this paper is as follows. In Section 2, we introduce the stream functionformulation for the linearized problem of axisymmetric Navier-Stokes system and recall theexistence results obtained in [24]. Some good estimates for the derivatives in the axialdirection of the stream function and the swirl velocity are established in Section 3. Theseare the key ingredients to get the existence and uniqueness of large solutions when F is large.The existence of solutions for the nonlinear problem is obtained via standard iteration inSection 4 . Section 5 devotes to the study on the convergence rates of the flows at far fields,where the key observation is that u − ¯ U must be small at far fields when the condition(14) holds so that the estimate for the linearized problem can be used. Some importantinequalities are collected in Appendix A.2. Stream function formulation and existence results
To get the existence of the Navier-Stokes equations, we start from the following linearizedsystem around Hagen-Poiseuille flows,(18) ( ¯ U · ∇ v + v · ∇ ¯ U − ∆ v + ∇ P = F , in Ω , div v = 0 , in Ω , supplemented with no-slip boundary conditions and the flux constraint,(19) v = 0 on ∂ Ω , Z B (0) v z ( · , · , z ) dS = 0 for any z ∈ R . Stream function formulation.
In terms of the cylindrical coordinates, an axisym-metric solution v can be written as v = v r ( r, z ) e r + v z ( r, z ) e z + v θ ( r, z ) e θ . Then the linearized equation for the Navier-Stokes system (5) around Hagen-Poiseuille flowscan be written as(20) ¯ U ( r ) ∂v r ∂z + ∂P∂r − (cid:20) r ∂∂r (cid:18) r ∂v r ∂r (cid:19) + ∂ v r ∂z − v r r (cid:21) = F r in D,v r ∂ ¯ U∂r + ¯ U ( r ) ∂v z ∂z + ∂P∂z − (cid:20) r ∂∂r (cid:18) r ∂v z ∂r (cid:19) + ∂ v z ∂z (cid:21) = F z in D,∂ r v r + ∂ z v z + v r r = 0 in D and(21) ¯ U ( r ) ∂ z v θ − (cid:20) r ∂∂r (cid:18) r ∂v θ ∂r (cid:19) + ∂ v θ ∂z − v θ r (cid:21) = F θ in D. YUN WANG AND CHUNJING XIE
Here F r , F z , and F θ are the radial, axial, and azimuthal component of F , respectively, and D = { ( r, z ) : r ∈ (0 , , z ∈ R } . The no-slip boundary conditions and the flux constraint(19) become(22) v r (1 , z ) = v z (1 , z ) = 0 , Z rv z ( r, z ) dr = 0 , and(23) v θ (1 , z ) = 0 . It follows from the third equation in (20) that there exists a stream function ψ ( r, z )satisfying(24) v r = ∂ z ψ and v z = − ∂ r ( rψ ) r . The azimuthal vorticities of v and F are defined as ω θ = ∂ z v r − ∂ r v z = ∂∂r (cid:18) r ∂∂r ( rψ ) (cid:19) + ∂ z ψ and f = ∂ z F r − ∂ r F z , respectively. It follows from the first two equations in (20) that(25) ¯ U ( r ) ∂ z ω θ − (cid:18) ∂ r + ∂ z + 1 r ∂ r (cid:19) ω θ + ω θ r = f. Denote L = ∂∂r (cid:18) r ∂∂r ( r · ) (cid:19) = ∂ ∂r + 1 r ∂∂r − r . Hence ω θ = ( L + ∂ z ) ψ and ψ satisfies the following fourth order equation,(26) ¯ U ( r ) ∂ z ( L + ∂ z ) ψ − ( L + ∂ z ) ψ = f. Next, we derive the boundary conditions for ψ . As discussed in [15], in order to getclassical solutions, some compatibility conditions at the axis should be imposed. Assumethat v and the vorticity ω are continuous so that v r (0 , z ) and ω θ (0 , z ) should vanish. Thisimplies ∂ z ψ (0 , z ) = ( L + ∂ z ) ψ (0 , z ) = 0 . Without loss of generality, one can assume that ψ (0 , z ) = 0. Hence, the following compati-bility condition holds at the axis,(27) ψ (0 , z ) = L ψ (0 , z ) = 0 . On the other hand, it follows from (22) that Z ∂ r ( rψ )( r, z ) dr = − Z rv z dr = 0 . XISTENCE AND ASYMPTOTIC BEHAVIOR OF LARGE AXISYMMETRIC SOLUTIONS 7
This, together with (27), gives(28) ψ (1 , z ) = lim r → ( rψ )( r, z ) = 0 . Moreover, according to the homogeneous boundary conditions for v , one has ∂∂r ( rψ ) | r =1 = rv z | r =1 = 0 . This, together with (28), implies(29) ∂ψ∂r (1 , z ) = 0 . Meanwhile, if v is continuous, then the compatibility conditions for v obtained in [15]implies v θ (0 , z ) = 0. Hence v θ satisfies the following problem(30) ¯ U ( r ) ∂ z v θ − (cid:20) r ∂∂r (cid:18) r ∂v θ ∂r (cid:19) + ∂ v θ ∂z − v θ r (cid:21) = F θ in D,v θ (1 , z ) = v θ (0 , z ) = 0 . Now let us introduce some notations and recall the existence results for ψ and v θ obtainedin [24]. For a given function g ( r, z ), define its Fourier transform with respect to z variableby ˆ g ( r, ξ ) = Z R g ( r, z ) e − iξz dz. ℜ g and ℑ g denote the real and imaginary part of a function or a number g , respectively. Definition 2.1.
Define a function space C ∞∗ ( D ) as follows C ∞∗ ( D ) = ϕ ( r, z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ϕ ∈ C ∞ c ([0 , × R ) , ϕ (1 , z ) = ∂ϕ∂r (1 , z ) = 0 , and lim r → L k ϕ ( r, z ) = lim r → ∂∂r ( r L k ϕ )( r, z ) = 0 , k ∈ N . The H r ( D ) -norm and H r ( D ) -norm are defined as follows, (31) k ϕ k H r ( D ) := Z + ∞−∞ Z "(cid:12)(cid:12)(cid:12)(cid:12) ∂∂r ( r L ˆ ϕ ) (cid:12)(cid:12)(cid:12)(cid:12) r + ξ |L ˆ ϕ | r + ξ (cid:12)(cid:12)(cid:12)(cid:12) ∂∂r ( r ˆ ϕ ) (cid:12)(cid:12)(cid:12)(cid:12) r + ξ | ˆ ϕ | r dr + Z + ∞−∞ Z " |L ˆ ϕ | r + ξ (cid:12)(cid:12)(cid:12)(cid:12) ∂∂r ( r ˆ ϕ ) (cid:12)(cid:12)(cid:12)(cid:12) r + ξ | ˆ ϕ | r dr + Z + ∞−∞ Z "(cid:12)(cid:12)(cid:12)(cid:12) ∂∂r ( r ˆ ϕ ) (cid:12)(cid:12)(cid:12)(cid:12) r + ξ | ˆ ϕ | r + | ˆ ϕ | r dr. YUN WANG AND CHUNJING XIE H ∗ ( D ) denotes the closure of C ∞∗ ( D ) under the H r ( D ) -norm. Furthermore, L r ( D ) is thecompletion of C ∞ ( D ) under the L r ( D ) -norm defined as follows k f k L r ( D ) = Z + ∞−∞ Z | f | r drdz. The existence of solutions for the problems (26)–(29) and (30) has been established in [24].
Proposition 2.1. [24, Theorem 1.1] Assume that F ∗ = F r e r + F z e z ∈ L (Ω) and F ∗ isaxisymmetric. There exists a unique solution ψ ∈ H ∗ ( D ) to the linear system (26) – (29) ,and a positive constant C independent of F ∗ and Φ , such that k v ∗ k H (Ω) ≤ C k F ∗ k L (Ω) , and k v ∗ k H (Ω) ≤ C (1 + Φ ) k F ∗ k L (Ω) , where v ∗ = v r e r + v z e z = ∂ z ψ e r − ∂ r ( rψ ) r e z . Proposition 2.2. [24, Proposition 4.6] Assume that F θ = F θ e θ ∈ L (Ω) and F θ = F θ ( r, z ) is axisymmetric. There exist a unique solution v θ to the linear problem (30) and a positiveconstant C independent of F θ and Φ , such that k v θ k H (Ω) ≤ C k F θ k L (Ω) , where v θ = v θ e θ . Some Refined estimates for solutions of Linearized problem
Propositions 2.1-2.2 provide some uniform estimates for ψ and v θ . They are the keyingredients to get the existence and uniqueness of solutions of the steady Navier-Stokessystem, when F is uniformly small [24]. In this section, we give some refined estimates,especially for the z -derivatives of ψ and v θ , which yield the existence of large solutions ofsteady Navier-Stokes system even when the external force is large. These estimates alsoshow the stabilizing effect of the linearized convection term when Φ is large.We take the Fourier transform with respect to z for the equation (26) . For every fixed ξ ,ˆ ψ satisfies(32) iξ ¯ U ( r )( L − ξ ) ˆ ψ − ( L − ξ ) ˆ ψ = ˆ f = iξ c F r − ddr c F z . Furthermore, the boundary conditions (27)-(29) can be written as(33) ˆ ψ (0) = ˆ ψ (1) = ˆ ψ ′ (1) = L ˆ ψ (0) = 0 . First, let us recall the a priori estimates obtained in [24] which hold for every ξ ∈ R . XISTENCE AND ASYMPTOTIC BEHAVIOR OF LARGE AXISYMMETRIC SOLUTIONS 9
Proposition 3.1. [24, Section 6] Let ˆ ψ ( r, ξ ) be a smooth solution of the problem (32) – (33) .Then it holds (34) Z |L ˆ ψ | r dr + ξ Z (cid:12)(cid:12)(cid:12)(cid:12) ddr ( r ˆ ψ ) (cid:12)(cid:12)(cid:12)(cid:12) r dr + ξ Z | ˆ ψ | r dr ≤ C Z ( | F r | + | F z | ) r dr. The next two propositions give some further estimates for ψ , especially the z -derivativesof ψ . Proposition 3.2.
Assume that F r = F r e r ∈ L (Ω) , the solution ψ of the problem (35) ¯ U ( r ) ∂ z ( L + ∂ z ) ψ − ( L + ∂ z ) ψ = ∂ z F r ,ψ (0) = ψ (1) = ∂∂r ψ (1) = L ψ (0) = 0 , satisfies k v ∗ k L (Ω) ≤ C Φ − k F r k L (Ω) and k v ∗ k H (Ω) ≤ C Φ − k F r k L (Ω) . Proof.
Taking the Fourier transform with respect to z for the system (35) yields that forevery fixed ξ , ˆ ψ ( r, ξ ) satisfies(36) iξ ¯ U ( r )( L − ξ ) ˆ ψ − ( L − ξ ) ˆ ψ = iξ c F r . Multiplying (36) by r ˆ ψ and integrating the resulting equation over [0 ,
1] give Z iξ ¯ U ( r )( L − ξ ) ˆ ψ ˆ ψr dr − Z ( L − ξ ) ˆ ψ ˆ ψr dr = iξ Z c F r ˆ ψr dr. It follows from the direct computations and integration by parts that one has(37) ξ Z ¯ U ( r ) r (cid:12)(cid:12)(cid:12)(cid:12) ddr ( r ˆ ψ ) (cid:12)(cid:12)(cid:12)(cid:12) dr + ξ Z ¯ U ( r ) | ˆ ψ | r dr = −ℜ Z ξ c F r ˆ ψr dr. This, together with together with Lemma A.2, givesΦ Z (cid:12)(cid:12)(cid:12)(cid:12) ddr ( r ˆ ψ ) (cid:12)(cid:12)(cid:12)(cid:12) − r r dr ≤ C Z | c F r || ˆ ψ | r dr ≤ C (cid:18)Z | c F r | r dr (cid:19) Z (cid:12)(cid:12)(cid:12)(cid:12) ddr ( r ˆ ψ ) (cid:12)(cid:12)(cid:12)(cid:12) − r r dr ! . Hence, we have(38) Z (cid:12)(cid:12)(cid:12)(cid:12) ddr ( r ˆ ψ ) (cid:12)(cid:12)(cid:12)(cid:12) − r r dr ≤ C Φ − Z | c F r | r dr. It follows from A.3, Lemma A.1, and Proposition 3.1 that(39) Z (cid:12)(cid:12)(cid:12)(cid:12) ddr ( r ˆ ψ ) (cid:12)(cid:12)(cid:12)(cid:12) r dr ≤ C Z (cid:12)(cid:12)(cid:12)(cid:12) ddr ( r ˆ ψ ) (cid:12)(cid:12)(cid:12)(cid:12) − r r dr ! (cid:18)Z |L ˆ ψ | r dr (cid:19) ≤ C Φ − Z | c F r | r dr. This implies(40) k v z k L (Ω) = Z + ∞−∞ Z (cid:12)(cid:12)(cid:12)(cid:12) ddr ( r ˆ ψ ) (cid:12)(cid:12)(cid:12)(cid:12) r drdξ ≤ C Φ − k F r k L (Ω) . Similarly, the equality (37), together with Lemma A.2, gives(41) Φ ξ Z (1 − r ) | ˆ ψ | r dr ≤ C (cid:18)Z | c F r | r dr (cid:19) Z (cid:12)(cid:12)(cid:12)(cid:12) ddr ( r ˆ ψ ) (cid:12)(cid:12)(cid:12)(cid:12) − r r dr ! ≤ C Φ − Z | c F r | r dr. It follows from Lemma A.3, Lemma A.1, and Proposition 3.1 again that one has(42) ξ Z | ˆ ψ | r dr ≤ C (cid:18) ξ Z (1 − r ) | ˆ ψ | r dr (cid:19) ξ Z (cid:12)(cid:12)(cid:12)(cid:12) ddr ( r ˆ ψ ) (cid:12)(cid:12)(cid:12)(cid:12) r dr ! ≤ C Φ − Z | c F r | r dr. This implies(43) k v r k L (Ω) = Z + ∞−∞ Z ξ | ˆ ψ | r drdξ ≤ C Φ − k F r k L (Ω) . By the interpolation between L (Ω) and H (Ω), one has(44) k v ∗ k H (Ω) ≤ C k v ∗ k L (Ω) k v ∗ k H (Ω) ≤ C Φ − k F r k L (Ω) . This finishes the proof of Proposition 3.2. (cid:3)
Next, we study the case when F has only axial component. Proposition 3.3.
Assume that F z = F z e z ∈ L (Ω) , the solution ψ of the following problem (45) ¯ U ( r ) ∂ z ( L + ∂ z ) ψ − ( L + ∂ z ) ψ = − ∂ r F z ,ψ (0) = ψ (1) = ∂∂r ψ (1) = L ψ (0) = 0 , XISTENCE AND ASYMPTOTIC BEHAVIOR OF LARGE AXISYMMETRIC SOLUTIONS 11 satisfies k v r k L (Ω) ≤ C Φ − k F z k L (Ω) and k v r k H (Ω) ≤ C Φ − k F z k L (Ω) , where v r = v r e r = ∂ z ψ e r .Proof. Taking the Fourier transform with respect to z for the system (45) yields that forfixed ξ , ˆ ψ ( r, ξ ) satisfies(46) iξ ¯ U ( r )( L − ξ ) ˆ ψ − ( L − ξ ) ˆ ψ = − ddr c F z . Multiplying (46) by r ˆ ψ and integrating the resulting equation over [0 ,
1] give(47) ξ Z ¯ U ( r ) r (cid:12)(cid:12)(cid:12)(cid:12) ddr ( r ˆ ψ ) (cid:12)(cid:12)(cid:12)(cid:12) dr + ξ Z ¯ U ( r ) | ˆ ψ | r dr = ℑ Z c F z ddr ( r ˆ ψ ) dr. This, together with Proposition 3.1, implies(48) Φ ξ Z (cid:12)(cid:12)(cid:12)(cid:12) ddr ( r ˆ ψ ) (cid:12)(cid:12)(cid:12)(cid:12) − r r dr ≤ C | ξ | Z | c F z | (cid:12)(cid:12)(cid:12)(cid:12) ddr ( r ˆ ψ ) (cid:12)(cid:12)(cid:12)(cid:12) dr ≤ C Z | c F z | r dr. Applying Lemma A.2 yields(49) ξ Z | ˆ ψ | r dr ≤ Cξ Z (cid:12)(cid:12)(cid:12)(cid:12) ddr ( r ˆ ψ ) (cid:12)(cid:12)(cid:12)(cid:12) − r r dr ≤ C Φ − Z | c F z | r dr. Therefore, one has(50) k v r k L (Ω) ≤ C Φ − k F z k L (Ω) . It follows from the interpolation and Proposition 2.1 that one has(51) k v r k H (Ω) ≤ C Φ − k F z k L (Ω) . This finishes the proof of Proposition 3.3. (cid:3)
Now we are in position to analyze v θ . Taking the Fourier transform with respect to z forthe problem (30) gives(52) iξ ¯ U ( r ) b v θ − ( L − ξ ) b v θ = c F θ , b v θ (1) = b v θ (0) = 0 . Let us first recall the uniform estimate for v θ obtained in [24]. Proposition 3.4. [24, Proposition 3.1] Assume that b v θ is a smooth solution to the problem (52) . For every fixed ξ , it holds that (53) Z |L b v θ | r dr + ξ Z (cid:12)(cid:12)(cid:12)(cid:12) ddr ( r b v θ ) (cid:12)(cid:12)(cid:12)(cid:12) r dr + ξ Z | b v θ | r dr ≤ C Z | c F θ | r dr. The next two propositions give some further estimates for v θ . Proposition 3.5.
Assume that F θ = F θ e θ ∈ L (Ω) . The solution v θ to the linear problem (30) satisfies that k ∂ z v θ k L (Ω) ≤ C Φ − k F θ k L (Ω) . Proof.
Multiplying the equation in (52) by r b v θ and integrating the resulting equation over[0 ,
1] yield(54) Z iξ ¯ U ( r ) | b v θ | r dr + Z (cid:12)(cid:12)(cid:12)(cid:12) ddr ( r b v θ ) (cid:12)(cid:12)(cid:12)(cid:12) r dr + ξ Z | b v θ | r dr = Z c F θ b v θ r dr. It follows from H¨older inequality, Lemma A.1, and (54) that one has(55) Z (cid:12)(cid:12)(cid:12)(cid:12) ddr ( r b v θ ) (cid:12)(cid:12)(cid:12)(cid:12) r dr + ξ Z | b v θ | r dr ≤ Z | c F θ | r dr and(56) Φ | ξ | Z (1 − r ) | b v θ | r dr ≤ C Z | c F θ || b v θ | r dr. The equality (56), together with (53), gives(57) | ξ | Z (1 − r ) | b v θ | r dr ≤ C Φ − (cid:18)Z | c F θ | r dr (cid:19) (cid:18) ξ Z | b v θ | r dr (cid:19) ≤ C Φ − Z | c F θ | r dr. By Lemma A.3 and (55), one has(58) | ξ | Z (1 − r ) | b v θ | r dr ≤ C (cid:18) | ξ | Z (1 − r ) | b v θ | r dr (cid:19) Z (cid:12)(cid:12)(cid:12)(cid:12) ddr ( r b v θ ) (cid:12)(cid:12)(cid:12)(cid:12) r dr ! ≤ C Φ − Z | c F θ | r dr. This implies(59) k ∂ z v θ k L (Ω) ≤ C Φ − k F θ k L (Ω) . Hence the proof of Proposition 3.5 is completed. (cid:3)
XISTENCE AND ASYMPTOTIC BEHAVIOR OF LARGE AXISYMMETRIC SOLUTIONS 13
Proposition 3.6.
Assume that G θ = G θ e θ ∈ H (Ω) . The solution v θ to the following linearproblem (60) ¯ U ( r ) ∂ z v θ − (cid:20) r ∂∂r (cid:18) r ∂v θ ∂r (cid:19) + ∂ v θ ∂z − v θ r (cid:21) = ∂ z G θ in D,v θ (1 , z ) = v θ (0 , z ) = 0 , satisfies k v θ k L (Ω) ≤ C Φ − k G θ k L (Ω) and k v θ k H (Ω) ≤ C Φ − k ∂ z G θ k L (Ω) k G θ k L (Ω) . Proof.
Similar computations as that in the proof of Proposition 3.5 yield(61) Z iξ ¯ U ( r ) | b v θ | r dr + Z (cid:12)(cid:12)(cid:12)(cid:12) ddr ( r b v θ ) (cid:12)(cid:12)(cid:12)(cid:12) r dr + ξ Z | b v θ | r dr = iξ Z c G θ b v θ r dr. It follows from H¨older inequality and Lemma A.1 that one has(62) Z (cid:12)(cid:12)(cid:12)(cid:12) ddr ( r b v θ ) (cid:12)(cid:12)(cid:12)(cid:12) r dr + ξ Z | b v θ | r dr ≤ C Z | c G θ | r dr and(63) Φ Z (1 − r ) | b v θ | r dr ≤ C Z | c G θ | r dr. Hence, it holds that(64) Z | b v θ | r dr ≤ C (cid:18)Z (1 − r ) | b v θ | r dr (cid:19) Z (cid:12)(cid:12)(cid:12)(cid:12) ddr ( r b v θ ) (cid:12)(cid:12)(cid:12)(cid:12) r dr ! ≤ C Φ − Z | c G θ | r dr. This implies(65) k v θ k L (Ω) ≤ C Φ − k G θ k L (Ω) . By the interpolation and Proposition 2.2, one has(66) k v θ k H (Ω) ≤ C k v θ k L (Ω) k v θ k H (Ω) ≤ C Φ − k G θ k L (Ω) k ∂ z G θ k L (Ω) . Hence the proof of Proposition 3.6 is finished. (cid:3) Existence and Uniqueness of solutions for nonlinear problem
In this section, we prove the existence and uniqueness of strong axisymmetric solution ofthe nonlinear problem (5)-(7), in particular, when F and Φ are large. The stream function ψ satisfies the following equation(67) ¯ U ( r ) ∂ z ( L + ∂ z ) ψ − ( L + ∂ z ) ψ = ∂ z F r − ∂ r F z − ∂ r ( v r ω θ ) − ∂ z ( v z ω θ ) + ∂ z " (cid:0) v θ (cid:1) r , supplemented with the boundary condition,(68) ψ (0 , z ) = ψ (1 , z ) = ∂ r ψ (1 , z ) = L ψ (0 , z ) = 0 . Here ω θ = − ∂ r v z + ∂ z v r = ( L + ∂ z ) ψ .The swirl velocity v θ = v θ e θ satisfies the equation(69) ¯ U ( r ) ∂ z v θ − ∆ v θ = F θ − ( v r ∂ r + v z ∂ z ) v θ − v r r v θ supplemented with the homogeneous boundary condition(70) v θ = 0 on ∂ Ω . Proof of Theorem 1.1.
We divide the proof into three steps.
Step 1. Iteration scheme.
The existence of solutions is proved via an iteration method.Let F ∗ = F r e r + F z e z . For any given F = (cid:0) F ∗ , F θ (cid:1) ∈ L (Ω) × L (Ω), there exists a uniquesolution ( ψ, v θ ) to the combined linear problem (21), (23), and (26)–(29), and we denotethis solution by T F . Let Ψ = T F , Ψ n = ( ψ n , v θn ) , and(71) Ψ n +1 = Ψ + T ( F ∗ n , F θn )with(72) F ∗ n = (cid:20) − v zn ω θn + ( v θn ) r (cid:21) e r + v rn ω θn e z , F θn = − ( v rn ∂ r + v zn ∂ z ) v θn − v rn r v θn , where v ri = ∂ z ψ i , v zi = − ∂ r ( rψ i ) r , v θi = v θi · e θ , ω θi = − ∂ r v zi + ∂ z v ri , i ∈ N . Set S = ( ψ, v θ ) ∈ H ∗ ( D ) × H (Ω) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k v ∗ k H (Ω) ≤ C Φ , k v r k L (Ω) ≤ Φ − k v θ k H (Ω) ≤ C Φ , k ∂ z v θ k L (Ω) ≤ Φ − . XISTENCE AND ASYMPTOTIC BEHAVIOR OF LARGE AXISYMMETRIC SOLUTIONS 15
Here C and C are the two constants indicated in Propositions 2.1-2.2. Under the assump-tion that k F k L (Ω) ≤ Φ , according to Propositions 2.1-2.2, Propositions 3.2-3.5, one has Ψ ∈ S , when Φ is large enough. Assume that Ψ n ∈ S , our aim is to prove Ψ n +1 ∈ S . Step 2. Estimate for the velocity field and existence.
Denote v ∗ i = v ri e r + v zi e z . It followsfrom Sobolev embedding inequalities that the estimates(73) k v rn ω θn k L (Ω) ≤ C k v rn k L (Ω) k ω θn k L (Ω) ≤ C k v rn k L (Ω) k v rn k L ∞ (Ω) k v ∗ n k H (Ω) ≤ C k v rn k L (Ω) k v ∗ n k H (Ω) ≤ C Φ − · Φ · and(74) k v zn ω θn k L (Ω) ≤ k v zn k L ∞ (Ω) k ω θn k L (Ω) ≤ C k v ∗ n k H (Ω) ≤ C Φ hold. Moreover, one has(75) (cid:13)(cid:13)(cid:13)(cid:13) v θn v θn r (cid:13)(cid:13)(cid:13)(cid:13) L (Ω) ≤ C k v θn k L ∞ (Ω) (cid:13)(cid:13)(cid:13)(cid:13) v θn r (cid:13)(cid:13)(cid:13)(cid:13) L (Ω) ≤ C k v θn k H (Ω) k v θn k H (Ω) ≤ C Φ . Similarly, using Sobolev embedding inequalities gives(76) k v rn ∂ r v θn k L (Ω) ≤ C k v rn k L ∞ (Ω) k ∂ r v θn k L (Ω) ≤ C k v rn k L (Ω) k v ∗ k H (Ω) k v θn k H (Ω) ≤ C Φ − · Φ · and(77) k v zn ∂ z v θn k L (Ω) ≤ k v zn k L ∞ (Ω) k ∂ z v θn k L (Ω) ≤ C k v ∗ n k H (Ω) k ∂ z v θn k L (Ω) ≤ C Φ Φ − . Furthermore, it holds that(78) (cid:13)(cid:13)(cid:13)(cid:13) v rn v θn r (cid:13)(cid:13)(cid:13)(cid:13) L (Ω) ≤ C k v rn k L ∞ (Ω) (cid:13)(cid:13)(cid:13)(cid:13) v θn r (cid:13)(cid:13)(cid:13)(cid:13) L (Ω) ≤ C Φ − · Φ · . Combining the estimates (73)–(78) and applying Propositions 2.1-2.2, 3.2 yield(79) k v ∗ n +1 k H (Ω) ≤ C Φ + C (cid:16) Φ − · Φ · + Φ − Φ (cid:17) and(80) k v θn +1 k H (Ω) ≤ C Φ + C (cid:16) Φ − · Φ · + Φ Φ − (cid:17) . Moreover, it follows from Propositions 3.2–3.5 that one has(81) k v rn +1 k L (Ω) ≤ C max n Φ − , Φ − o (cid:16) Φ + Φ − · Φ · + Φ − Φ (cid:17) and(82) k ∂ z v θn +1 k L (Ω) ≤ C Φ − (cid:16) Φ + Φ − · Φ · + Φ Φ − (cid:17) . Choose a constant Φ large enough such that C (cid:16) Φ − · Φ · + Φ − Φ (cid:17) < C Φ , C (cid:16) Φ − · Φ · + Φ Φ − (cid:17) < C Φ , and C Φ − (cid:16) Φ + Φ − · Φ · + Φ − Φ (cid:17) < Φ − ,C Φ − (cid:16) Φ + Φ − · Φ · + Φ Φ − (cid:17) < Φ − . When Φ ≥ Φ , the estimates (79)–(82) imply that ( ψ n +1 , v θn +1 ) ∈ S . By mathematicalinduction, Ψ n ∈ S for every n ∈ N . Note that v n = v ∗ n + v θn . The above estimates show that(83) k v n k H (Ω) ≤ C Φ for every n ∈ N . Since { v n } is uniformly bounded in H (Ω), there exists a vector-valued function v ∈ H (Ω)such that v n ⇀ v in H (Ω) and k v k H (Ω) ≤ C Φ . Meanwhile, as proved in [24], the equation (71) implies that(84) curl (cid:0) ( ¯ U · ∇ ) v n +1 + ( v n +1 · ∇ ) ¯ U − ∆ v n +1 + ( v n · ∇ ) v n − F (cid:1) = 0 . Taking the limit of the equation (84) yields(85) curl (cid:0) ( ¯ U · ∇ ) v + ( v · ∇ ) ¯ U − ∆ v + ( v · ∇ ) v − F (cid:1) = 0 . Hence, there exists a function P with ∇ P ∈ L (Ω), satisfying(86) ( ¯ U · ∇ ) v + ( v · ∇ ) ¯ U − ∆ v + ( v · ∇ ) v + ∇ P = F . Moreover, according to the regularity estimates in Propositions 2.1-(2.2), it holds that(87) k v k H (Ω) ≤ C (1 + Φ ) k F k L (Ω) + C (1 + Φ ) k ( v · ∇ ) v k L (Ω) ≤ C Φ Φ + C Φ k v k H (Ω) ≤ C Φ . Let u = v + ¯ U . Then u is a solution of the problem (5)-(7). Step 3. Uniqueness.
Suppose there are two axisymmetric solutions u and ˜ u of the problem(5)–(7), satisfying(88) k u − ¯ U k H (Ω) ≤ C Φ , k ˜ u − ¯ U k H (Ω) ≤ C Φ , and(89) k ˜ u r k L (Ω) ≤ Φ − . Let v = u − ¯ U = v r e r + v θ e θ + v z e z , ˜ v = ˜ u − ¯ U = ˜ v r e r + ˜ v θ e θ + ˜ v z e z . XISTENCE AND ASYMPTOTIC BEHAVIOR OF LARGE AXISYMMETRIC SOLUTIONS 17
Suppose that ψ and ˜ ψ are the stream functions associated with the vector fields v and ˜ v ,respectively. Then ψ − ˜ ψ satisfies the following equation,¯ U ( r ) ∂ z ( L + ∂ z )( ψ − ˜ ψ ) − ( L + ∂ z ) ( ψ − ˜ ψ )= − ∂ r ( v r ω θ − ˜ v r ˜ ω θ ) − ∂ z ( v z ω θ − ˜ v z ˜ ω θ ) + 2 ∂ z (cid:18) v θ r v θ − ˜ v θ r ˜ v θ (cid:19) . It follows from Sobolev’s embedding inequalities that one has(90) k v r ω θ − ˜ v r ˜ ω θ k L (Ω) ≤ k v r − ˜ v r k L (Ω) k ω θ k L (Ω) + k ˜ v r k L (Ω) k ω θ − ˜ ω θ k L (Ω) ≤ C k v r − ˜ v r k H (Ω) k v ∗ k H (Ω) + C k ˜ v r k L (Ω) k ˜ v r k H (Ω) k v ∗ − ˜ v ∗ k H (Ω) ≤ C k v r − ˜ v r k H (Ω) Φ + C k v ∗ − ˜ v ∗ k H (Ω) Φ − · + · and(91) k v z ω θ − ˜ v z ˜ ω θ k L (Ω) ≤ C k v z − ˜ v z k L (Ω) k ω θ k L (Ω) + C k ˜ v z k L (Ω) k ω θ − ˜ ω θ k L (Ω) ≤ C k v ∗ − ˜ v ∗ k H (Ω) (cid:16) k v ∗ k H (Ω) + k ˜ v ∗ k H (Ω) (cid:17) ≤ C k v ∗ − ˜ v ∗ k H (Ω) Φ . Similarly, it holds that(92) (cid:13)(cid:13)(cid:13)(cid:13) v θ r v θ − ˜ v θ r ˜ v θ (cid:13)(cid:13)(cid:13)(cid:13) L (Ω) ≤ C k v θ − ˜ v θ k H (Ω) (cid:16) k v θ k H (Ω) + k ˜ v θ k H (Ω) (cid:17) ≤ C k v θ − ˜ v θ k H (Ω) Φ . Combining the above estimates (90)–(92), it follows from Propositions 3.2–3.3 that onehas(93) Φ k v r − ˜ v r k H (Ω) ≤ C Φ Φ − h Φ k v r − ˜ v r k H (Ω) + Φ − · k v ∗ − ˜ v ∗ k H (Ω) +Φ k v ∗ − ˜ v ∗ k H (Ω) + Φ k v θ − ˜ v θ k H (Ω) i and(94) k v z − ˜ v z k H (Ω) ≤ C Φ − Φ k v r − ˜ v r k H (Ω) + C Φ − · k v ∗ − ˜ v ∗ k H (Ω) + C Φ − Φ h k v ∗ − ˜ v ∗ k H (Ω) + k v θ − ˜ v θ k H (Ω) i . On the other hand, v θ − ˜ v θ satisfies¯ U ( r ) ∂ z ( v θ − ˜ v θ ) − ∆( v θ − ˜ v θ )= − (cid:0) v r ∂ r v θ − ˜ v r ∂ r ˜ v θ (cid:1) − (cid:18) v r v θ r − ˜ v r ˜ v θ r (cid:19) − (cid:0) ∂ r v r v θ − ∂ r ˜ v r ˜ v θ (cid:1) − ∂ z (cid:0) v z v θ − ˜ v z ˜ v θ (cid:1) . It follows from Sobolev’s embedding inequalities that one has(95) k v r ∂ r v θ − ˜ v r ∂ r ˜ v θ k L (Ω) ≤ k v r − ˜ v r k L ∞ (Ω) k ∂ r v θ k L (Ω) + k ˜ v r k L ∞ (Ω) k ∂ r v θ − ∂ r ˜ v θ k L (Ω) ≤ C Φ k v r − ˜ v r k H (Ω) + C k ˜ v r k L (Ω) k ˜ v r k H (Ω) k v θ − ˜ v θ k H (Ω) ≤ C Φ − Φ k v r − ˜ v r k H (Ω) + C Φ − · Φ · k v θ − ˜ v θ k H (Ω) . The similar computations give(96) (cid:13)(cid:13)(cid:13)(cid:13) v r v θ r − ˜ v r ˜ v θ r (cid:13)(cid:13)(cid:13)(cid:13) L (Ω) ≤ C Φ − Φ k v r − ˜ v r k H (Ω) + C Φ − · Φ · k v θ − ˜ v θ k H (Ω) and(97) k ∂ r v r v θ − ∂ r ˜ v r ˜ v θ k L (Ω) ≤ C k ∂ r v r − ∂ r ˜ v r k L (Ω) k v θ k L ∞ (Ω) + C k ∂ r ˜ v r k L (Ω) k v θ − ˜ v θ k L ∞ (Ω) ≤ C k v r − ˜ v r k H (Ω) k v θ k H (Ω) + C k ˜ v r k L (Ω) k ˜ v r k H (Ω) k v θ − ˜ v θ k L ∞ (Ω) ≤ C Φ − Φ k v r − ˜ v r k H (Ω) + C Φ − · + · k v θ − ˜ v θ k H (Ω) . Moreover,(98) k v z v θ − ˜ v z ˜ v θ k L (Ω) ≤ C k v z − ˜ v z k H (Ω) k v θ k H (Ω) + C k ˜ v z k H (Ω) k v θ − ˜ v θ k H (Ω) ≤ C Φ (cid:16) k v z − ˜ v z k H (Ω) + k v θ − ˜ v θ k H (Ω) (cid:17) , and(99) k ∂ z ( v z v θ ) − ∂ z (˜ v z ˜ v θ ) k L (Ω) ≤ C Φ (cid:16) k v z − ˜ v z k H (Ω) + k v θ − ˜ v θ k H (Ω) (cid:17) . Combining the estimates (95)–(99), it follows from Propositions 3.5–3.6 that(100) k v θ − ˜ v θ k H (Ω) ≤ C Φ − Φ k v r − ˜ v r k H (Ω) + C Φ − · Φ · k v θ − ˜ v θ k H (Ω) + C Φ − k v θ − ˜ v θ k H (Ω) + C Φ − (cid:16) k v z − ˜ v z k H (Ω) + k v θ − ˜ v θ k H (Ω) (cid:17) . Combining the three estimates (93)–(94) and (100) together gives the uniqueness of thesolution when Φ is large enough. Thus the proof of Theorem 1.1 is finished. (cid:3)
XISTENCE AND ASYMPTOTIC BEHAVIOR OF LARGE AXISYMMETRIC SOLUTIONS 19 Asymptotic Behavior
In this section, we investigate the asymptotic behavior of solutions to (5)–(7) and proveTheorem 1.2. The proof consists of two steps. In the first step, the asymptotic behaviorof the solution which is a small perturbation of Hagen-Poiseuille flow is established. In thesecond step, the smallness requirement is removed since the solution be a small perturbationof Hagen-Poiseuille flow at far fields when the condition (14) is satisfied.Before giving the detailed proof of Theorem 1.2, we first state the uniform estimate of ψ for the linear problem (26)–(29) when f ∈ L r ( D ). Proposition 5.1.
Assume that f ( r, z ) ∈ L r ( D ) , the solution ψ obtained in Proposition 2.1belongs to H ∗ ( D ) and satisfies (101) k v ∗ k H (Ω) ≤ C k f k L r ( D ) , k v ∗ k H (Ω) ≤ C (1 + Φ ) k f k L r ( D ) where the constant C is independent of Φ .Proof. Let F ( r, z ) = − R r f ( r, z ) dz. Hence, f ( r, z ) = ∂ r F ( r, z ). Moreover, for every ( r, z ) ∈ D , by H¨older inequality, | F ( r, z ) | ≤ (cid:18)Z | f ( s, z ) | s ds (cid:19) | ln r | , which implies that k F k L r ( D ) ≤ C Z + ∞−∞ Z | f | s ds · Z | ln r | r dr ≤ C k f k L r ( D ) . This, together with the regularity estimates in Proposition 2.1, finishes the proof of Propo-sition 5.1. (cid:3)
The following lemma on the estimate between the stream function and the velocity fieldis needed in the proof of Theorem 1.2.
Lemma 5.2.
Assume that v ∗ = v r e r + v z e z ∈ H (Ω) and v ∗ is axisymmetric. Let ψ ∈ H ∗ ( D ) be the corresponding stream function of the velocity field v ∗ . It holds that (102) kL ψ k L r ( D ) + k ∂ z ψ k L r ( D ) + (cid:13)(cid:13)(cid:13)(cid:13) r ∂∂r ( rψ ) (cid:13)(cid:13)(cid:13)(cid:13) L r ( D ) + k ∂ z ψ k L r ( D ) + k ψ k L r ( D ) ≤ C k v ∗ k H (Ω) and (103) k ψ e r k H (Ω) ≤ C k v ∗ k H (Ω) . Proof.
Recall that the stream function ψ of the vector field v ∗ satisfies(104) ( L + ∂ z ) ψ = ∂ z v r − ∂ r v z in D. Multiplying (104) by ( L + ∂ z ) ψr and integrating over D , integration by parts gives(105) Z + ∞−∞ Z (cid:20) |L ψ | r + 2 | ∂ r ∂ z ( rψ ) | r + | ∂ z ψ | r (cid:21) drdz ≤ C Z + ∞−∞ Z | ∂ z v r − ∂ r v z | r drdz ≤ C k v ∗ k H (Ω) . Furthermore, by Lemma A.1 and (105), one has(106) (cid:13)(cid:13)(cid:13)(cid:13) r ∂∂r ( rψ ) (cid:13)(cid:13)(cid:13)(cid:13) L r ( D ) + k ∂ z ψ k L r ( D ) + k ψ k L r ( D ) ≤ C kL ψ k L r ( D ) + C (cid:13)(cid:13)(cid:13)(cid:13) r ∂ ∂r∂z ( rψ ) (cid:13)(cid:13)(cid:13)(cid:13) L r ( D ) ≤ C k v ∗ k H (Ω) . The straightforward computations yield ( ∆( ψ e r ) = ( L + ∂ z ) ψ e r , in Ω ,ψ = 0 , on ∂ Ω . It follows from the regularity theory for elliptic equations [8] and (105) that one has(107) k ψ e r k H (Ω) ≤ C k ( L + ∂ z ) ψ e r k L (Ω) ≤ C k v ∗ k H (Ω) . The proof of Lemma 5.2 is completed. (cid:3)
Proposition 5.3.
Assume that F ∈ L (Ω) , F = F ( r, z ) is axisymmetric, and u ∈ H (Ω) is an axisymmetric solution to the problem (5) – (7) . There exist a constant ǫ , independentof F and Φ , and a constant α ( ≤ depending on Φ , such that if (108) k e α | z | F k L (Ω) < + ∞ with some α ∈ (0 , α ) and (109) k u − ¯ U k H (Ω) ≤ ǫ , then the solution u satisfies (110) k e α | z | ( u − ¯ U ) k H (Ω) ≤ C k e α | z | F k L (Ω) . Here C is a uniform constant independent of F and Φ . XISTENCE AND ASYMPTOTIC BEHAVIOR OF LARGE AXISYMMETRIC SOLUTIONS 21
Proof.
Let v = u − ¯ U and ψ be the stream function associated with the vector field v . Then( ψ, v θ ) satisfies the problem (67)–(70). Multiplying (67) and (69) by e αz gives(111) ¯ U ( r ) ∂ z ( L + ∂ z )( e αz ψ ) − ( L + ∂ z ) ( e αz ψ )= ∂ z ( e αz F r ) − ∂ r ( e αz F z ) − ∂ r (cid:2) v r ( L + ∂ z )( e αz ψ ) (cid:3) − ∂ z (cid:2) v z ( L + ∂ z )( e αz ψ ) (cid:3) + ∂ z (cid:20) v θ r ( e αz v θ ) (cid:21) + ¯ U ( r ) (cid:2) α L ( e αz ψ ) + α e αz ψ − α ∂ z ( e αz ψ ) + 3 α∂ z ( e αz ψ ) (cid:3) − ∂ z (cid:2) α∂ z ( e αz ψ ) − α ∂ z ( e αz ψ ) + 4 α ( e αz ψ ) (cid:3) + α e αz ψ − (cid:2) α L ∂ z ( e αz ψ ) − α L ( e αz ψ ) (cid:3) − αe αz F r + ∂ r (cid:8) v r (cid:2) α∂ z ( e αz ψ ) − α e αz ψ (cid:3)(cid:9) + ∂ z (cid:8) v z (cid:2) α∂ z ( e αz ψ ) − α e αz ψ (cid:3)(cid:9) + αv z ( L + ∂ z )( e αz ψ ) − αv z (cid:2) α∂ z ( e αz ψ ) − α e αz ψ (cid:3) − α v θ r ( e αz v θ ) , and(112) ¯ U ( r ) ∂ z ( e αz v θ ) − ∆( e αz v θ ) = e αz F θ − ( v ∗ · ∇ )( e αz v θ ) − v r r ( e αz v θ )+ ¯ U ( r ) αe αz v θ − α∂ z ( e αz v θ ) + α e αz v θ + v z αe αz v θ . Denote v rα = ∂ z ( e αz ψ ) , v zα = − ∂ r ( re αz ψ ) r , v ∗ α = v rα e r + v zα e z , v θα = e αz v θ e θ . Regarding the terms on the right hand of (111), by Sobolev embedding inequalities andLemma 5.2, one has(113) k v r ( L + ∂ z )( e αz ψ ) k L r ( D ) ≤ C k v r k L ∞ (Ω) k ( L + ∂ z )( e αz ψ ) k L r ( D ) ≤ C k v k H (Ω) k v ∗ α k H (Ω) ≤ C k v k H (Ω) k v ∗ α k H (Ω) and(114) k v z ( L + ∂ z )( e αz ψ ) k L r ( D ) ≤ C k v z k L ∞ (Ω) k ( L + ∂ z )( e αz ψ ) k L r ( D ) ≤ C k v k H (Ω) k v ∗ α k H (Ω) ≤ C k v k H (Ω) k v ∗ α k H (Ω) . Similarly, one has(115) (cid:13)(cid:13)(cid:13)(cid:13) v θ r e αz v θ (cid:13)(cid:13)(cid:13)(cid:13) L r ( D ) ≤ (cid:13)(cid:13)(cid:13)(cid:13) v θ r (cid:13)(cid:13)(cid:13)(cid:13) L (Ω) k e αz v θ k L ∞ (Ω) ≤ C k v k H (Ω) k v θα k H (Ω) . Note that(116) k ¯ U ( r ) (cid:2) α L ( e αz ψ ) + α e αz ψ − α ∂ z ( e αz ψ ) + 3 α∂ z ( e αz ψ ) (cid:3) k L r ( D ) ≤ C Φ | α |k v ∗ α k H (Ω) ≤ C Φ | α |k v ∗ α k H (Ω) . and(117) k α∂ z ( e αz ψ ) − α ∂ z ( e αz ψ ) + 4 α ( e αz ψ ) k L r ( D ) + k α e αz ψ k L r ( D ) + k α L ( e αz ψ ) k L r ( D ) + k α L ( e αz ψ ) k L r ( D ) ≤ C | α |k v ∗ α k H (Ω) + C | α |k v ∗ α k H (Ω) ≤ C | α |k v ∗ α k H (Ω) . Furthermore, it holds that(118) k v r (cid:2) α∂ z ( e αz ψ ) − α e αz ψ (cid:3) k L r ( D ) + k v z (cid:2) α∂ z ( e αz ψ ) − α e αz ψ (cid:3) k L r ( D ) + k αv z ( L + ∂ z )( e αz ψ ) − αv z (cid:2) α∂ z ( e αz ψ ) − α e αz ψ (cid:3) k L r ( D ) ≤ C | α |k v r k L ∞ (Ω) k v ∗ α k H (Ω) + C | α |k v k H (Ω) k v ∗ α k H (Ω) ≤ C | α |k v k H (Ω) k v ∗ α k H (Ω) . Similar to (115), one has(119) (cid:13)(cid:13)(cid:13)(cid:13) α v θ r ( e αz v θ ) (cid:13)(cid:13)(cid:13)(cid:13) L r ( D ) ≤ C | α |k v k H (Ω) k v θα k H (Ω) . It follows from the Sobolev embedding inequalities that one has(120) k ( v ∗ · ∇ )( e αz v θ ) k L (Ω) ≤ C k v ∗ k L ∞ (Ω) k e αz v θ k H (Ω) ≤ C k v k H (Ω) k v θα k H (Ω) and(121) (cid:13)(cid:13)(cid:13)(cid:13) v r r ( e αz v θ ) (cid:13)(cid:13)(cid:13)(cid:13) L (Ω) ≤ C k ∂ r v r + ∂ z v z k L (Ω) k e αz v θ k L (Ω) ≤ C k v k H (Ω) k v θα k H (Ω) . Furthermore, it holds that(122) k − α∂ z ( e αz v θ ) + α e αz v θ k L (Ω) ≤ C | α |k v θα k H (Ω) and(123) k ¯ U ( r ) αe αz v θ k L (Ω) + k v z αe αz v θ k L (Ω) ≤ C | α | Φ k e αz v θ k H (Ω) + C | α |k v z k L ∞ (Ω) k e αz v θ k H (Ω) ≤ C | α | (cid:16) Φ + k v k H (Ω) (cid:17) k v θα k H (Ω) . Hence, collecting the estimates (113)–(123) and applying Propositions 2.1-2.2, Proposition5.1 yield(124) k v ∗ α k H (Ω) + k v θα k H (Ω) ≤ C k e αz F k L (Ω) + C k v k H (Ω) (cid:16) k v ∗ α k H (Ω) + k v θα k H (Ω) (cid:17) + C | α | (cid:16) Φ + 1 + k v k H (Ω) (cid:17) (cid:16) k v ∗ α k H (Ω) + k v θα k H (Ω) (cid:17) . XISTENCE AND ASYMPTOTIC BEHAVIOR OF LARGE AXISYMMETRIC SOLUTIONS 23
Choose a small constant ǫ such that C ǫ ≤ . If u and α satisfy k v k H (Ω) = k u − ¯ U k H (Ω) ≤ ǫ and | α | ≤ α ≤ C (1 + Φ + ǫ ) , then the inequality (124) implies that(125) k v ∗ α k H (Ω) + k v θα k H (Ω) ≤ C k e αz F k L (Ω) . Note that e αz v r = ∂ z ( e αz ψ ) − αe αz ψ = v rα − αe αz ψ and e αz v z = − ∂ r ( re αz ψ ) ∂r = v zα . Decompose e αz v ∗ into two parts as follows e αz v ∗ = v ∗ α − αe αz ψ e r . It follows from Lemma 5.2 that one has(126) k e αz ψ e r k H (Ω) ≤ k e αz ψ e r k H (Ω) ≤ C k v ∗ α k H (Ω) ≤ C k v ∗ α k H (Ω) . This, together with (125), implies(127) k e αz ( u − ¯ U ) k H (Ω ∩{ z ≥ } ) ≤ C k e α | z | F k L (Ω) . Similarly, one has(128) k e − αz ( u − ¯ U ) k H (Ω ∩{ z ≤ } ) ≤ C k e α | z | F k L (Ω) . Hence the proof the proof of Proposition 5.3 is completed. (cid:3)
Next, we remove the smallness requirement for u − ¯ U in Proposition 5.3. The key observa-tion is that k u − ¯ U k H (Ω L ) with Ω L = B (0) × ( L, ∞ ) is sufficiently small for sufficiently large L , provided that k u − ¯ U k H (Ω) is bounded. This implies that u satisfies the assumptions ofProposition 5.3 in the domain Ω L , and hence Theorem 1.2 can be proved in the similar way. Proof of Theorem 1.2.
Let v = u − ¯ U and ψ be the corresponding stream function. Let L be a positive constant to be determined. Denote Ω L = B (0) × ( L, + ∞ ). Choose a smoothcut-off function η ( z ) satisfying η ( z ) = ( , z ≤ L, , z ≥ L + 1 . Note that ( ψ, v θ ) is a solution to the problem (67)–(70). Multiplying (67) by η ( z ) yields(129) ¯ U ( r ) ∂ z ( L + ∂ z )( ηψ ) − ( L + ∂ z ) ( ηψ )= ∂ z ( ηF r + ˜ F r ) − ∂ r ( ηF z + ˜ F z ) + ˜ f − ∂ r (cid:2) v r ( L + ∂ z )( ηψ ) (cid:3) − ∂ z (cid:2) v r ( L + ∂ z )( ηψ ) (cid:3) + ∂ z (cid:20) v θ r ηv θ (cid:21) , where˜ F r = ¯ U ( r ) [2 η ′ ( z ) ∂ z ψ + η ′′ ( z ) ψ ] − η ′ ( z ) L ψ − η ′ ( z ) ∂ z ψ + 2 v z η ′ ( z ) ∂ z ψ + v z η ′′ ( z ) ψ, ˜ F z = − [2 v r η ′ ( z ) ∂ z ψ + v r η ′′ ( z ) ψ ] , and˜ f = − η ′ ( z ) F r + ¯ U ( r ) η ′ ( z )( L + ∂ z ) ψ + 2 η ′′ ( z ) L ψ − (cid:2) η (4) ( z ) ψ + 4 η (3) ( z ) ∂ z ψ + 2 η ′′ ( z ) ∂ z ψ (cid:3) + v z η ′ ( z )( L + ∂ z ) ψ − v θ r η ′ ( z ) v θ . Similarly, one has(130) ¯ U ( r ) ∂ z ( η v θ ) − ∆( η v θ ) = η F θ + ˜ F θ − ( v r ∂ r + v z ∂ z )( η v θ ) − v r r ( η v θ ) , where ˜ F θ = ¯ U ( r ) η ′ ( z ) v θ − (cid:2) η ′ ( z ) ∂ z v θ + η ′′ ( z ) v θ (cid:3) + v z η ′ ( z ) v θ . Denote v rα,η = ∂ z ( e αz ηψ ) , v zα,η = − ∂ r ( re αz ηψ ) r , and v ∗ α,η = v α,η e r + v α,η e z . It follows from the same lines as in the proof of Proposition 5.3 that one has(131) k v ∗ α,η k H (Ω) + k e αz ( η v θ ) k H (Ω) ≤ C (1 + Φ )( k e αz η F k L (Ω) + k e αz ˜ F r k L r ( D ) + k e αz ˜ F z k L r ( D ) + k e αz ˜ f k L r ( D ) )+ C (1 + Φ ) k v k H (Ω L ) (cid:0) k v ∗ α,η k H (Ω) + k e αz ( η v θ ) k H (Ω) (cid:1) + C (1 + Φ ) | α | (cid:16) Φ + 1 + k v k H (Ω L ) (cid:17) (cid:0) k v ∗ α,η k H (Ω) + k e αz ( η v θ ) k H (Ω) (cid:1) . Since k v k H (Ω) < + ∞ , there exists an L large enough such that k v k H (Ω L ) ≤ min ( C (1 + Φ ) , ) . Choose an α > C (1 + Φ )(Φ + 2) α ≤ . XISTENCE AND ASYMPTOTIC BEHAVIOR OF LARGE AXISYMMETRIC SOLUTIONS 25
For every α ∈ (0 , α ), it holds that k v ∗ α,η k H (Ω) + k e αz ( η v θ ) k H (Ω) ≤ C (1 + Φ ) (cid:16) k e αz ˜ F r k L r ( D ) + k e αz ˜ F z k L r ( D ) + k e αz ˜ f k L r ( D ) + k e αz ˜ F θ k L (Ω) (cid:17) + C (1 + Φ ) k e αz η F k L (Ω) . Note that supp ˜ F r , supp ˜ F z , supp ˜ f , supp ˜ F θ ⊆ B (0) × [ L, L + 1], then(132) k e αz ˜ F r k L r ( D ) + k e αz ˜ F z k L r ( D ) + k e αz ˜ f k L r ( D ) + k e αz ˜ F θ k L (Ω) ≤ C (1 + Φ) k v k H (Ω) . Hence, one has(133) k v ∗ α,η k H (Ω) + k e αz ( η v θ ) k H (Ω) < + ∞ . Similar to the proof of Proposition 5.3, one can rewrite e αz η v ∗ as follows, e αz η v ∗ = v ∗ α,η − ( αe αz ηψ + e αz η ′ ( z ) ψ ) e r . Thus it follows from Lemma 5.2 that one has(134) k ( αe αz ηψ ) e r k H (Ω) + k e αz η ′ ( z ) ψ e r k H (Ω) ≤ C k v ∗ α,η k H (Ω) + C k v k H (Ω) < + ∞ . This, together with (133), completes the proof of Theorem 1.2. (cid:3)
Appendix A. Some elementary lemmas
In this appendix, we collect some basic lemmas which play important roles in the paper.Their proofs can be found in [24, Appendix A], so we omit the details here.The following lemma is about Poincar´e type inequalities.
Lemma A.1.
For a function g ∈ C ([0 , it holds that (135) Z | g | r dr ≤ Z (cid:12)(cid:12)(cid:12)(cid:12) ddr ( rg ) (cid:12)(cid:12)(cid:12)(cid:12) r dr. If, in addition, g (0) = g (1) = 0 , then one has (136) Z (cid:12)(cid:12)(cid:12)(cid:12) ddr ( rg ) (cid:12)(cid:12)(cid:12)(cid:12) r dr ≤ (cid:18)Z |L g | r dr (cid:19) (cid:18)Z | g | r dr (cid:19) ≤ Z |L g | r dr. The following lemma is a variant of Hardy-Littlewood-P´olya type inequality [9], whichplays an important role in many estimates in the paper.
Lemma A.2.
Let g ∈ C ([0 , satisfy g (0) = 0 , one has (137) Z | g ( r ) | dr ≤ Z | g ′ ( r ) | (1 − r ) dr, and (138) Z | g | r dr ≤ C Z (cid:12)(cid:12)(cid:12)(cid:12) d ( rg ) dr (cid:12)(cid:12)(cid:12)(cid:12) − r r dr. The following lemma is about a weighted interpolation inequality, which is quite similarto [7, (3.28)].
Lemma A.3.
Let g ∈ C [0 , , then one has (139) Z | g | r dr ≤ C (cid:18)Z (1 − r ) | g | r dr (cid:19) Z r (cid:12)(cid:12)(cid:12)(cid:12) ddr ( rg ) (cid:12)(cid:12)(cid:12)(cid:12) dr ! + C Z (1 − r ) | g | r dr, and (140) Z (cid:12)(cid:12)(cid:12)(cid:12) ddr ( rg ) (cid:12)(cid:12)(cid:12)(cid:12) r dr ≤ C "Z − r r (cid:12)(cid:12)(cid:12)(cid:12) ddr ( rg ) (cid:12)(cid:12)(cid:12)(cid:12) dr (cid:18)Z |L g | r dr (cid:19) + C Z − r r (cid:12)(cid:12)(cid:12)(cid:12) ddr ( rg ) (cid:12)(cid:12)(cid:12)(cid:12) dr. Acknowledgement.
The research of Wang was partially supported by NSFC grant11671289. The research of Xie was partially supported by NSFC grants 11971307 and11631008, and Young Changjiang Scholar of Ministry of Education in China. The authorswould like to thank Professor Yasunori Maekawa for helpful discussions.
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