Decay properties of Axially Symmetric D-solutions to the Steady Incompressible Magnetohydrodynamic Equations
aa r X i v : . [ m a t h . A P ] F e b Decay properties of Axially Symmetric D-solutions to theSteady Incompressible Magnetohydrodynamic Equations
Shangkun Weng ∗ Yan Zhou † Abstract
In this paper, we investigate the decay properties of axially symmetric solutions tothe steady incompressible magnetohydrodynamic equations in R with finite Dirichletintegral. We first derive the decay rates of general D-solutions to the axisymmetric MHDequations. In the special case where the magnetic field only has the swirl component h ( r, z ) = h θ ( r, z ) e θ , we obtain better decay rates. The last result examines the decayrates along the axis Oz also within the special class of D-solutions with only swirl magneticfield. The main tool in this paper is the combination of the scaling argument, the Brezis-Gallouet inequality and the weighted energy estimate.
Mathematics Subject Classifications 2010: Primary 35Q35; Secondary 76W05.Key words: Magnetohydrodynamics equations, decay rate, axially symmetric, scal-ing argument, weighted energy estimates.
In this paper, we investigate the decay properties of axially symmetric solutions to thesteady Magnetohydrodynamics equations (MHD), which are listed as follows. u · ∇ u + ∇ p = h · ∇ h + △ u + f , u · ∇ h − h · ∇ u = △ h + curl g , ∇ · u = ∇ · h = 0 , (1.1)with the additional condition at infinitylim | x |→∞ u ( x ) = lim | x |→∞ h ( x ) = 0 , (1.2)and the finite Dirichlet integral Z R |∇ u ( x ) | + |∇ h ( x ) | dx < + ∞ . (1.3)Here u is the velocity field, h is the magnetic field, p is the pressure, f and ∇ × g arethe external forces on the magnetically charged fluid flows. It is well known that Leray ∗ School of mathematics and statistics, Wuhan University, Wuhan, Hubei Province, 430072, People’s Re-public of China. Email: [email protected] † School of mathematics and statistics, Wuhan University, Wuhan, Hubei Province, 430072, People’s Re-public of China. Email: [email protected] r = q x + x , tan θ = x x , z = x , and the basis vectors e r , e θ , e z are e r = (cos θ, sin θ, , e θ = ( − sin θ, cos θ, , e z = (0 , , . For convenience, we write x ′ = ( x , x ) and x = z. We call a function f is axially symmetric if it does not depend on θ . A vector-valuedfunction u = ( u r , u θ , u z ) is called axially symmetric if u r , u θ , u z do not depend on θ . Andwe call a vector-valued function u = ( u r , u θ , u z ) is axially symmetric without swirl if u θ = 0while u r and u z do not depend on θ .Assume that the vector field u ( x ) = u r ( r, z ) e r + u θ ( r, z ) e θ + u z ( r, z ) e z and the magneticfield h ( x ) = h r ( r, z ) e r + h θ ( r, z ) e θ + h z ( r, z ) e z is the axially symmetric solution of (1.1), thenwe have the following asymmetric steady MHD equations, ( u r ∂ r + u z ∂ z ) u r − u θ r + ∂ r p = ( h r ∂ r + h z ∂ z ) h r − h θ r + ( ∂ r + r ∂ r + ∂ z − r ) u r + f r , ( u r ∂ r + u z ∂ z ) u θ + u r u θ r = ( h r ∂ r + h z ∂ z ) h θ + h r h θ r + ( ∂ r + r ∂ r + ∂ z − r ) u θ + f θ , ( u r ∂ r + u z ∂ z ) u z + ∂ z p = ( h r ∂ r + h z ∂ z ) h z + ( ∂ r + r ∂ r + ∂ z ) u z + f z , ( u r ∂ r + u z ∂ z ) h r = ( h r ∂ r + h z ∂ z ) u r + ( ∂ r + r ∂ r + ∂ z − r ) h r − ∂ z g θ , ( u r ∂ r + u z ∂ z ) h θ + h r u θ r = ( h r ∂ r + h z ∂ z ) u θ + u r h θ r + ( ∂ r + r ∂ r + ∂ z − r ) h θ + ( ∂ z g r − ∂ r g z ) , ( u r ∂ r + u z ∂ z ) h z = ( h r ∂ r + h z ∂ z ) u z + ( ∂ r + r ∂ r + ∂ z ) h z + r ∂ r ( rg θ ) ,∂ r u r + u r r + ∂ z u z = 0 ,∂ r h r + h r r + ∂ z h z = 0 . (1.4)The vorticity ω u and the current density ω h are defined as ω u ( x ) = ∇ × u ( x ) = ω ur ( r, z ) e r + ω uθ ( r, z ) e θ + ω uz ( r, z ) e z and ω h ( x ) = ∇× h ( x ) = ω hr ( r, z ) e r + ω hθ ( r, z ) e θ + ω hz ( r, z ) e z ,where ω ur ( r, z ) = − ∂ z u θ , ω uθ = ∂ z u r − ∂ r u z , ω uz = r ∂ r ( ru θ ) ,ω hr ( r, z ) = − ∂ z h θ , ω hθ = ∂ z h r − ∂ r h z , ω hz = r ∂ r ( rh θ ) . (1.5)2hen we have ( u r ∂ r + u z ∂ z ) ω ur − ( ω ur ∂ r + ω uz ∂ z ) u r = ( h r ∂ r + h z ∂ z ) ω hr − ( ω hr ∂ r + ω hz ∂ z ) h r +( ∂ r + r ∂ r + ∂ z − r ) ω ur − ∂ z f θ , ( u r ∂ r + u z ∂ z ) ω uθ − u r ω uθ r + 2 u θ r ω ur = ( h r ∂ r + h z ∂ z ) ω hθ − h r ω hθ r + 2 h θ r ω hr +( ∂ r + r ∂ r + ∂ z − r ) ω uθ + ( ∂ z f r − ∂ r f z ) , ( u r ∂ r + u z ∂ z ) ω uz − ( ω ur ∂ r + ω uz ∂ z ) u z = ( h r ∂ r + h z ∂ z ) ω hz − ( ω hr ∂ r + ω hz ∂ z ) h z +( ∂ r + r ∂ r + ∂ z ) ω uz + r ∂ r ( rf θ ) , ( u r ∂ r + u z ∂ z ) ω hr − ( ω hr ∂ r + ω hz ∂ z ) u r + r ∂ z ( u r h θ ) = ( h r ∂ r + h z ∂ z ) ω ur − ( ω ur ∂ r + ω uz ∂ z ) h r + r ∂ z ( u θ h r ) + ( ∂ r + r ∂ r + ∂ z − r ) ω hr − ∂ z ( ∂ z g r − ∂ r g z ) , ( u r ∂ r + u z ∂ z ) ω hθ − u r r ω hθ + 2 ∂ z u r ∂ r h r + 2 ∂ z u z ∂ r h z = ( h r ∂ r + h z ∂ z ) ω uθ − h r r ω uθ +2 ∂ z h r ∂ r u r + 2 ∂ z h z ∂ r u z + ( ∂ r + r ∂ r + ∂ z − r ) ω hθ − ∂ z ( ∂ z g θ ) − ∂ r ( r ∂ r ( rg θ )) , ( u r ∂ r + u z ∂ z ) ω hz + ( ∂ r u r ∂ r + ∂ r u z ∂ z ) h θ + ∂ r ( u θ h r r ) = ( h r ∂ r + h z ∂ z ) ω uz +( ∂ r h r ∂ r + ∂ r h z ∂ z ) u θ + ∂ r ( u r h θ r ) + ( ∂ r + r ∂ r + ∂ z ) ω hz + r ∂ r ( r ( ∂ z g r − ∂ r g z )) . (1.6)Finn [14] have studied the physically reasonable solutions to the steady Navier-Stokeswith O ( | x | − ) decay at infinity. Borchers-Miyakawa [1], Galdi-Simader [17] and Novotny-Padula [30] obtained the existence of such u with decay rate O ( | x | − ) at infinity providedthe forcing term is sufficiently small in different function spaces. Under the assumption that | u ( x ) | = O ( | x | − ) as | x | → ∞ , Sverak and Tsai [33] had showed that |∇ k u ( x ) | ≤ O ( | x | − k − )for any k ≥ u ≡ u ∈ L / ( R ), which was logarithmically improved in [7]. Many authors have identifieddifferent integrability or decay conditions on u , which lead to many interesting Liouville typeresults, one may refer to [4, 5, 6, 8, 9, 22, 24, 31] for more details. The Liouville theoremin the case of the axisymmetric three dimensional flows without swirl can be derived from[21]. Chae [5] explored the maximum principle for the total head pressure, and proved thetriviality of u by assuming ∆ u ∈ L ( R ). Seregin [31, 32] applied Caccioppoli type inequalityto show that any smooth solution to the stationary Navier-Stokes system in R , belonging to L ( R ) and BM O − , must be zero. Kozono, Terasawa and Wakasugi [24] proposed a decaycondition on the vorticity to guarantee that u ≡
0. For the MHD system, Liu and Zhang [26]proved the Liouville theorem for the bounded ancient solution. In [34], it was shown thatthere are not non-trivial solutions to MHD equations under the finite Dirichlet integral orthe L p integrability of the velocity and magnetic fields. In their proof, the smallness of themagnetic field plays a vital role.For the investigation of the decay properties of D-solutions to the exterior stationaryNavier-Stokes equations with large external force, Gilbarg and Weinberger [18] had made3reat progress in the two dimensional exterior domain, showed that the weak solution con-structed by Leray [25] was bounded and converged to a limit u in a mean square sense, whilethe pressure converged pointwise. In [19], they further found that the weak solution withfinite Dirichlet integral may not be bounded, but it must grow more slowly than (ln r ) / . Byfurther adapting the ideas in [18, 19] to the 3D axisymmetric setting, the authors in [10, 35]obtained some decay rates for smooth axially symmetric solutions to steady Navier-Stokesequations. On the other hand, it is well-known that if ( u , p ) solves stationary Navier-Stokesequations, so does ( u λ , p λ ) for all λ >
0, where u λ ( x ) = λ u ( λ x ) and p λ ( x ) = λ p ( λ x ). Ac-cording to this, the authors in [3] utilized the Brezis-Gallouet inequality to improve the decayrate of the vorticity in [10, 35]. Recently, the decay properties of the second order derivativeswere investigated in [27]. Combining the results in [3, 10, 35], one has: for x ∈ R , | u ( x ) | ≤ C ( ln rr ) / , | ω ur ( x ) | + | ω uz ( x ) | ≤ C (ln r ) / r / , | ω uθ ( x ) | + |∇ ω uθ ( x ) | ≤ C (ln r ) / r / , |∇ u r | + |∇ u z | ≤ C (ln r ) / r / , where C is a positive constant.As was noticed in [3], the derivation of the decay of the velocity itself almost had nouse of the Navier-Stokes equations, it is straightforward to gain the same decay rates for themagnetic field. However, the estimates of the vorticity does depend on the structure of thevorticity equations. For the steady MHD equations, the equations for the electric current ∇ × h are more complicate than those of the vorticity, therefore the decay rates of ω u and ω h obtained here are weaker than those of Navier-Stokes equations. Our first result is statedas follows. Theorem 1.1.
Support f ∈ H ( R ) and g ∈ H ( R ) be axially symmetric vector field with k f k H − ( R ) + k f k H ( R ) + k curl g k L ( R ) + k r curl g k L ( R ) ≤ M (1.7) for some constant M . Let ( u , h , p ) be a smooth axially symmetric solution to the steady MHDequations (1.1) with (1.2)-(1.3), then there exist a constant C ( M ) > , such that | u | + | h | ≤ C ( M )( ln rr ) , (1.8) | ω uθ | + | ω hθ | ≤ C ( M ) ln rr , (1.9) | ω ur | + | ω uz | + | ω hr | + | ω hz | ≤ C ( M ) ln rr , (1.10) for large r . Next we consider a special class of axisymmetric D-solutions to the steady MHD equationswith h ( r, z ) = h θ ( r, z ) e θ . In this situation, we obtain better decay results for the vorticityfield by employing the Biot-Savart law as in [3]. However, this argument does not improvethe decay rates of ω hr and ω hz . Here we resort to the weighted energy estimates in [35] to geta better decay for ω hr and ω hz . 4 heorem 1.2. Support f ∈ H ( R ) and g ∈ H ( R ) be axially symmetric vector fieldwithout swirl, and satisfying (1.7) and k ( r ln r ) f θ k L ( R ) + k r curl f k L ( R ) ≤ M, (1.11) k r ∂ r ( ∂ z g r − ∂ r g z ) k L ( R ) ≤ M, (1.12) for some constant M . Let ( u , h , p ) be a smooth axially symmetric solution to the steady MHDequations (1.1) with (1.2)-(1.3), where h ( r, z ) = h θ ( r, z ) e θ . Then we have better estimatesas follows: | ω uθ | ≤ C ( M ) (ln r ) / r / , (1.13) | ω ur | + | ω uz | ≤ C ( M ) (ln r ) / r / , (1.14) |∇ u r | + |∇ u z | ≤ C ( M ) (ln r ) / r / , (1.15) | ω hr | + | ω hz | + |∇ h θ | ≤ C ( M ) r − ( ) − , (1.16) where a − denotes any number less than a . The last result examines the decay rate in the Oz -direction for the axisymmetric D-solution of steady MHD equations with the magnetic field having only the swirl component h ( r, z ) = h θ ( r, z ) e θ . The quantity Π := h θ r satisfies an elliptic equation, from which weightedenergy estimates are available for the weight ρ = √ r + z . Furthermore, if the swirl of thevelocity u θ ≡
0, we can deduce some decay estimates for the velocity as in steady axisymmetricNavier-Stokes case [35]. However, the decay rates are worse than the Navier-Stokes case.
Theorem 1.3.
Suppose f ∈ H ( R ) and g ∈ H ( R ) be axially symmetric vector field withoutswirl, and satisfying (1.7), (1.12) and k ρ ∂ z f r − ∂ r f z r k L ( R ) ≤ M, (1.17) k ρ ∂ z g r − ∂ r g z r k L ( R ) ≤ M (1.18) k ( r ln r ) g θ k L ( R ) + k r curl g k L ( R ) ≤ M, (1.19) for some constant M . Let ( u , h , p ) be a smooth axially symmetric solution to the steady MHDequations (1.1) with (1.2)-(1.3), where h ( r, z ) = h θ ( r, z ) e θ . Then there holds | h θ ( r, z ) | ≤ C ( M )( ρ + 1) − , ρ = p r + z . Moreover, if the swirl velocity u θ ≡ , then | u r ( r, z ) | + | u z ( r, z ) | ≤ C ( M )( ρ + 1) − ( ) − , (1.20) | ω θ ( r, z ) | ≤ C ( M )( ρ + 1) − ( ) − , (1.21) | h θ ( r, z ) | ≤ C ( M )( ρ + 1) − ( ) − . (1.22)5his paper is organized as follows. In section 2 we introduce some preliminary tools. InSection 3, we prove the decay rates of ( u , h ) and ( ω u , ω h ) by applying the scaling argumentand the Brezis-Gallouet inequality . In Section 4, we will use the
Biot-Savart law and alsoweighted energy estimates for the magnetic field to get some improved decay rate estimatesfor the special class of solutions with h ( r, z ) = h θ ( r, z ) e θ . In Section 5, we will investigatethe decay properties of ( u , h ) in the O z -direction also within the special class of solutionswith h ( r, z ) = h θ ( r, z ) e θ . Lemma 2.1. [2, Theorem 1]
Let f ∈ H (Ω) where Ω ⊂ R . Then there exists a constant C Ω , depending only on Ω , such that k f k L ∞ (Ω) ≤ C Ω (1 + k f k H (Ω) ) ln / (cid:0) e + k ∆ f k L (Ω) (cid:1) . Lemma 2.2. [3, Lemma 3.2]
Assume that K ( x, y ) be a Calderon-Zygmund kernel and f isa smooth axisymmetric function satisfying, for x = ( x ′ , z ) ∈ R | f ( x ) | + |∇ f ( x ) | ≤ ln b ( e + | x ′ | )(1 + | x ′ | ) a f or < a < , b > . Define
T f ( x ) := R K ( x, y ) f ( y ) dy . Then there exists a constant c such that | T f ( x ) | ≤ c ln b +1 ( e + | x ′ | )(1 + | x ′ | ) a . Lemma 2.3. [35, Lemma 2.1]
Suppose a smooth axially symmetric function f ( x ) satisfiesthe following weighted energy estimates Z R (cid:0) r e | f ( r, z ) | + r e |∇ f ( r, z ) | + r e | ∂ z ∇ f ( r, z ) | (cid:1) dx ≤ C, with nonnegative constants e , e , e . Then for any r > , we have Z ∞−∞ | f ( r, z ) | dz ≤ Cr − ( e + e ) − , Z ∞−∞ | ∂ z f ( r, z ) | dz ≤ Cr − ( e + e ) − , | f ( r, z ) | ≤ Cr − ( e +2 e + e ) − , ∀ z ∈ R . We start to prove Theorem 1.1. Utilizing the similar argument as in [3], we fix any point x ∈ R such that | x ′ | = λ is large, and consider the scaled solutions ˜ u (˜ x ), ˜ h (˜ x ) and the twodimensional domain ˜ D , ˜ u (˜ x ) = λ u ( λ ˜ x ) , ˜ h (˜ x ) = λ h ( λ ˜ x ) , ˜ D = { (˜ r, ˜ z ) | / ≤ ˜ r ≤ , | ˜ z | ≤ } , x = λ ˜ x .The argument developed in [3] for the steady axisymmetric Navier-Stokes equations alsoworks in this case with obvious modification, we obtain | u ( x ) | + | h ( x ) | ≤ C ( ln rr ) .It remains to derive the decay rates of ω u and ω h . According to previous scaling, we have˜ ω u (˜ x ) = λ ω u ( λ ˜ x ) = λ ω u ( x )˜ ω h (˜ x ) = λ ω h ( λ ˜ x ) = λ ω h ( x )For simplification of notation, we will drop the “ ∼ ” when computations take place underthe scaled sense. Select the domains C = { ( r, θ, z ) : < r < , ≤ θ ≤ π, | z | ≤ } , C = { ( r, θ, z ) : < r < , ≤ θ ≤ π, | z | ≤ } . Let φ ( y ) be a cut-off function satisfying sup φ ( y ) ⊂ C and φ ( y ) = 1 for y ∈ C suchthat the gradient of φ is bounded. Now testing the vorticity equation (1.6) with ω ur φ , ω uθ φ , ω uz φ , ω hr φ , ω hθ φ and ω hz φ respectively, and integrating over C , after direct computationswe obtain Z C (cid:18) |∇ ( ω uθ φ ) | + ( ω uθ φ ) r (cid:19) dy = Z C (cid:18) | ω uθ | |∇ φ | + 12 | ω uθ | ( u r ∂ r + u z ∂ z )( φ )+ u r r ( ω uθ φ ) − r u θ ω ur ω uθ φ + 2 r h θ ω hr ω uθ φ + ω uθ φ ( h r ∂ r + h z ∂ z )( ω hθ φ ) − h r r ω uθ ω hθ φ − ω uθ ω hθ φ ( h r ∂ r + h z ∂ z ) φ + φ ( f z ∂ r − f r ∂ z )( ω uθ φ ) + ω uθ φ ( f z ∂ r − f r ∂ z ) φ + f z ω uθ φr φ (cid:19) dy ≤ C (1 + k ( u , h ) k L ∞ ( C ) ) k ( ω ur , ω uθ , ω hr , ω hθ ) k L ( C ) + 18 k ( ∇ ( ω uθ φ ) , ∇ ( ω hθ φ )) k L ( C ) , Z C (cid:18) |∇ ( ω hθ φ ) | + ( ω hθ φ ) r (cid:19) dy = Z C (cid:18) |∇ φ | | ω hθ | + 12 | ω hθ | ( u r ∂ r + u z ∂ z )( φ )+2 ∂ z ( ω hθ φ )( u z ∂ r h z − h r ∂ r u r ) + 2 ∂ r ( ω hθ φ )( h r ∂ z u r − u z ∂ z h z ) + u r r ( ω hθ φ ) + ω hθ φ ( h r ∂ r + h z ∂ z )( ω uθ φ ) − ω uθ ω hθ φ ( h r ∂ r + h z ∂ z ) φ − h r r ω uθ φ · ω hθ φ +2 ω hθ φr ( h r ∂ z u r − u z ∂ z h z ) φ + ∂ z g θ ( ∂ z ( ω hθ φ )) + ω hθ φ∂ z φ )+ ∂ r ( rg θ )( φ∂ r ( ω hθ φ ) + ω hθ φ∂ r φ + ω hθ φr φ ) (cid:19) dy ≤ C (1 + k ( u r , u z , h r , h z ) k L ∞ ( C ) ) k ( ω uθ , ω hθ ) k L ( C ) + C k ( u r , u z , h r , h z ) k L ∞ ( C ) k ( ∇ u r , ∇ h z ) k L ( C ) + 18 k ( ∇ ( ω uθ φ ) , ∇ ( ω hθ φ )) k L ( C ) . Combining these two estimates, we find k ( ∇ ω uθ , ∇ ω hθ ) k L ( C ) ≤ C (1 + k ( u , h ) k L ∞ ( C ) ) k ( ω ur , ω uθ , ω hr , ω hθ ) k L ( C ) + C k ( u r , u z , h r , h z ) k L ∞ ( C ) k ( ∇ u r , ∇ h z ) k L ( C ) . (3.1)7here also holds Z C (cid:18) |∇ ( ω ur φ ) | + ( ω ur φ ) r (cid:19) dy = Z C (cid:18) |∇ φ | | ω ur | + 12 | ω ur | ( u r ∂ r + u z ∂ z )( φ ) − u r r ( ω ur φ ) − u r ( ∂ r ( ω ur φ ) + ∂ z ( ω ur φ · ω uz φ )) + ω ur φ ( h r ∂ r + h z ∂ z )( ω hr φ ) − ω ur ω hr φ ( h r ∂ r + h z ∂ z ) φ + h r ∂ r ( ω ur φ · ω hr φ ) + h r ∂ z ( ω ur φ · ω hz φ )+ h r r ω ur φ · ω hr φ + f θ φ ( ∂ z ( ω ur φ ) + ω ur ∂ z φ ) (cid:19) dy ≤ C (1 + k ( u r , u z , h r , h z ) k L ∞ ( C ) ) k ( ω ur , ω uz , ω hr , ω hz ) k L ( C ) + 18 k ( ∇ ( ω ur φ ) , ∇ ( ω uz φ ) , ∇ ( ω hr φ ) , ∇ ( ω hz φ ) k L ( C ) , Z C |∇ ( ω uz φ ) | dy = Z C (cid:18) |∇ φ | | ω uz | + 12 | ω uz | ( u r ∂ r + u z ∂ z )( φ ) − u z ∂ r ( ω ur φ · ω uz φ ) − u z r ω ur φ · ω uz φ + ω uz φ ( h r ∂ r + h z ∂ z )( ω hz φ ) − ω uz ω hz φ ( h r ∂ r + h z ∂ z ) φ + h z ( ∂ r ( ω uz φ · ω hr φ ) + ∂ z ( ω uz φ · ω hz φ )) − u z ∂ z ( ω uz φ ) + h z r ω uz φ · ω hr φ − f θ φ ( ∂ r ( ω uz φ ) + ω uz ∂ r φ ) (cid:19) dy ≤ C (1 + k ( u r , u z , h r , h z ) k L ∞ ( C ) ) k ( ω ur , ω uz , ω hr , ω hz ) k L ( C ) + 18 k ( ∇ ( ω ur φ ) , ∇ ( ω uz φ ) , ∇ ( ω hr φ ) , ∇ ( ω hz φ )) k L ( C ) , Z C (cid:18) |∇ ( ω hr φ ) | + ( ω hr φ ) r (cid:19) dy = Z C (cid:18) |∇ φ | | ω hr | + 12 | ω hr | ( u r ∂ r + u z ∂ z )( φ ) − u r ∂ r ( ω hr φ ) − u r ∂ z ( ω hr φ · ω hz φ ) − u r r | ω hr φ | + ω hr φ ( h r ∂ r + h z ∂ z )( ω ur φ ) − ω ur ω hr φ ( h r ∂ r + h z ∂ z ) φ + h r ∂ r ( ω ur φ · ω hr φ ) + h r ∂ z ( ω uz φ · ω hr φ )+ h r r ω ur φ · ω hr φ − r ∂ z ( u r h θ ) ω hr φ + 2 r ∂ z ( u θ h r ) ω hr φ + ∂ z ( ω hr φ )( ∂ z g r − ∂ r g z ) (cid:19) dy ≤ C (1 + k ( u r , u z , h r , h z ) k L ∞ ( C ) ) k ( ω ur , ω uz , ω hr , ω hz ) k L ( C ) + C k ( u r , u θ , h r , h θ ) k L ∞ ( C ) ×k∇ ( u r , u θ , h r , h θ ) k L ( C ) + 18 k ( ∇ ( ω ur φ ) , ∇ ( ω uz φ ) , ∇ ( ω hr φ ) , ∇ ( ω hz φ )) k L ( C ) , Z C |∇ ( ω hz φ ) | dy = Z C (cid:18) | ω hz | |∇ φ | + 12 ( ω hz ) ( u r ∂ r + u z ∂ z )( φ )+ ∂ r ( ω hz φ )( h θ ∂ r u r − u θ ∂ r h r ) + ∂ z ( ω hz φ )( h θ ∂ r u z − u θ ∂ r h z )+ ω hz φ r ( ∂ r ( u r h θ ) − ∂ r ( u θ h r )) + ω hz φ ( h r ∂ r + h z ∂ z )( ω uz φ ) − ω uz ω hz φ ( h r ∂ r + h z ∂ z ) φ − ∂ r ( ω hz φ )( ∂ z g r − ∂ r g z ) (cid:19) dy ≤ C (1 + k ( u r , u z , h r , h z ) k L ∞ ( C ) ) k ( ω uz , ω hz ) k L ( C ) + 18 k ( ∇ ( ω uz φ ) , ∇ ( ω hz φ )) k L ( C ) + C k ( u r , u θ , h r , h θ ) k L ∞ ( C ) k ( ∇ u r , ∇ u θ , ∇ u z , ∇ h r , ∇ h θ , ∇ h z ) k L ( C ) . k ( ∇ ω ur , ∇ ω uz , ∇ ω hr , ∇ ω hz ) k L ( C ) ≤ C (1 + k ( u r , u z , h r , h z ) k L ∞ ( C ) ) k ( ω ur , ω uz , ω hr , ω hz ) k L ( C ) + C k ( u r , u θ , h r , h θ ) k L ∞ ( C ) k∇ ( u r , u θ , u z , h r , h θ , h z ) k L ( C ) . (3.2)Set ¯ C := { ( r, z ) : 34 < r < , | z | ≤ / } . Utilizing the
Brezis-Gallouet’s inequality in Lemma 2.1 and the localized energy estimates(3.1) and (3.2), we can conclude k ( ω ur , ω uz , ω hr , ω hz ) k L ∞ ( ¯ C ) ≤ C (cid:18) k ( u r , u z , h r , h z ) k L ∞ ( C ) ) k ( ω ur , ω uz , ω hr , ω hz ) k L ( C ) + k ( u r , u θ , h r , h θ ) k L ∞ ( C ) k∇ ( u r , u θ , u z , h r , h θ , h z ) k L ( C ) (cid:19) × ln / (cid:18) e + k ∆( ω ur , ω uz , ω hr , ω hz ) k L ( C ) (cid:19) , k ( ω uθ , ω hθ ) k L ∞ ( ¯ C ) ≤ C (cid:18) k ( u , h ) k L ∞ ( C ) ) k ( ω ur , ω uθ , ω hr , ω hθ ) k L ( C ) + k ( u r , u z , h r , h z ) k L ∞ ( C ) · k ( ∇ u r , ∇ h z ) k L ( C ) (cid:19) × ln / (cid:18) e + k (∆ ω uθ , ∆ ω hθ ) k L ( ¯ C ) (cid:19) . Then scaling back to the domains C ,λ = { ( r, θ, z ) : λ < r < λ , ≤ θ ≤ π, | z | ≤ λ } , C ,λ = { ( r, θ, z ) : 3 λ < r < λ , ≤ θ ≤ π, | z | ≤ λ } , we derive λ k ( ω ur , ω uz , ω hr , ω hz ) k L ∞ ( C ,λ ) ≤ C (cid:18) (1 + λ k ( u , h ) k L ∞ ( C ,λ ) ) λ / k ( ω ur , ω uz , ω hr , ω hz ) k L ( C ,λ ) + λ / k ( u r , u θ , h r , h θ ) k L ∞ ( C ,λ ) k∇ ( u r , u θ , u z , h r , h θ , h z ) k L ( C ,λ ) + 1 (cid:19) × ln / (cid:18) e + λ / k ∆( ω ur , ω uz , ω hr , ω hz ) k L ( C ,λ ) (cid:19) ,λ k ( ω uθ , ω hθ ) k L ∞ ( C ,λ ) ≤ C (cid:18) (1 + λ k ( u , h ) k L ∞ ( C ,λ ) ) λ / k ( ω ur , ω uθ , ω hr , ω hθ ) k L ( C ,λ ) + λ / k ( u z , h r ) k L ∞ ( C ,λ ) k ( ∇ u r , ∇ h z ) k L ( C ,λ ) (cid:19) × ln / (cid:18) e + λ / k (∆ ω uθ , ∆ ω hθ ) k L ( C ,λ ) (cid:19) . By the finite Dirichlet integral assumption (1.3) and the a priori decay rate estimate on( u , h ), we conclude that k ( ω uθ , ω hθ ) k L ∞ ( C ,λ ) ≤ C ln λλ , k ( ω ur , ω uz , ω hr , ω hz ) k L ∞ ( C ,λ ) ≤ C ln λλ . Proof of Theorem 1.2
In this section, we will consider the special class of axisymmetric D-solution to the steadyMHD equations, where the magnetic field has only the swirl component h ( r, z ) = h θ ( r, z ) e θ .In this case, ω h ( r, z ) = ∇ × h(x) = ω hr ( r, z ) e r + ω hz ( r, z ) e z , the equations (1.6) reduce to ( u r ∂ r + u z ∂ z ) ω ur − ( ω ur ∂ r + ω uz ∂ z ) u r = ( ∂ r + r ∂ r + ∂ z − r ) ω ur − ∂ z f θ , ( u r ∂ r + u z ∂ z ) ω uθ − u r ω uθ r + r ∂ z ( h θ − u θ ) = ( ∂ r + r ∂ r + ∂ z − r ) ω uθ + ( ∂ z f r − ∂ r f z ) , ( u r ∂ r + u z ∂ z ) ω uz − ( ω ur ∂ r + ω uz ∂ z ) u z = ( ∂ r + r ∂ r + ∂ z ) ω uz + r ∂ r ( rf θ ) , ( u r ∂ r + u z ∂ z ) ω hr − ( ω hr ∂ r + ω hz ∂ z ) u r + r ∂ z ( u r h θ ) = ( ∂ r + r ∂ r + ∂ z − r ) ω hr − ∂ z ( ∂ z g r − ∂ r g z ) , ( u r ∂ r + u z ∂ z ) ω hz − ( ω hr ∂ r + ω hz ∂ z ) u z − r ∂ r ( u r h θ ) = ( ∂ r + r ∂ r + ∂ z ) ω hz + r ∂ r ( r ( ∂ z g r − ∂ r g z )) . (4.1)Comparing with (1.6), (4.1) has a simpler form, from which we derive better localizedenergy estimates and it turns out that we can obtain better decay rates. Step 1:
Proof of ω uθ and a weaker decay of ( ω ur , ω uz ).Same as in section 3, consider the scaled solution ˜ u (˜ x ), ˜ h (˜ x ) and ˜ ω (˜ x ), where ˜ x = λ x .Drop the ” ∼ ” for simplification of notation when computations take place under the scaledsense. Taking the same cut-off function φ ( y ) as previous, and testing the vorticity equation(4.1) with ω ur φ , ω uθ φ , ω uz φ , ω hr φ and ω hz φ respectively, we deduce that Z C (cid:18) |∇ ( ω uθ φ ) | + ( ω uθ φ ) r (cid:19) dy = Z C (cid:18) | ω uθ | |∇ φ | + 12 | ω uθ | ( u r ∂ r + u z ∂ z )( φ )+ u r r ( ω uθ ) φ − ω uθ φ r ( − ω hr h θ + ω ur u θ ) + f z ω uθ φr φ + φ ( f z ∂ r − f r ∂ z )( ω uθ φ ) + ω uθ φ ( f z ∂ r − f r ∂ z ) φ (cid:19) dy ≤ C (1 + k ( u r , u θ , u z , h θ ) k L ∞ ( C ) ) k ( ω ur , ω uθ , ω hr ) k L ( C ) + 18 k∇ ( ω uθ φ ) k L ( C ) , Z C (cid:18) |∇ ( ω ur φ ) | + ( ω ur φ ) r (cid:19) dy = Z C (cid:18) ( ω ur ) |∇ φ | + 12 | ω ur | ( u r ∂ r + u z ∂ z )( φ ) − u r ( ∂ r (( ω ur φ ) ) + ∂ z ( ω ur φ · ω uz φ )) − ( ω ur φ ) r u r + f θ φ ( ∂ z ( ω ur φ ) + ω ur ∂ z φ ) (cid:19) dy = Z C (cid:18) ( ω ur ) |∇ φ | + 12 ( ω ur ) ( u r ∂ r + u z ∂ z )( φ ) + ( | ω ur | φ ∂ r + ω ur φ · ω uz φ∂ z ) u r + f θ φ ( ∂ z ( ω ur φ ) + ω ur ∂ z φ ) (cid:19) dy ≤ ( C (1 + k ( u r , u z ) k L ∞ ( C ) ) k ( ω ur , ω uz ) k L ( C ) + k ( ∇ ( ω ur φ ) , ∇ ( ω uz φ ) k L ( C ) ,C (1 + k ( u r , u z ) k L ∞ ( C ) + k∇ u r k L ∞ ( C ) ) k ( ω ur , ω uz ) k L ( C ) + k∇ ( ω ur φ ) k L ( C ) , C |∇ ( ω uz φ ) | dy = Z C (cid:18) ( ω uz ) |∇ φ | + 12 ( ω uz ) ( u r ∂ r + u z ∂ z )( φ ) − u z ∂ r ( ω ur φ · ω uz φ ) − u z ∂ z ( ω uz φ ) − u z r ω ur ω uz φ − f θ φ ( ∂ r ( ω uz φ ) + ω uz ∂ r φ ) (cid:19) dy = Z C (cid:18) ( ω uz ) |∇ φ | + 12 ( ω uz ) ( u r ∂ r + u z ∂ z )( φ ) + ( ω uz ) φ ∂ z u z + ω ur ω uz φ ∂ r u z − f θ φ ( ∂ r ( ω uz φ ) + ω uz ∂ r φ ) (cid:19) dy ≤ C (1 + k ( u r , u z ) k L ∞ ( C ) ) k ( ω ur , ω uz ) k L ( C ) + k ( ∇ ( ω ur φ ) , ∇ ( ω uz φ )) k L ( C ) C (cid:18) k ( u r , u z ) k L ∞ ( C ) + k∇ u z k L ∞ ( C ) (cid:19) k ( ω ur , ω uz ) k L ( C ) + k∇ ( ω uz φ ) k L ( C ) , Z C (cid:18) |∇ ( ω hr φ ) | + ( ω hr φ ) r (cid:19) dy = Z C (cid:18) | ω hr | |∇ φ | + 12 ( ω hr ) ( u r ∂ r + u z ∂ z )( φ ) − u r ∂ r ( ω hr φ ) − u r ∂ z ( ω hr φ · ω hz φ ) − u r r ( ω hr φ ) + 2 u r h θ r ∂ z ( ω hr φ )+ ∂ z ( ω hr φ )( ∂ z g r − ∂ r g z ) (cid:19) dy = Z C (cid:18) | ω hr | |∇ φ | + 12 | ω hr | ( u r ∂ r + u z ∂ z )( φ ) + (( ω hr ) φ ∂ r + ω hr ω hz φ ∂ z ) u r − ∂ r u r + ∂ z u z ) h θ ∂ z ( ω hr φ ) + ∂ z ( ω hr φ )( ∂ z g r − ∂ r g z ) (cid:19) dy ≤ C (1 + k ( u r , u z ) k L ∞ ( C ) ) k ( ω hr , ω hz ) k L ( C ) + k ( ∇ ( ω hr φ ) , ∇ ( ω hz φ )) k L ( C ) + C k u r k L ∞ ( C ) R R h θ r dyC (1 + k ( u r , u z ) k L ∞ ( C ) + k∇ u r k L ∞ ( C ) ) k ( ω hr , ω hz ) k L ( C ) + k∇ ( ω hr φ ) k L ( C ) + C k h θ k L ∞ ( C ) k ( ∇ u r , ∇ u z ) k L ( C ) , Z C |∇ ( ω hz φ ) | dy = Z C (cid:18) | ω hz | |∇ φ | + 12 | ω hz | ( u r ∂ r + u z ∂ z )( φ ) − u z ∂ r ( ω hr φ · ω hz φ ) − u z r ω hr ω hz φ − u z ∂ z ( ω hz φ ) − u r h θ r ∂ r ( ω hz φ ) − ∂ r ( ω hz φ )( ∂ z g r − ∂ r g z ) (cid:19) dy = Z C (cid:18) | ω hz | |∇ φ | + 12 | ω hz | ( u r ∂ r + u z ∂ z )( φ ) + ( ω hr ω hz φ ∂ r + ( ω hz φ ) ∂ z ) u z +2 ω hz φ r ∂ r ( u r h θ ) − ∂ r ( ω hz φ )( ∂ z g r − ∂ r g z ) (cid:19) dy ≤ C (1 + k ( u r , u z ) k L ∞ ( C ) ) k ( ω hr , ω hz ) k L ( C ) + k ( ∇ ( ω hr φ ) , ∇ ( ω hz φ )) k L ( C ) + C k u r k L ∞ ( C ) R R h θ r dy,C (1 + k ( u r , u z ) k L ∞ ( C ) + k∇ u z k L ∞ ( C ) ) k ( ω hr , ω hz ) k L ( C ) + k∇ ( ω hz φ ) k L ( C ) + C k ( ∇ u r , ∇ h θ ) k L ( C ) k ( u r , h θ ) k L ∞ ( C ) . k∇ ω uθ k L ( C ) ≤ C (1 + k ( u r , u θ , u z , h θ ) k L ∞ ( C ) ) k ( ω ur , ω uθ , ω hr ) k L ( C ) , (4.2) k ( ∇ ω ur , ∇ ω uz ) k L ( C ) ≤ C (1 + k ( u r , u z ) k L ∞ ( C ) ) k ( ω ur , ω uz ) k L ( C ) , (4.3) k ( ∇ ω hr , ∇ ω hz ) k L ( C ) ≤ C (1 + k ( u r , u z ) k L ∞ ( C ) ) k ( ω hr , ω hz ) k L ( C ) + C k u r k L ∞ ( C ) , (4.4)and k∇ ( ω ur , ω uz ) k L ( C ) ≤ C (1 + k ( u r , u z ) k L ∞ ( C ) + k∇ ( u r , u z ) k L ∞ ( C ) ) k ( ω ur , ω uz ) k L ( C ) , (4.5) k∇ ( ω hr , ω hz ) k L ( C ) ≤ C (1 + k ( u r , u z ) k L ∞ ( C ) + k ( ∇ u r , ∇ u z ) k L ∞ ( C ) ) k ( ω hr , ω hz ) k L ( C ) + C k ( ∇ u r , ∇ h θ ) k L ( C ) k ( u r , h θ ) k L ∞ ( C ) . (4.6)As previous, utilizing the Brezis-Gallouet’s inequality and (4.2)-(4.6), we are led to k ω uθ k L ∞ ( ¯ C ) ≤ C (cid:18) k ( u r , u θ , u z , h θ ) k / L ∞ ( C ) ) k ( ω ur , ω uθ , ω hr ) k L ( C ) (cid:19) × ln / (cid:18) e + k ∆ ω uθ k L ( ¯ C ) (cid:19) , k ( ω ur , ω uz ) k L ∞ ( ¯ C ) ≤ C (cid:18) k ( u r , u z ) k L ∞ ( C ) ) k ( ω ur , ω uz ) k L ( C ) (cid:19) × ln / (cid:18) e + k (∆ ω ur , ∆ ω uz ) k L ( ¯ C ) (cid:19) , k ( ω hr , ω hz ) k L ∞ ( ¯ C ) ≤ C (cid:18) k ( u r , u z ) k L ∞ ( C ) ) k ( ω hr , ω hz ) k L ( C ) + k u r k L ∞ ( C ) (cid:19) × ln / (cid:18) e + k (∆ ω ur , ∆ ω uz ) k L ( ¯ C ) (cid:19) . and k ( ω ur , ω uz ) k L ∞ ( ¯ C ) ≤ C (cid:18) k ( u r , u z ) k / L ∞ ( C ) + k ( ∇ u r , ∇ u z ) k / L ∞ ( C ) ) k ( ω ur , ω uz ) k L ( C ) (cid:19) × ln / (cid:18) e + k (∆ ω ur , ∆ ω uz ) k L ( ¯ C ) (cid:19) , k ( ω hr , ω hz ) k L ∞ ( ¯ C ) ≤ C (cid:18) k ( u r , u z ) k / L ∞ ( C ) + k ( ∇ u r , ∇ u z ) k / L ∞ ( C ) ) k ( ω hr , ω hz ) k L ( C ) + k ( ∇ u r , ∇ h θ ) k L ( C ) k ( u r , h θ ) k L ∞ ( C ) (cid:19) × ln / (cid:18) e + k (∆ ω ur , ∆ ω uz ) k L ( ¯ C ) (cid:19) . Then scaling back to find λ k ω uθ k L ∞ ( C ,λ ) ≤ C (cid:18) λ / k ( u r , u θ , u z , h θ ) k / L ∞ ( C ,λ ) ) λ / k ( ω ur , ω uθ , ω hr ) k L ( C ,λ ) (cid:19) × ln / (cid:18) e + λ / k ∆ ω uθ k L ( C ,λ ) (cid:19) , k ( ω ur , ω uz ) k L ∞ ( C ,λ ) ≤ Cλ / (cid:18) λ k ( u r , u z ) k L ∞ ( C ,λ ) ) k ( ω ur , ω uz ) k L ( C ,λ ) (cid:19) × ln / (cid:18) e + λ / k (∆ ω ur , ∆ ω uz ) k L ( C ,λ ) (cid:19) ,λ k ( ω hr , ω hz ) k L ∞ ( C ,λ ) ≤ C (cid:18) λ k ( u r , u z ) k L ∞ ( C ,λ ) ) λ / k ( ω hr , ω hz ) k L ( C ,λ ) + λ k u r k L ∞ ( C ,λ ) (cid:19) ln / (cid:18) e + λ / k (∆ ω hr , ∆ ω hz ) k L ( C ,λ ) (cid:19) , and λ k ( ω ur , ω uz ) k L ∞ ( C ,λ ) ≤ Cλ / (cid:18) λ / k ( u r , u z ) k / L ∞ ( C ,λ ) (4.7)+ λ k ( ∇ u r , ∇ u z ) k / L ∞ ( C ,λ ) ) k ( ω ur , ω uz ) k L ( C ,λ ) (cid:19) ln / (cid:18) e + λ / k (∆ ω ur , ∆ ω uz ) k L ( C ,λ ) (cid:19) ,λ k ( ω hr , ω hz ) k L ∞ ( C ,λ ) ≤ C (cid:18) λ / k ( u r , u z ) k / L ∞ ( C ,λ ) + λ k ( ∇ u r , ∇ u z ) k / L ∞ ( C ,λ ) ) × λ / k ( ω hr , ω hz ) k L ( C ,λ ) + λ k ( ∇ u r , ∇ h θ ) k L ( C ,λ ) k ( u r , h θ ) k L ∞ ( C ,λ ) (cid:19) × ln / (cid:18) e + λ / k (∆ ω hr , ∆ ω hz ) k L ( ¯ C ) (cid:19) . (4.8)By the a priori bound on ( u , h ), we have k ω uθ k L ∞ ( C ,λ ) ≤ C (ln λ ) / λ / , k ( ω ur , ω uz ) k L ∞ ( C ,λ ) ≤ C ln λλ , (4.9) k ( ω hr , ω hz ) k L ∞ ( C ,λ ) ≤ C ln λλ . This verifies (1.13).In addition, carrying out a similar estimate on the unit ball centered at x ∈ ¯ C , we canshow that k∇ ω uθ k L ∞ ( C ,λ ) ≤ C (ln λ ) / λ / . (4.10) Step 2: we employ the
Biot-Savart law to get better decay estimates of | ω ur | + | ω uz | and | ω hr | + | ω hz | .In order to get more decay estimates of ω r and ω z , as argued in [3], we consider further L ∞ estimates of ∇ u r , ∇ u z and ∇ h θ by applying Lemma 2.2.It is well-known that ∇ ( u r + u z e z ) = C · ω θ e θ + K ∗ ( ω θ e θ ) , (4.11)where C is a constant matrix and K is a Calderon-Zymund kernel . One may refer to the book[28] for the details. 13pplying Lemma 2.2 to (4.11) and note the decay of ω uθ , we can get |∇ u r | + |∇ u z | ≤ C (ln λ ) / λ / . (4.12)Now go back to (4.7) and (4.8), we get better decay on | ω ur | + | ω uz | and | ω hr | + | ω hz | , | ω ur | + | ω uz | ≤ Cλ − / (cid:18) λ / ( λ − / (ln λ ) / ) / + λ ( λ − / (ln λ ) / ) / (cid:19) (ln λ ) / ≤ Cλ − / (ln λ ) / , for large λ, (4.13) | ω hr | + | ω hz | ≤ Cλ − λ (ln λ ) / (ln λ ) / ≤ Cλ − (ln λ ) / , for large λ. (4.14) Step 3:
Improved decay rate of ∇ h θ by weighted energy estimates. By slightly modifyingthe proof in Lemma 3.2 in [35] (taking δ = ( ) − in that lemma), we obtain the followingestimates, which will be used in the proof of Theorem 1.3. Lemma 4.1.
Let ( u , h , p ) be an axially symmetric smooth D-solutions to inhomogeneousstationary MHD equations with f satisfying (1.7) and (1.11), where h ( r, z ) = h θ ( r, z ) e θ .Then the following estimates hold Z R | ω uθ | dx + Z R r ( ) − |∇ ω uθ | dx + Z R r ( ) − | ∂ z ∇ ω uθ | dx ≤ C ( M ) . Lemma 4.2.
Let ( u , h , p ) be an axially symmetric smooth D-solutions to inhomogeneousstationary MHD equations with f satisfying (1.7), (1.11) and g satisfying (1.12), where h ( r, z ) = h θ ( r, z ) e θ , suppose that | u r ( r, z ) | + | u θ ( r, z ) | + | u z ( r, z ) | + | h θ ( r, z ) | ≤ C (1 + r ) − δ , (4.15) |∇ u r ( r, z ) | + |∇ u z ( r, z ) | ≤ C (1 + r ) − − γ , (4.16) holds for some δ, γ ∈ [0 , . Then the following estimates holds Z R r δ ∧ γ ( |∇ ω ur | + |∇ ω uz | ) dx ≤ C ( M ) , (4.17) Z R r δ ∧ γ ( |∇ ω hr | + |∇ ω hz | ) dx ≤ C ( M ) , (4.18) Z R r δ ∧ γ +2 δ ( | ∂ z ∇ ω ur | + | ∂ z ∇ ω uz | ) dx ≤ C ( M ) , (4.19) Z R r δ ∧ γ +2 δ ( | ∂ z ∇ ω hr | + | ∂ z ∇ ω hz | ) dx ≤ C ( M ) , (4.20) where δ ∧ γ = min { δ, γ } .In particular, by Lemma 2.3, we obtain the following decay rate: | ω ur ( r, z ) | + | ω uz ( r, z ) | ≤ C ( M ) r − − (3( δ ∧ γ )+2 δ ) , | ω hr ( r, z ) | + | ω hz ( r, z ) | ≤ C ( M ) r − − (3( δ ∧ γ )+2 δ ) . roof. According to Lemma 3.8 in [35], we have (4.17) and (4.19). Similarly, we can prove(4.18). It remains to prove (4.20).We start to derive the weighted estimates of ∇ ∂ rz h θ . ∂ rz (cid:20) ( u r ∂ r + u z ∂ z ) h θ − u r r h θ (cid:21) = (cid:18) ∂ r + 1 r ∂ r + ∂ z − r (cid:19) ∂ rz h θ (4.21) − r ∂ rz h θ + 2 r ∂ z h θ + ∂ rz ( ∂ z g r − ∂ r g z )take η r a ∂ rz h θ as a test function to (4.21) and integrating over R , we obtain0 = Z R η r a |∇ ∂ rz h θ | dx + 12 Z R (cid:20) ∂ r ( η r a − ) − ∂ r ( η r a ) (cid:21) | ∂ rz h θ | dx +2 Z R η r a − | ∂ rz h θ | dx − Z R η r a ∂ rzz h θ ( u r ∂ r + u z ∂ z ) ∂ r h θ dx − Z R η r a ∂ rzz h θ ( ∂ r u r ∂ r + ∂ r u z ∂ z ) h θ dx − Z R η r a ∂ rzz h θ (cid:18) u r r − ∂ r u r r (cid:19) h θ r dx + Z R η r a ∂ rzz h θ u r r ∂ r h θ dx + 2 Z R η r a − ∂ rzz h θ h θ r dx − Z R η r a ∂ rzz h θ ∂ r ( ∂ z g r − ∂ r g z ) dx =: − Z R η r a |∇ ∂ rz h θ | dx + X i =1 D i + X j =1 E j . Using previous estimates, we can bound these terms as follows. | D | ≤ C Z r ≥ r r a − | ∂ rz h θ | dx ≤ C Z R r γ | ∂ rz h θ | dx, if a ≤ γ, | D | ≤ C Z R η r a − | ∂ rz h θ | dx ≤ C Z R r γ | ∂ rz h θ | dx, if a ≤ γ, | E | ≤ Z R η r a |∇ ∂ rz h θ | dx + Z R η r a ( | u r | + | u z | ) | ∂ rz h θ | dx ≤ Z R η r a |∇ ∂ rz h θ | dx + C Z R r γ | ∂ rz h θ | dx, if a ≤ γ + 2 δ, | E | ≤ Z R η r a |∇ ∂ rz h θ | dx + Z R η r a ( |∇ u r | + |∇ u z | ) |∇ h θ | dx ≤ Z R η r a |∇ ∂ rz h θ | dx + C k∇ h θ k L , if a ≤ γ ) , | E | ≤ Z R η r a |∇ ∂ rz h θ | dx + Z R η r a (cid:18) | u r | r + |∇ u r | (cid:19) | h θ r | dx ≤ Z R η r a |∇ ∂ rz h θ | dx + C Z R h θ r , if a ≤ { γ, δ } , | E | ≤ Z R η r a |∇ ∂ rz h θ | dx + C Z R η r a u r r |∇ h θ | dx, if a ≤ δ ) , E | ≤ Z R η r a |∇ ∂ rz h θ | dx + C Z R η r a − h θ r dx ≤ Z R η r a |∇ ∂ rz h θ | dx + C Z R h θ r dx, if a ≤ , | E | ≤ Z R η r a |∇ ∂ rz h θ | dx + C Z R η r a | ∂ r ( ∂ z g r − ∂ r g z ) | dx. We infer that Z R r δ + γ |∇ ∂ rz h θ | dx < ∞ . (4.22)Similarly, there also holds Z R r δ + γ |∇ ∂ zz h θ | dx < ∞ . (4.23)Therefore Z R r δ + γ ( |∇ ∂ z ω hr | + |∇ ∂ z ω hz | ) dx < ∞ . By (1.8) and (4.12), choose δ = ( ) − and γ = ( ) − in Lemma 4.2, and we are led to Z R r ( ) − ( |∇ ω hr | + |∇ ω hz | ) dx < ∞ , (4.24) Z R r ( ) − ( |∇ ∂ z ω hr | + |∇ ∂ z ω hz | ) dx < ∞ , (4.25)and | ω hr ( r, z ) | + | ω hz ( r, z ) | ≤ C ( M ) r − ( ) − . (4.26)Note that ω hr = − ∂ z h θ and ∂ r h θ = ω hz − h θ r , we also have (1.16). The proof of Theorem (1.2)is finished. Proof of Theorem 1.3.
Note that the scaling technique by employing the
Brezis-Gallouet in-equality can not be applied to get any decay along the axis, the only way we know is to dothe weighted energy estimates with the weight ρ = √ r + z . Step 1.
We first notice that the quantity Π := h θ r satisfies the following elliptic equation( u r ∂ r + u z ∂ z )Π = (cid:18) ∂ r + 3 r ∂ r + ∂ z (cid:19) Π + 1 r ( ∂ z g r − ∂ r g z ) . (5.1)Following the argument developed in [35], we have the following weighted estimates for Π.16 uppose that | u r ( r, z ) | + | u z ( r, z ) | ≤ C (1 + ρ ) − τ , ρ = p r + z (5.2) for some τ ∈ [0 , , and g satisfies (1.18) and (1.19), then we have Z R | Π( r, z ) | rdrdz < ∞ , (5.3) Z R ρ τ |∇ Π( r, z ) | rdrdz < ∞ , (5.4) Z R ρ τ |∇ ∂ z Π( r, z ) | rdrdz < ∞ . (5.5) Step 2.
Derive the decay rate for h θ .By a priori estimates, we have Z R | h θ ( r, z ) | r + |∇ h θ ( r, z ) | dx < ∞ . (5.6)Combining the results in (4.18) and (5.3)-(5.6), then Z R | Π( r, z ) | dx < ∞ , (5.7) Z R ( r + | z | τ ) |∇ Π( r, z ) | dx < ∞ , (5.8) Z R ( r γ + | z | τ ) |∇ ∂ z Π( r, z ) | dx < ∞ , (5.9)where γ can be any constant less than . Fix d >
1, then for each n ∈ N , Z n +1 n Z ∞ d | Π( r, z ) | rdrdz < ∞ . By mean value theorem, there exists z n ∈ [2 n , n +1 ] such that Z ∞ d | Π( r, z n ) | rdr ≤ Cz n . Then for any z , choose z n > z and Z ∞ d | Π( r, z ) | rdr = Z ∞ d | Π( r, z n ) | rdr − Z ∞ d Z z n z Π( r, t ) ∂ t Π( r, t ) rdrdt =: I + I , | I | ≤ (cid:18) Z ∞ d Z z n z | Π( r, t ) | rdrdt (cid:19) / (cid:18) Z ∞ d Z z n z | ∂ t Π( r, t ) | rdrdt (cid:19) / ≤ C | z | (1+ τ ) . Letting z n → ∞ , then I → Z ∞ d | Π( r, z ) | rdr ≤ C | z | (1+ τ ) . (5.10)17imilarly, one can find z n ∈ [2 n , n +1 ] such that Z ∞ d |∇ Π( r, z n ) | rdr ≤ Cz n , Z ∞ d |∇ Π( r, z ) | rdr = Z ∞ d |∇ Π( r, z n ) | rdr − Z ∞ d Z z n z ∇ Π( r, t ) · ∂ t ∇ Π( r, t ) rdrdt =: J + J , | J | ≤ (cid:18) Z ∞ d Z z n z |∇ Π( r, t ) | rdrdt (cid:19) / (cid:18) Z ∞ d Z z n z | ∂ t ∇ Π( r, t ) | rdrdt (cid:19) / ≤ Cd γ , C | z | τ . Letting n → ∞ , J →
0. Recall that γ = ( ) − , then J ≤ min (cid:26) Cd γ , C | z | τ (cid:27) and byinterpolation Z ∞ d |∇ Π( r, z ) | rdr ≤ (cid:18) Cd ( ) − (cid:19) (cid:18) C | z | τ (cid:19) ≤ Cd | z | ( (1+2 τ )) − . Finally, | Π( d, z ) | = 1 r − d Z r d | Π( r, z ) | dr + ( | Π( r, z ) | − r − d Z r d | Π( r, z ) | dr )=: H + H , | H | = (cid:12)(cid:12)(cid:12)(cid:12) | Π( d, z ) | − | Π( d ∗ , z ) | (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z r d | Π( r, z ) ∂ r Π( r, z ) | dr ≤ Cd (cid:18) Z ∞ d | Π( r, z ) | rdr (cid:19) / (cid:18) Z ∞ d |∇ Π( r, z ) | rdr (cid:19) / ≤ Cd (cid:18) C | z | (1+ τ ) (cid:19) / (cid:18) Cd | z | ( (1+2 τ )) − (cid:19) ≤ Cd | z | ( + τ ) − , It follows from (5.11) that lim r →∞ H = 0 and | h θ ( d, z ) | ≤ C ( M ) | z | − ( + τ ) − . (5.11)Together with previous decay results on r , we have | h θ ( r, z ) | ≤ C ( M )( ρ + 1) − ( + τ ) − , ∀ ( r, z ) ∈ R + × R , ρ = p r + z . (5.12)In general case where u θ = 0, we only have τ = 0, which implies that | h θ | ≤ C ( M )( ρ + 1) − .We remark here that we do not obtain any decay rate for Π, since the weighted estimate for ∇ ∂ r Π is not available.
Step 3.
For the special case where the flow has zero swirl u θ ≡
0, we can further derivethe decay rates of u . 18n this case, one has( u r ∂ r + u z ∂ z ) ω uθ − u r r ω uθ + 1 r ∂ z ( h θ ) = (cid:18) ∂ r + 1 r ∂ r + ∂ z − r (cid:19) ω uθ − ( ∂ z f r − ∂ r f z ) , and ( u r ∂ r + u z ∂ z )Ω + ∂ z Π = (cid:18) ∂ r + 3 r ∂ r + ∂ z (cid:19) Ω + 1 r ( ∂ z f r − ∂ r f z ) , for Ω := ω uθ ( r,z ) r . We run the same argument as in [35] for Ω to derive the estimate R R ρ d |∇ Ω | dx , there is an additional term (cid:12)(cid:12)(cid:12)(cid:12) Z R φ ρ d Ω ∂ z Π dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z R φ ρ d | Π || Ω || ∂ z Π | dx ≤ C k Π k L ∞ k Ω k L ( Z R ρ d | ∂ z Π | dx ) / < ∞ , if d ≤ τ , where φ is a cut-off function satisfying φ ∈ C ∞ ( R ), φ = φ ( ρ ), 0 ≤ φ ≤ φ ( ρ ) = 1 on2 ρ ≤ ρ ≤ ρ , φ ( ρ ) = 0 on ρ ≤ ρ or ρ ≥ ρ , such that |∇ φ | ≤ Cρ on ρ < ρ < ρ and |∇ φ | ≤ Cρ on ρ < ρ < ρ , and ∇ φ = 0 elsewhere. Thus Z R ρ τ |∇ Ω( r, z ) | dx < ∞ . (5.13)To derive the estimate of ∇ ∂ z Ω, we see that ∂ z Ω satisfies ∂ z [( u r ∂ r + u z ∂ z )Ω + ∂ z Π ] = (cid:18) ∂ r + 3 r ∂ r + ∂ z (cid:19) ∂ z Ω + ∂ z ( 1 r ( ∂ z f r − ∂ r f z )) . (5.14)Multiplying (5.14) by φ ρ d ∂ z Ω and integrating over R , then we get0 = Z R φ ρ d |∇ ∂ z Ω | dx − Z R [ ∂ r ( φ ρ d ) + ∂ z ( φ ρ d )] | ∂ z Ω | dx + 12 Z R ∂ r ( φ ρ d ) | ∂ z Ω | r dx − Z R ∂ z ( φ ρ d ) ∂ z Ω( u r ∂ r + u z ∂ z )Ω dx − Z R φ ρ d ∂ z Ω( u r ∂ r + u z ∂ z )Ω dx − Z R ∂ z ( φ ρ d ) ∂ z Ω · ∂ z Π dx − Z R φ ρ d ∂ z Ω · ∂ z Π dx + Z R ∂ z ( φ ρ d ) ∂ z Ω · ( ( ∂ z f r − ∂ r f z ) r ) dx + Z R φ ρ d ∂ z Ω · ( ∂ z f r − ∂ r f z ) r dx =: Z R φ ρ d |∇ ∂ z Ω | dx + X i =1 K i + X j =1 Q j . | K | + | K | ≤ C Z ρ ≥ ρ ρ d − |∇ Ω | dx ≤ ∞ , if d ≤ τ , | K | ≤ C Z ρ ≥ ρ ρ d − | ( u r , u z ) ||∇ Ω | dx ≤ ∞ , if d ≤
32 (1 + τ ) , | K | ≤ Z R φ ρ d |∇ ∂ z Ω | dx + C Z R φ ρ d | ( u r , u z ) | |∇ Ω | dx, if d ≤
12 + 5 τ , | K | ≤ C Z R φρ d − |∇ Ω || Π || ∂ z Π | dx ≤ C Z R ρ τ |∇ Ω | dx + C Z R ρ d − − τ | Π | | ∂ z Π | dx ≤ C Z R ρ τ |∇ Ω | dx + C Z R ρ τ | ∂ z Π | dx, if d ≤
74 + 34 τ, | K | ≤ Z R φ ρ d | ∂ z Ω | dx + C Z R φ ρ d Π | ∂ z Π | dx ≤ Z R φ ρ d | ∂ z Ω | dx + C Z R ρ τ | ∂ z Π | dx, if d ≤ τ, and | Q | ≤ C Z ρ ≥ ρ ρ d − − (1+ τ ) | ∂ z f r − ∂ r f z r | dx + C Z ρ ≥ ρ ρ τ |∇ Ω | dx, if d ≤ τ , | Q | ≤ Z R φ ρ d |∇ ∂ z Ω | dx + C Z R φ ρ d | ∂ z f r − ∂ r f z r | dx. Letting ρ → ∞ , we obtain Z R ρ τ |∇ ∂ z Ω | dx < ∞ . (5.15)Combining the weighted energy estimate in Lemma 4.1 and (5.13)-(5.15), we find Z R r | Ω | dx < ∞ , Z R ( r δ + | z | τ ) |∇ Ω | dx < ∞ , Z R ( r δ + | z | τ ) |∇ ∂ z Ω | dx < ∞ . Same as
Step 2 , fix any d >
1, we have Z ∞ d | Ω( r, z ) | rdr ≤ Cd | z | τ . For each n ∈ N , Z n +1 n Z ∞ d | z | τ |∇ Ω( r, z ) | rdrdz < ∞ .
20y mean value theorem, one can find z n ∈ [2 n , n +1 ] such that Z ∞ d |∇ Ω( r, z n ) | rdr ≤ C | z n | / , Z ∞ d |∇ Ω( r, z ) | rdr = Z ∞ d |∇ Ω( r, z n ) | rdr − Z ∞ d Z z n z ∇ Ω( r, t ) · ∂ t ∇ Ω( r, t ) rdrdt := L + L , | L | ≤ (cid:18) Z ∞ d Z z n z |∇ Π( r, t ) | rdrdt (cid:19) / (cid:18) Z ∞ d Z z n z | ∂ t ∇ Π( r, t ) | rdrdt (cid:19) / ≤ Cd δ , C | z |
34 (1+ τ ) . Letting n → ∞ , L →
0. Take δ = ( ) − and L ≤ min { Cd δ , C | z |
34 (1+ τ ) } , Z ∞ d |∇ Ω( r, z ) | rdr ≤ (cid:18) Cd − (cid:19) (cid:18) C | z | (1+ τ ) (cid:19) ≤ Cd | z | (1+ τ ) − . Hence, | Ω( d, z ) | = 1 r − d Z r d | Ω( r, z ) | dr + (cid:18) | Ω( d, z ) | − r − d Z r d | Ω( r, z ) | dr (cid:19) =: G + G . | G | = (cid:12)(cid:12)(cid:12)(cid:12) | Ω( d, z ) | − | Ω( d ∗ , z ) | (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z r d | Ω( r, z ) · ∂ r Ω( r, z ) | dr ≤ Cd (cid:18) Z ∞ d | Ω( r, z ) | rdr (cid:19) / (cid:18) Z ∞ d |∇ Ω( r, z ) | rdr (cid:19) / ≤ Cd (cid:18) Cd | z | (1+ τ ) − (cid:19) / (cid:18) Cd | z | (1+ τ ) − (cid:19) / ≤ Cd | z | (1+ τ ) − . Therefore we have | ω uθ ( r, z ) | ≤ C ( M )(1 + ρ ) − ( (1+ τ )) − , ∀ ( r, z ) ∈ R + × R , ρ = p r + z . Step 4.
Decay rate of u .Following the argument developed in [35], fix any x ∈ R \ { } , define a cut-off function ϕ ∈ C ∞ ( R ) satisfying ϕ ( y ) ≡ ∀ y ∈ B ρ/ ( x ) and ϕ ( y ) ≡ ∀ y / ∈ B ρ/ ( x ), where ρ = | x | and |∇ ϕ ( y ) | ≤ C | y | , |∇ ϕ ( y ) | ≤ C | y | for y ∈ D := B ρ/ ( x ) \ B ρ/ ( x ). For u ( x ) = u r e r + u z e z , since curl u = ω uθ e θ , then u ( x ) = − Z R ∇ y Γ( x , y ) × ( ω uφ ( y ) ϕ ( y ) e φ ) d y − Z R Γ( x , y )( ∇ y ϕ ( y ) × e φ ) ω uφ ( y ) d y Z R Γ( x , y )(∆ y ϕ )( y ) u ( y ) d y + 2 Z R ( ∇ y Γ)( x , y ) · ( ∇ y ϕ )( y ) u ( y ) d y := X i =1 R i | R i | ≤ Cρ / , i = 2 , ,
4. For the estimate of R , fix a d ∈ (0 , ρ ),which will be determined later, then | R | ≤ sup y ∈ B d ( x ) | ω uφ ( y ) | Z B d ( x ) |∇ Γ( x − y ) | d y + (cid:18) Z B ρ/ ( x ) \ B d ( x ) |∇ Γ( x − y ) | d y (cid:19) (cid:18) Z B ρ/ ( x ) \ B d ( x ) | ω uφ ( y ) | d y (cid:19) ≤ Cρ − ( (1+ τ )) − d + Cd − . By choosing d = ρ (1+ τ ) − , we obtain the bound for | R | ≤ Cρ τ ) − . Hence we have | ( u r , u z )( r, z ) | ≤ C ( M )(1 + ρ ) − (1+ τ ) − . (5.16) Step 5.
Iteration.At the beginning, we have τ = 0 in (5.2), then by using the arguments developed in Step1 to Step 4 , we have a new τ in (5.2), which will be denoted by τ = ( ) − . Running asecond iteration of these three steps, we get a new τ = τ + τ , and after n iteration, weget τ n = τ + 13192 τ n − = τ n − X i =0 ( 13192 ) i . Let n → ∞ , τ n → ( ) − as n → ∞ . In a word, we infer the following decay rates | u r ( r, z ) | + | u z ( r, z ) | ≤ C ( M )( ρ + 1) − ( ) − , (5.17) | ω θ ( r, z ) | ≤ C ( M )( ρ + 1) − ( ) − , (5.18) | h θ ( r, z ) | ≤ C ( M )( ρ + 1) − ( ) − . (5.19) Acknowledgement.
Weng is partially supported by National Natural Science Founda-tion of China 11701431, 11971307, 12071359, the grant of Project of Thousand Youth Talents(No. 212100004).
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