A Review of results on axially symmetric Navier-Stokes equations, with addendum by X. Pan and Q. S. Zhang
aa r X i v : . [ m a t h . A P ] J a n A REVIEW OF RESULTS ON AXIALLY SYMMETRIC NAVIER-STOKESEQUATIONS, WITH ADDENDUM BY X. PAN AND Q. S. ZHANG
QI S. ZHANGA bstract . In this paper, we give a brief survey of recent results on axially symmetricNavier-Stokes equations (ASNS) in the following categories: regularity criterion, Liou-ville property for ancient solutions, decay and vanishing of stationary solutions. Somediscussions also touch on the full 3 dimensional equations. Two results, closing of thescaling gap for ASNS and vanishing of homogeneous D solutions in 3 dimensional slabswill be described in more detail.In the addendum, two new results in the 3rd category will also be presented, which aregeneralizations of recently published results by the author and coauthors. C ontents
1. Introduction 12. Regularity Criterion 52.1. Critical and slightly super critical regularity conditions 52.2. Criticality of ASNS and closing of the scaling gap 83. Ancient Solutions 114. Stationary Solutions 174.1. D solutions 174.2. Self-similar solutions 274.3. Addendum: Extra results on D solutions, by Xinghong Pan and Q. S. Zhang 30Acknowledgments 37References 371. I ntroduction
The Cauchy problem of Navier-Stokes equations (NS) describing the motion of viscousincompressible fluids in R is µ ∆ v − v ∇ v − ∇ P − ∂ t v = , on R × (0 , ∞ ) div v = , v ( x , = v ( x ) . (1.1)Here v is the velocity field, P is the pressure, both of which are the unknowns; v is thegiven initial velocity; µ > Mathematics Subject Classification.
Key words and phrases.
Regularity, Liouville theorem; ancient solutions, D-solutions; Axially symmetricNavier-Stokes equations.Xinghong Pan is supported by Natural Science Foundation of Jiangsu Province (No. BK20180414), DoubleInnovation Scheme of Jiangsu Province and National Natural Science Foundation of China (No. 11801268). stated otherwise. One can also add a forcing term on the righthand side, then it becomes anonhomogeneous problem.Thanks to Leray’s work [62] in 1934, we know the above problem has a weak solution v ∈ L ∞ ((0 , ∞ ) , L ( R )) such that |∇ v | ∈ L ((0 , ∞ ) , L ( R )) provided that the initial condi-tion has finite kinetic energy. Moreover, k v ( · , t ) − v ( · ) k L ( R ) → t → ∀ T > Z | v ( x , T ) | dx + Z T Z |∇ v ( x , t ) | dxdt ≤ Z | v ( x ) | dx < ∞ . (1.2)See Theorem 3.10 in Tsai’s book [110] for a modern and concise proof e.g. Solutions satis-fying (1.2) are often referred to as Leray-Hopf solutions, in order to distinguish them fromeven weaker solutions. In general, one does not know if a Leray-Hopf solution is smooth,except for a few special cases, usually as a perturbation of a special smooth solution. Sta-bility of NS under small perturbation is well studied. A general result of such kind canbe found in [94] e.g. Over the years several su ffi cient conditions under which Leray-Hopfsolutions are smooth have been obtained. For example the Ladyzhenskaya-Prodi-Serrincondition: | v | ∈ L p , qx , t with p + q ≤ < p < ∞ and the end point result p = , q = ∞ by Escauriaza, Seregin and Sverak [33]. See also [31] by Dong and Wang in higher dimen-sional cases, including both interior and boundary regularity. Here and later, a measurablefunction f = f ( x , t ) is said to be in L p , qx , t if k f k L p , qx , t ≡ (cid:18)R ∞ (cid:16)R R | f | p dx (cid:17) q / p dt (cid:19) / q < ∞ . If p + q =
1, these conditions are scaling invariant or critical under the natural scaling of theNavier Stokes equations: for λ >
0, if ( v , P ) solves (1.1), then ( v λ , P λ ) defined by v λ ( x , t ) ≡ λ v ( λ x , λ t ) , P λ ( x , t ) ≡ λ P ( λ x , λ t ) (1.3)also solves (1.1). It is easy to see that k v k L p , qx , t = k v λ k L p , qx , t for the above p , q . Sometimesthese conditions can be improved logarithmically, even for endpoint cases. See the articlesX.H. Pan [88], T. Tao [106], Barker and Prange [9], e.g. A partial regularity result forthe so-called ”suitable weak solutions” was found by Ca ff arelli, Kohn and Nirenberg [17],building on earlier work of Sche ff er [95, 96]. These solutions are Leray-Hopf solutionswith an extra integrability condition on the pressure term P . It is proven that the singular setof suitable weak solutions, if exists, has one dimensional parabolic Hausdor ff measure 0.The proof utilizes a blow up argument to deduce an ǫ regularity result: smallness of certainscaling invariant integral quantities involving the velocity or vorticity implies boundednessof solutions. Then the size estimate of the possible singular set follows from a coveringargument. See also the papers of F. H. Lin [65], A. Vasseur [113], J. Wolf [117] for similarresults and shorter proofs, some of which employ a De Giogi type (refined energy) methodinstead of blow up method. In [117], using the decomposition v ∇ v = ∇| v | − ( ∇ × v ) × v ,it is observed that one can also use a scaling invariant quantity involving ( ∇ × v ) × v | v | .One consequence is that small perturbations of Beltrami flows are regular. Recall that aflow or a vector field v is called a Beltrami flow if the vorticity ∇ × v is parallel to v . Theseconditional regularity results are consistent with the standard linear theory for second orderparabolic equations with lower order coe ffi cients, coming from the De Giorgi-Nash-Mosertheory. As far as the regularity conditions are concerned, the nonlinearity of the NSE onlyinduces a marginal improvement over the linear case. One can see the di ffi culty in provingregularity by observing that the energy inequality (1.2) only tells us, after using Sobolevinequality and interpolation, that v ∈ L / , / x , t . In a local space time domain, this a prioribound is much weaker than the regularity conditions mentioned above, such as L , x , t . If wehope to prove regularity of the solution, we need to study the behavior of v at micro orinfinitesimal scale. This is amount to studying the behavior of v λ at a fixed scale while EVIEW ON AXIALLY SYMMETRIC NS 3 letting λ →
0. Notice that k v λ k L / , / x , t = λ − / k v k L / , / x , t → ∞ , λ → . So the energy inequality does not furnish any information on micro scale. For this reasonNSE is considered as a super critical equation, i.e. a priori estimate is weaker than regu-larity condition at small scale. Another way to see the super criticality is to consider thevorticity ω = ∇ × v which satisfies the equation ∆ ω − v ∇ ω + ω ∇ v − ∂ t ω = . (1.4)The energy inequality (1.2) tells us |∇ v | ∈ L , x , t . This again is much weaker than the reg-ularity condition of L p , qx , t with p + q <
2, if ∇ v are regarded as potential functions for thevorticity equation. We note that the drift term v ∇ ω , can be handled even locally by anintegral argument since div v =
0, c.f. [120]. So the most dangerous and mysterious termin the vorticity equation is the so-called vortex stretching term ω ∇ v . An avenue of attackon the regularity problem is to exploit the structure on the sets where the vorticity has highvalue, such as angles, intermittency and sparseness. See [18], [32], [42] and a recent [7]e.g. In [7] Bradshaw, Farhat and Grujic reduce the scaling gap by using sparseness of thesuper level sets of the positive and negative parts of the vorticity components at a scalecomparable to the sup of 1 / | ω | nearby. There are also many activities on one componentregularity conditions and regularity conditions on directional derivatives of the velocity:[83], [124], [21], [30], [46]; [90], [57], [15] e.g. One-sided conditions can also be imposedon the eigenvalues of ∇ v or the middle eigenvalue of the strain tensor ∇ v + ( ∇ v ) T . See[121] and [78] e.g. However, unless something dramatic happens, such as the discovery ofa critical a priori estimate, a magic cancellation, or an ingenious construction of a blow upsolution, the regularity problem for the 3 dimensional NSE (1.1) will remain open. Evenif some Leray-Hopf solutions are found to blow up in finite time, it is still interesting tocharacterize the set of initial values that give rise to Leray-Hopf solutions that stay smoothall time, which is nonempty.In this paper, we will focus on a special case of (1.1), namely when v and P are indepen-dent of the angle in a cylindrical coordinate system ( r , θ, x ). That is, for x = ( x , x , x ) ∈ R , r = q x + x , θ = arctan( x / x ) , and the basis vectors e r , e θ , e : e r = ( x / r , x / r , , e θ = ( − x / r , x / r , , e = (0 , , v = v r ( r , x , t ) e r + v θ ( r , x , t ) e θ + v ( r , x , t ) e . By direct computation, v r , v and v θ satisfy the axially symmetric Navier-Stokes equa-tions (cid:0) ∆ − r (cid:1) v r − ( v r ∂ r + v ∂ x ) v r + ( v θ ) r − ∂ r P − ∂ t v r = , (cid:0) ∆ − r (cid:1) v θ − ( v r ∂ r + v ∂ x ) v θ − v θ v r r − ∂ t v θ = , ∆ v − ( v r ∂ r + v ∂ x ) v − ∂ x P − ∂ t v = , r ∂ r ( rv r ) + ∂ x v = , (1.5)which will be abbreviated as ASNS. It looks more complicated than the full 3 dimensionalequation. But simplifications happen in the 2nd equation where the pressure term disap-peared. A tip in carrying out vector calculations under the cylindrical system is to use Q. S. ZHANG tensor notations. For example ∇ v = ∂ r v ⊗ e r + r ∂ θ v ⊗ e θ + ∂ x v ⊗ e = ( ∂ r v r e r + ∂ r v θ e θ + ∂ r v e ) ⊗ e r + r ( v r e θ − v θ e r ) ⊗ e θ + ( ∂ x v r e r + ∂ x v θ e θ + ∂ x v e ) ⊗ e . Taking the inner product with the second entry, the convection terms become − v ∇ v = − ( v r e r + v θ e θ + v e ) · ( ∂ r v r e r + ∂ r v θ e θ + ∂ r v e ) ⊗ e r − ( v r e r + v θ e θ + v e ) · r ( v r e θ − v θ e r ) ⊗ e θ − ( v r e r + v θ e θ + v e ) · ( ∂ x v r e r + ∂ x v θ e θ + ∂ x v e ) ⊗ e = [ − ( v r ∂ r + v ∂ x ) v r + ( v θ ) r ] e r − [( v r ∂ r + v ∂ x ) v θ + v θ v r r ] e θ − ( v r ∂ r + v ∂ x ) v e . This gives rise to the most complicated terms in (1.5).In 2015, it was observed by Lei-Zhang [73] that ASNS is essentially critical under thestandard scaling. So the aforementioned scaling gap is 0. This observation has the e ff ect ofmaking ASNS looks less formidable than the full 3 dimensional case which has a positivescaling gap. Nevertheless all major open problems for the later are still open for the former.In the next three sections, we will describe some recent research results on the followingtopics: regularity conditions (Section 2), ancient solutions (Section 3) and stationary so-lutions (Section 4). These topics are closely related. The study of possible singularity ofsolutions leads to the study of ancient solutions, i.e. solutions whose existence time ex-tends to −∞ . Stationary solutions are ancient solutions but are even more special. Theaforementioned topics have an apparently decreasing order in terms of level of di ffi culties: regularity problem > classification of ancient solutions > classification of homogeneousstationary solutions > classification of homogeneous D solutions. The last type of solutions are defined as stationary solutions to (1.1) with finite Dirich-let energy: R R |∇ v | dx < ∞ , which also vanish at infinity. See Definition 4.1. They werestudied in Leray’s first paper [61]. However as we shall see, even our understanding of sta-tionary solutions are still very primitive. For example, we still do not know if homogeneousD solutions in R are zero, even in the axially symmetric case.Due to the large number of papers in the literature, we need to make a selection on whatto present. This selection only reflects personal interest and knowledge. Some importantresults may be missed. For example, we will not address any papers on boundary valueproblems seriously, which as well known, have their own complications and complexity.Nor will we touch non-uniquess of very weak solutions. Over the decades, in conjunctionwith the development of research on NS, many books have been written, and it is safe tosay that the trend will continue. Let us list a few of them for comprehensive informationand the history of the field: [59, 107, 27, 80, 56, 14, 100, 39, 63, 64, 10, 99, 108, 93, 110].Any suggestions on missing information or improvement are very welcome.We end the introduction by listing a number of notations and conventions to be usedthroughout, which are more or less standard. The velocity field is usually called v and thevorticity ∇ × v is called ω . We use superscripts to denote their components in coordinates.Given a point x = ( x , x , x ) ∈ R , we write x ′ = ( x , x , x h = ( x , x ), r = ( x + x ) / and θ = arctan( x / x ). L p ( D ), p ≥
1, denotes the usual Lebesgue space on a domainD which may be a spatial, temporal or space-time domain. Let X be a Banach space EVIEW ON AXIALLY SYMMETRIC NS 5 defined for functions on D ⊂ R . L p (0 , T ; X ) is the Banach space of space-time functions f on the space time domain D × [0 , T ] with the norm (cid:18)R T k f ( · , t ) k pX dt (cid:19) / p . If no confusionarises, we will also use L p X to abbreviate L p (0 , T ; X ). Sometimes we will also use L px L qt or L qt L px to denote the mixed p , q norm in space time. Let D ⊂ R be an open domain,then H ( D ) = W , ( D ) = { f | f , |∇ f | ∈ L ( D ) } and H ( D ) = W , ( D ) = { f | f , |∇ f | , |∇ f | ∈ L ( D ) } , the standard Sobolev spaces on D. Also, interchangeable notations div v = ∇ · v , v ∇ v = P v i ∂ x i v = v · ∇ v will be used. If there is no confusion, the vertical variable x maybe replaced with z . Also B r ( x ) denotes the ball of radius r centered at x in a Euclideanspace. If s is a number, then s − means any number which is close but strictly less than s .2. R egularity C riterion Critical and slightly super critical regularity conditions.
If the swirl v θ =
0, thenit is known for long time ( O. A. Ladyzhenskaya [58], M. R. Uchoviskii and B. I. Yudovich[112]), that finite energy solutions to (1.5) are smooth for all time. See also the paper by S.Leonardi, J. Malek, J. Necas, and M. Pokorny [75]. By finite energy, we mean (1.2) holds,i.e., the solution is a Leray-Hopf solution.In the presence of swirl, it is still not known in general if finite energy solutions blowup in finite time.However a lower bound for the possible blow up rate is known by the results of C.-C.Chen, R. M. Strain, T.-P.Tsai, and H.-T. Yau in [28], [29], G. Koch, N. Nadirashvili, G.Seregin, and V. Sverak in [51], which appeared around 2008. See also the work by G.Seregin and V. Sverak [102] for a localized version. These authors prove that if | v ( x , t ) | ≤ Cr , (2.1)then solutions are smooth for all time. Here C is any positive constant.The proof is based on the fact that the scaling invariant quantity Γ = rv θ : satisfies theequation ∆Γ − b ∇ Γ − r ∂ r Γ − ∂ t Γ = , (2.2)where b = v r e r + v e . The bound (2.1) says that the equation is essentially scaling invariantand the classical linear regularity theory can be applied after some nontrivial modification.In [29], the authors use this approach to prove that Γ is H¨older continuous first. This implies | v θ | is bounded by r − + α near the z axis, which makes v θ subcritical under the standardscaling. Here α is a small positive constant. Thus v θ is small in micro scale. The smallnessenters into the equation for ω θ in (2.5) after a scaling argument. The authors then manageto prove that ω θ is bounded which in turn proves the whole velocity field v is boundedby the Biot-Savart law. In contrast, a blow up method is used in [51]. The first step is toshow that if a solution of the ASNS blows up in finite time, then after a suitable scalingand limiting procedure, one obtains a nonzero bounded, mild solution of the ASNS, whichexists in the time interval ( −∞ , v ∞ , is called a mild ancientsolution. Here the word ”mild” means that a solution, in addition to being a pointwiseor weak solution of the Navier-Stokes equation, must also satisfy an integral equationinvolving the Stokes kernel. See (3.1) or [51] for a precise definition e.g. The purposeis to rule out solutions of the form a ( t ) ∇ h ( x ) where a = a ( t ) is a di ff erentiable functionand h is a harmonic function. The equation for Γ = rv θ ∞ again plays an essential role.Using the bound (2.1), which survives the scaling, and an integration argument involvingthe maximum principle for equation (2.2), it is shown that v θ ∞ = Γ / r =
0. Therefore v ∞ is a Q. S. ZHANG swirl free, bounded, mild, ancient solution. Now the equation for ω θ / r (see (2.4)) satisfiesthe maximum principle. Using the aforementioned integration argument for the equationof ω θ / r , exploiting the interplay between the velocity and vorticity, one can show that v ∞ is a constant. The bound (2.1) then forces v ∞ =
0. But we already know that v ∞ , v r , v are inthe space of L ∞ ([0 , ∞ ) , BMO − ( R )) and rv θ ( · , ∈ L ∞ , then the solution is regular. Here BMO is the space of functions with bounded mean oscillation, c.f. [47], and
BMO − isthe space of tempered distributions which can be written as partial derivatives of BMOfunctions. Well-posedness and other properties of solutions to NS have been studied byKoch-Tataru [52], Miura [79] and Germain-Pavlovic-Sta ffi lani [40]. Note the function C / r is contained in BMO − . Hence this result extends the one described in the previousparagraph. See also [97], [118]. At the first glance, it seems that the main improvement isjust the relaxation of a pointwise condition to an integral type condition. However, there isan additional feature in that one only needs to impose a condition on the vertical velocity v to gain regularity. See Theorem 2.5 below. The precise statement is: Theorem 2.1. ( [71] ) Let v = v ( x , t ) be a Leray-Hopf solution to (1.5) in the space timeregion R × [0 , T ] . Assume that the initial value satisfies, | rv θ ( x , | < C. Suppose alsov ( · , t ) = ∇ × B ( · , t ) with sup < t < T k B ( · , t ) k BMO ≤ C ∗ . Then v is smooth in R × (0 , T ] . HereC and C ∗ are arbitrary positive constants. In the original paper, the pertinent solutions are stated as suitable weak solutions ex-plained in Section 1. However, no such restriction is actually needed in the proof. Anotherregularity condition proposed in [71] only involves a region outside a paraboloid with thevertex at a given space-time point. This suggests that regularity of a solution at a space-time point only depends on the behavior of the solution in a small part of a space-timecube with vertex at the same point. Such a phenomenon was later proven for the full 3dimensional NS by Neustupa in [82].Recently Seregin and Zhou [105] have relaxed the L ∞ BMO − assumption further to L ∞ ˙ B − ∞ , ∞ assumption. Let us recall ˙ B − ∞ , ∞ is the Besov space consisted of tempered distribu-tions f such that the norm k f k ˙ B − ∞ , ∞ = sup t > t / sup x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z R G ( x , t , y ) f ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) is finite. Here G ( x , t , y ) = (1 / (4 π t ) / ) exp( −| x − y | / (4 t )) is the standard heat kernel on R . Theorem 2.2. ( [105] ) Any axially symmetric suitable weak solution of (1.5) , belonging toL ∞ ˙ B − ∞ , ∞ , is smooth. Using localized energy inequalities coupled with interpolation of L between ˙ B − ∞ , ∞ andthe homogeneous Sobolev space ˙ H ( R ), they prove the following. If v is a suitable weaksolution to the three dimensional NS from the space L ∞ (0 , T ; ˙ B − ∞ , ∞ ), then a number ofscaled energy quantities of v are bounded. Consequently only type I blow up can occur. Inthe axially symmetric case, this has been ruled out in [97]. Therefore these solutions aresmooth. We mention that although BMO − ⊂ ˙ B − ∞ , ∞ , the result in [71] goes beyond suitableweak solutions since there is no need for the local energy inequality. EVIEW ON AXIALLY SYMMETRIC NS 7
Without knowing if blow up happens in general, it is desirable to find an upper boundfor the growth of velocity. It is expected that the solutions are smooth away from the axis,with certain growing bound when approaching the axis. The next theorem confirms thisintuitive idea. Although it did not give the bound (2.1) which is required for smoothness,it reveals the exact gap between what we have and what we need.This seems to be the first pointwise bound for the speed (velocity) for the axially sym-metric Navier-Stokes equation. We mention that a less accurate a priori upper bound forthe vorticity has been found in Burke-Zhang [12] a few years earlier. Also, in the paper[72], Lei-Zhang proved that if the scaling invariant quantity r | v ( x , t ) | is su ffi ciently large ata point ( x , t ), then the solution is close, in C , sense, to a nonzero constant vector after asuitable scaling. Theorem 2.3. (Lei-Navas-Zhang [76] ) (velocity bound). Suppose v is a smooth, axi-ally symmetric solution of the three-dimensional Navier-Stokes equations in R × ( − T , with initial data v = v ( · , − T ) ∈ L ( R ) . Assume further rv θ ∈ L ∞ ( R ) and let R = min { , √ T / } .Then for all ( x , t ) ∈ R × ( − R , , it holds | v r ( x , t ) | + | v ( x , t ) | ≤ C √| ln r | r , < r ≤ min { / , R } . Here r is the distance from x to the z axis, and C is a constant depending only on the initialdata.
In the same paper, the following results are also proven.
Theorem 2.4.
Under the same assumption as the previous theorem, there exists a constantC, depending only on the initial data, such that, | L θ ( x , t ) | ≤ C | ln r | / r / , r ≤ min { / , R } . Here L θ is the angular part of the stream function, which gives rise to v r and v by thefollowing relations v = r ∂ r ( rL θ ) , v r = ∂ x L θ . So it is the most important component of the stream function (vector).
Theorem 2.5.
Let v be a Leray-Hopf solution to (1.5) in R × (0 , ∞ ) such that rv θ ( · , ∈ L ∞ ( R ) . Suppose, for a given constant C > , and all x ∈ R and t ≥ , | v ( x , t ) | ≤ Cr . Then v is regular for all time.Proof.
From the relation v = r ∂ r ( rL θ ), using the notation | x ′ | = r = q x + x , we have, (cid:12)(cid:12)(cid:12) | x ′ | L θ ( x , t ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z | x ′ | ∂ r ( rL θ ) dr (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z | x ′ | (cid:12)(cid:12)(cid:12) rv (cid:12)(cid:12)(cid:12) dr ≤ C | x ′ | . Here we just used the assumption on v . Therefore L θ is a bounded function. Then fromthe main result in [71], we know that v is regular for all time. (cid:3) Q. S. ZHANG
Comparing with the previously mentioned results of Chen, Strain, Tsai and Yau[29] andKoch, Nadirashvilli, Sverak and Seregin [51], there is no restriction on v r in our case. Seealso the paper by Chen, Fang, T.Zhang [16].There are also regularity condition on one component of the velocity and / or vorticity.J. Neustupa and M. Pokorny [84] proved that the regularity of one component (either v r or v θ ) implies regularity of the other components of the solution. See more refined resultsin P. Zhang and T. Zhang [123]. Also proving regularity is the work of Q. Jiu and Z. Xin[50] under an assumption of su ffi ciently small zero-dimension scaled norms. D. Chae andJ. Lee [22] also proved regularity results assuming finiteness of another zero-dimensionalintegral. A pointwise critical blow up criterion: | ω θ | ≤ Cr was also given in Z. Li and X.Pan [77].As mentioned earlier, X. H. Pan [88] recently obtained a loglog improvement of themain result in Chen-Strain-Tsai-Yau and Koch, Nadirashvili, Seregin, and V. Sverak. Al-though it looks like a small improvement, it is a slightly super critical result based on theargument in [71]. The proof replies on the robustness of De Giorgi-Nash-Moser method toprove the function Γ = rv θ has a modulus of continuity at the z axis, under the slightly su-percritical condition on b = v r e r + v e . The vector b controls the drift term in the equationfor Γ : (2.2).There are also global regularity result in special cases. G. Tian and Z. Xin [111] con-structed a family of singular axially symmetric solutions with singular initial data. T. Houand C. Li [43] found a special class of global smooth solutions. See also a recent extension:T. Hou, Z. Lei and C. Li [45].2.2. Criticality of ASNS and closing of the scaling gap.
Despite these e ff orts, there isstill a finite scaling gap between the regularity condition and a priori bounds. In almostall the literature, the regularity conditions are critical and hence are scaling invariant understandard scaling. Improvements are at most logarithmic in nature, except for resorting tofurther requirements such as sparseness of the sets where the vorticity is high. We havementioned the Ladyzhenskaya-Prodi-Serrin condition for regularity requires the velocityto be bounded in suitable function space whose norm is invariant under standard scaling,such as L p , q with p + q =
1. However the energy bound scales as − /
2. So there is a finitegap which makes the equation supercritical.However in a recent paper [73], Lei and Zhang made the following observation.
The vortex stretching term of the ASNS is critical.
Previously it was believed to be super-critical, which means in micro scales the equationbecomes chaotic and intractable by current method. Critical equations are still very di ffi cultbut more tools are available to study them. In the next few pages we describe the result inmore details.Let ω = ∇ × v = ω r e r + ω θ e θ + ω e be the vorticity. Define J = ω r r , Ω = ω θ r . (2.3)Then the triple J , Ω , ω satisfy the system: for b = v r e r + v e , ∆ J − ( b · ∇ ) J + r ∂ r J + ( ω r ∂ r + ω ∂ x ) v r r − ∂ t J = , ∆Ω − ( b · ∇ ) Ω + r ∂ r Ω − v θ r J − ∂ t Ω = , ∆ ω − ( b · ∇ ) ω + ω r ∂ r v + ω ∂ x v − ∂ t ω = . (2.4) EVIEW ON AXIALLY SYMMETRIC NS 9
These follow from direct computation based on the vorticity equation (cid:0) ∆ − r (cid:1) ω r − ( b · ∇ ) ω r + ω r ∂ r v r + ω ∂ x v r − ∂ t ω r = , (cid:0) ∆ − r (cid:1) ω θ − ( b · ∇ ) ω θ + v θ r ∂ x v θ + ω θ v r r − ∂ t ω θ = , ∆ ω − ( b · ∇ ) ω + ω ∂ x v + ω r ∂ r v − ∂ t ω = , (2.5)and the relations ω r = − ∂ x v θ , ω θ = ∂ x v r − ∂ r v , ω = ∂ r v θ + v θ r . (2.6)We mention that the function J was introduced in the recent paper by H. Chen-D.Y.Fang-T. Zhang [16]. By carrying out an energy estimate on the first two equations, they provedthe following result: if | v θ ( x , t ) | ≤ Cr − ǫ , for all x and t >
0, then solutions are regular everywhere. Here ǫ > C are positiveconstants. This result gives a hint that the ASNS is a little super-critical. The reason is that v θ has the well known a priori bound | v θ ( x , t ) | ≤ r k rv θ ( · , k ∞ , (2.7)which comes from equation (2.2) via the maximum principle.Now we observe that the vortex stretching terms in all three equations in (2.4) are criticalwhen viewed in a suitable way. The key is to treat (2.4) as a closed system. Therefore thevorticity equation of 3 dimensional axially symmetric Navier-Stokes equations are criticalinstead of supercritical as commonly believed.Here are the details. From its a priori bound, we know that v θ at worst scales as − ω r and ω in the vortex stretching termsin (2.4) at worst scale as −
2. The key observation is to treat ω r and ω as potential functionsrather than unknowns. It is well known that in a second order reaction di ff usion equation,potentials which scale as − ∂ r v r r , ∂ x v r r , ∂ r v and ∂ x v ? It turns out that they can allbe converted to J , Ω and ω which are treated as unknown functions in (2.4).In fact one has the following inequalities k∇ v r r k ≤ k Ω k , k∇ v r r k ≤ k ∂ Ω k . (2.8)These can be seen from the identities v r = − ∂ x L θ , ( ∆ + r ∂ r ) L θ r = − Ω which imply ( ∆ + r ∂ r ) v r r = ∂ x Ω . Then one can use v r r and ∆ v r r as test functions respectively to deduce (2.8).Moreover from the relation ∆ ∂ i v = −∇ × ∂ i ω and integration by parts, we know that k∇ ∂ r v k + k∇ ∂ x v k ≤ C k∇ ω k . By direct computation, we also have the pointwise relation |∇ ω | ≤ r ( |∇ J | + |∇ Ω | ) + |∇ ω | + J + Ω ) . Therefore, even though ω r and ω are viewed as potential functions generated by v θ , thesystem (2.4) is still a closed system of J , Ω and ω .Note that in carrying out energy bound for equation (2.4) the drift terms (first orderterms) can be integrated out if functions decay su ffi ciently fast near infinity. One can alsocarry out a localized argument to take care of the drift term as in the paper [120]. Wetake the liberty to correct one misstatement in the text on p246 in that paper, where it wasstated that weak solutions are Lipschitz in the spatial direction. It should have been ”weaksolutions are locally bounded”.Now let us introduce the main result in [73]. Definition 2.6.
We say that the angular velocity v θ ( r , z , t ) is in ( δ ∗ , C ∗ )-critical class if Z | v θ | r | f | dx ≤ C ∗ Z | ∂ r f | dx + C Z r ≥ r | f | dx , (2.9) Z | v θ | | f | dx ≤ δ ∗ Z | ∂ r f | dx + C Z r ≥ r | f | dx , (2.10)holds for some r >
0, some C > t ≥ f ∈ H .Clearly, under the natural scaling of the Navier-Stokes equations: v λ ( t , x ) = λ v ( λ t , λ x ) , p λ ( t , x ) = λ p ( λ t , λ x ) , the above definition is invariant: ( v λ ) θ also satisfies (2.9)-(2.10) if v θ does. Theorem 2.7.
For arbitrary C ∗ > , there exists a constant δ ∗ > such that for all Leray-Hopf solutions v to the axially symmetric Navier-Stokes equations with initial value v satisfying k v k H < ∞ and k rv θ k L ∞ < ∞ , the following conclusion is true. If the angularvelocity field v θ is in ( δ ∗ , C ∗ ) -critical class, i.e. v θ satisfies the critical Form BoundednessCondition in (2.9) - (2.10) , then v is regular globally in time. An immediate corollary of the theorem is:
Corollary 2.8.
Let δ ∈ (0 , ) and C > . Let v be a Leray-Hopf solution to the axiallysymmetric Navier-Stokes equations with initial data v ∈ H / and k rv θ k L ∞ < ∞ . If sup ≤ t < T | rv θ ( r , x , t ) | ≤ C | ln r | − , r ≤ δ , (2.11) then v is regular globally in time. We mention that C ∗ in the theorem and C in the Corollary 2.8 are independent of neitherthe profile nor the norm of the given initial data. The point is that if (2.11) is satisfied, then(2.9)-(2.10) is true. Therefore one can apply the theorem to get the desired conclusion.After [73] was posted on the arxiv, in the paper by Dongyi Wei [115], the power in thelog term has been improved to − /
2. Namely, he proved
Theorem 2.9. ( [115] ) Let v be a Leray-Hopf solution to the axially symmetric Navier-Stokes equations with initial data v ∈ H and k rv θ k L ∞ < ∞ . If, for some δ ∈ (0 , / , sup ≤ t < T | rv θ ( r , x , t ) | ≤ C | ln r | − / , r ≤ δ , (2.12) then v is regular. EVIEW ON AXIALLY SYMMETRIC NS 11
The improvement is achieved by decomposing the space time in a dynamic way whencarrying out the energy estimate for the system of Ω and J in (2.4). More specifically,in (2.4), one multiplies the first equation by J and the second one by Ω and integrate inspace time. One can justify the integration by using a cut o ff function φ before a potentialsingular time t . After integration by parts, one deduces12 Z (cid:16) J + Ω (cid:17) φ dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t + Z t Z (cid:16) |∇ J | + |∇ Ω | (cid:17) φ dyds ≤ − Z t Z v θ r J Ω φ dyds | {z } T + Z t Z ( ω r ∂ r v r r + ω ∂ x v r r ) J φ dyds | {z } T + less singular terms . (2.13)If we can absorb T and T by the left hand side, then, we would know that ∇ J and ∇ Ω are L loc in space time. Since J = ω r / r and Ω = ω θ / r , we then know that ∇ ω θ and ∇ ω r are L loc in space time around the x axis. One can also argue, using the first term on the left handside that ω θ and ω r are in the space L ∞ (0 , t ; L loc ). With these information, it is well knownby Sobolev imbedding and bootstrapping that regularity of solutions follow. The term T is the most singular one on the right hand side. The term T , after using (2.6), (2.8) andintegration by parts, can be shown to be logarithmically less singular than T . So the maintask is to control T . Using Cauchy-Schwarz inequality, it is su ffi cient to use the left handside of (2.13) to bound the terms Z t Z | v θ | r J φ dyds , Z t Z | v θ | r Ω φ dyds Let us chose a positive function r = r ( t ) ≤ ǫ K ( ǫ ) a ( t ) − , where K is some exponentialfunction and ǫ is a small number to be chosen suitably; a ( t ) = k r − R r | v θ ( ρ, z , t ) | d ρ k L ∞ .Splitting the spatial domain along r = r ( t ) and using cut-o ff and integration by parts, oneshows that Z | v θ | r J φ dy ≤ ǫ − / Z | ∂ r J | dy + Cr − ( k Γ k L ∞ + ǫ − / ) Z r ≥ r / J dy , (2.14)provided that k Γ k L ∞ ( r ≤ r ) < ǫ <
1. The same bound holds when J is replaced by Ω . Nowusing the extra condition (2.12), one can substitute (2.14) into (2.13) and turn it into anordinary di ff erential inequality. Then the claimed bound for ω θ and ω r in L ∞ (0 , t ; L loc )space follows, giving us regularity.The appearance of the log term is due to the special property of the axially symmetricHardy’s inequality : for all ψ ∈ C ( R ), there is one positive constant C such that Z Z r | ln r | ψ ( r , x ) rdrdx ≤ C Z Z | ∂ r ψ | ( r , x ) rdrdx It would be interesting if one can lower the power on | ln r | even further in the regularitycriteria. However the drift term, which is almost harmless in the vortex equation so far,is the main obstacle. For instance, there is a dimension expansion trick in removing thelog term in the Hardy inequality. However the drift term no longer has the divergence freestructure viewing in high dimensions.3. A ncient S olutions Next we talk about another common way to study the Navier-Stokes equations and manyother nonlinear equations: blow up analysis.
Let v be a Leray-Hopf solution to the NS. Suppose a singularity happens in finite time T , we would like to know what is it? So we blow down the solution v or blow up thespace time near maximal points of | v | in the time interval [0 , t i ] ⊂ [0 , T ) where t i is asequence times approaching the singular time (like using a microscope). More specifically,let λ i = sup t ∈ [0 , t i ] | v | and pick points ( x i , s i ) with s i ≤ t i such that | v ( x i , s i ) | ≥ λ i / . Considerthe sequence of functions v i ( x , t ) ≡ λ − i v ( λ − i x + x i , λ − i t + t i ) , P i ( x , t ) ≡ λ − i P ( λ − i x + x i , λ − i t + t i ) . They are bounded solutions of the NS in a increasingly larger time interval. By standardregularity theory, v i sub-converges in C , loc topology to a limit function v ∞ . The resultingfunction is still a solution of NS. But it is a bounded solution existing on the time interval( −∞ , ff erent set of blow up points ( z i , s i ) where | v ( z i , s i ) | is large but is not comparable to the maximum of | v | before s i . However, sincethey have existed for a long time, they must be special. In other words, ancient solutionsare rigid. Even if it turns out that no finite time blow up occurs, ancient solutions still serveas approximation of the behavior of solutions in regions of high velocity.Can one classify all ancient solutions?The answer is not so easy in general without further assumptions, even for positive solu-tions to the heat equation. For example v = e x + t is a nonconstant, positive ancient solutionto the 1 dimensional heat equation in R . It is also not hard to see that v = (0 , e x + t ) is anancient solution to the 2 dimensional NS. Note this example shows a di ff erence betweenstationary and non-stationary Liouville property since it is well known that positive solu-tions to the Laplacian, namely, positive harmonic functions in R n are constants. Howeverthere are also similarities between the two. It is well known that harmonic functions on R n of sublinear growth are constants. The same conclusion was proven for ancient solutions tothe heat equation in Souplet-Zhang [104] in 2006, which can be extended to some noncom-pact manifold cases. For the NS, there is an additional twist. For any harmonic function h = h ( x ) on R and a = a ( t ) a C function of time, the function v = a ( t ) ∇ h ( x ) is an ancientsolution of NS. To rule out this kind of solutions, we usually consider the so-called mildsolutions only. For simplicity, we confine ourselves to solutions with locally finite energy,although the notation of mild solutions can be defined for other, more singular solutions. Definition 3.1.
A function v ∈ L ∞ loc (0 , T ; L loc ( R )) ∩ L loc (0 , T ; H loc ( R )) is called a mildsolution to the 3 dimensional NS if v ( x , t ) = Z R G ( x , t , y ) v ( y ) dy + Z t Z R K ( x , t − s , y ) v ∇ v ( y , s ) dyds , where G = G ( x , t , y ) is the standard heat kernel on R and K = K ( x , t − s , y ) is the Stokesheat kernel on R .By direct calculation, it can be shown that a mild solution in L ∞ (0 , T ; R ) is H¨oldercontinuous in R × [ δ, T ] for any δ >
0. This fact is useful in proving convergence resultsinvolving bounded mild solutions.See [101] for an earlier treatise and Chapter 5 of [110] for a recent discussion of theStokes heat kernel and mild solutions. The later is often called the Oseen kernel [87]. The
EVIEW ON AXIALLY SYMMETRIC NS 13 explicit formula is K ( x , t , y ) = K ( x − y , t , ≡ ( K i j ( z , t , z = x − y , i , j = , , K i j ( z , t , = G ( z , t , δ i j + π ∂ x i ∂ x j Z R G ( w , t , | z − w | dw . (3.1)So, a more reasonable question to ask would be:Are sublinear, mild ancient solutions of 3 dimensional NS constants? (3.2)We comment that this question is quite challenging without further restriction on thegrowth or decay of the ancient solutions. See a brief discussion at the end of the section. Ifthe answer to the above question is yes for certain class of ancient solutions, then we saythat the Liouville property or theorem holds for that class.Of course for the NS, there is the result of [51], which we have seen a little in theprevious section. Their results can be summarized as the following theorem. Note that adi ff erent proof for statement (b) in the stationary case was found in [53] later. Theorem 3.2. ( [51] Koch-Nadirashvili-Seregin-Sverak)(a). If v is a bounded, mild ancient solutions of the 2 dimensional NS, then v is constant.(b). If v is a bounded, mild ancient solutions of the 3 dimensional ASNS, then v isconstant if also v θ = .(c). If v is a bounded, mild ancient solutions of the 3 dimensional ASNS, then v is ifalso | v ( x ) | ≤ C / r. In the same paper, the authors also made the following conjecture:
The Liouville property is true for bounded, mild ancient solutions of the 3 dimensionalASNS .Next we describe further results from a recent paper by Lei, Zhang and Zhao [74]. Itis proven, in 2D and the 3D axially symmetric swirl free case, that the Liouville propertyholds for mild ancient solutions if the velocity fields are sublinear with respect to the spa-tial variable and the vorticity fields satisfy certain decay condition (see Theorem 3.3 andTheorem 3.4). We remark that, unlike the Liouville theorems in [51], there is no needfor the condition that solutions are bounded. Moreover, counterexamples are given whenthe velocity fields are linear with respect to the spatial variable. This shows that, underthe condition that solutions are sublinear with respect to the spatial variable, the Liouvilletheorems are sharp.The other main result in that paper is a Liouville property, under an extra decayingassumption, for bounded ancient solutions of the ASNS with general nontrivial swirl (seeTheorem 3.5). Let v be the bounded ancient mild solutions of the axially symmetric Navier-Stokes equations with v θ , Γ = rv θ . We prove that if Γ ∈ L ∞ t L px where 1 ≤ p < ∞ ,then v must be constants.Actually, in the 3D axially symmetric case, on the above conjecture, one can add theextra condition that Γ ∈ L ∞ t L ∞ x without losing much generality. The reason is that Γ isscaling invariant and it also satisfies the maximum principle. So if the initial value of asolution satisfies the bound, then it will persist over time. Therefore, any ancient solutionfrom blow up process will still satisfy this bound. As mentioned, when v θ ,
0, the Liouvilleproperty was proved in [51] under the condition | v | ≤ Cr . In comparison, in [74], one onlyneeds the condition on one component v θ of the velocity v while no additional conditionsare added on the other two components. Moreover, even though they haven’t totally provedthe conjecture in [51], the result can still be considered as a step forward in understandingthe conjecture. That is because the condition Γ ∈ L ∞ t L px with p being any finite numberseems not too far from Γ ∈ L ∞ . The following are the main results in [74]. We mention that solution v in the next twotheorems are also assumed to be locally bounded in space. Namely, if x is in a compactset, then | v ( x , t ) | is uniformly bounded for all t . This assumption was not stated in thecorresponding theorems in [74] although it was implicitly stated in the text. See also arelated result by Pan and Li [89] where v is allowed to grow at ( − t ) . − rate near −∞ . Theorem 3.3.
Let v be a smooth, locally bounded ancient solution of the 2D incompress-ible Navier-Stokes equations and let ω = ∇ × v be the vorticity. If lim | x |→ + ∞ | ω ( x , t ) | = , uniformly for all t ∈ ( −∞ , , then ω ≡ and v is harmonic.If, in addition, v satisfies lim | x |→ + ∞ | v ( x , t ) | / | x | = , (3.3) uniformly for all t ∈ ( −∞ , , then v must be a constant. Theorem 3.4.
Let v be a smooth, locally bounded ancient solution of the 3D axially sym-metric Navier-Stokes equations without swirl and let ω = ∇ × v = ω θ e θ be the vorticity.Define Ω = ω θ r , if lim r → + ∞ | Ω | = , (3.4) uniformly for all t ∈ ( −∞ , , then ω θ ≡ and v is harmonic.If, in addition, v satisfies lim | x |→ + ∞ | v ( x , t ) | / | x | = , (3.5) uniformly for all t ∈ ( −∞ , , then v must be a constant. Theorem 3.5.
Let v be a bounded ancient mild solution of the 3D axially symmetricNavier-Stokes equations with v θ , and let Γ = rv θ . If Γ ∈ L px L ∞ t ( R × ( −∞ , where ≤ p < ∞ , then v must be a constant. The condition v being sublinear (the condition (3.3)) in Theorem 3.3 can not be re-moved, even when ω ≡
0. Hence, the above 2 dimensional Liouville theorem is sharp.Here is a counterexample. Let v = ( x , − x ) , p = − x − x , then ω = ∂ u − ∂ u =
0, and ( v , p ) satisfies the 2D stationary Navier-Stokes equations.However, v is not a constant solution.The first conclusion in Theorem 3.4, the axially symmetric, swirl free case, shows thatif ω θ is sublinear with respect to r , then ω θ ≡
0. Here is a counterexample to show thatthe conclusion will be wrong if ω θ is linear with respect to r , and consequently one cannotprove v is harmonic. This infers that condition (3.4) is also important. Let v = ( − x x , − x x , x ) , p = − x + x , then v θ = v · e θ = v , p ) satisfies the stationary ASNS However ω θ = − r . ∆ v = (0 , , , v being sublinear to x ( condition (3.5)) is also necessary, evenwhen ω θ =
0. Moreover, one can give a counterexample to show that if v is linear in thespatial variable, then there exists a nontrivial ancient solution of ASNS without swirl. Itthen follows that Theorem 3.4 is sharp. For example, let v = ( − x , − x , x ) , p = x + x + x , EVIEW ON AXIALLY SYMMETRIC NS 15 then we have v θ = v · e θ = , v r = v · e r = − r , v = x . These imply that ω θ = ∂ x v r − ∂ r v = v , p ) satisfies the stationary ASNS equationswithout swirl. However, v is not a constant solution.So the remaining case for the Liouville property, which is also the most di ffi cult one, iswhen Γ = rv θ ( x , t ) does not decay near infinity. There are some partial results in the paper[69] by Lei-Ren-Zhang. Theorem 3.6.
Let v = v θ e θ + v r e r + v e be a bounded mild ancient solution to the ASNSsuch that Γ = rv θ is bounded. Suppose v is periodic in the x variable. Then v = ce wherec is a constant. Let us describe the general idea of the proof of this theorem. One will prove, by theDe Giorgi-Nash-Moser method that Γ satisfies a partially scaling invariant H¨older esti-mate which forces Γ ≡
0. Then the problem is reduced to the swirl free case that issolved in [51]. In general this method will break down in large scale, unless one im-poses scaling invariant decay conditions on v r and v . Although no decay conditions on v r or v are assumed in the theorem, one can demonstrate that the classical Nash-Moseriteration method can be carefully adapted to this situation. The key observation is thefollowing: the incompressibility condition ∇ · b = b = v r e r + v e , along withthe periodicity in x gives one extra information on v r . In fact one will essentially use v r ( r , θ, x ) = − ∂ x ( L θ ( r , θ, x ) − L θ ( r , θ, ∈ ( L ∞ ) − , where L θ is the angular stream func-tion. Another helpful factor is that the spatial domain R × S behaves like a 2 dimensionalEuclidean space in large scale, even though it really behaves 3 dimensionally near the axis.This shows that v r scales by the critical order −
1, which is quite helpful. But the same cannot be said for v . Fortunately, the role of v is not as important as v r in the x periodiccase.In contrast to the absence of nontrivial partially periodic ancient solutions for ASNS, inanother 3 dimensional parabolic flow, the Ricci flow, such a solution exists and representsa typical singularity: ( S × R ) × ( −∞ , Γ converges to its maximum at certain speed. Even though the result cannot yet reachthe full conjecture in [51], its proof utilizes a method of constructing a weight function bysolving an adapted PDE, which is then used in an energy estimate. It may be of independentvalue and use elsewhere. This result and the ones in [69] all were posted in the preprint[68]. In a review process, it was suggested by a reviewer to split that paper.Let lim sup r →∞ Γ = lim sup r →∞ sup x , t Γ ( r , x , t ) . (3.6)It can be shown that if v is any bounded ancient solution such that Γ is bounded, thenlim sup r →∞ Γ = sup Γ . Theorem 3.7. ( [70] ) Let v = v θ e θ + v r e r + v e be a bounded mild ancient solution to theASNS such that Γ = rv θ is bounded. There exists a small number ǫ ∈ (0 , , dependingonly on k v k ∞ , such that if | Γ ( r , x , t ) − lim sup r →∞ Γ | ≤ ǫ r lim sup r →∞ Γ (3.7) holds uniformly for x , t, and large r, then v = ce where c is a constant. Let us mention that to study the equation of Γ in an isolated manner is likely to fail. Forexample without the divergence free condition on the vector field b . There is no Liouvilleproperty for the equation ∆ f − r ∂ r f − b ∇ f = , f (0) = , lim r →∞ f = c in R , even when f = f ( r ) is a one variable function. One can just take b = a ( r ) e r with a ( r ) ≤ r exp( R a ( r ) dr ) being integrable on [0 , ∞ ). Then solve the ODE f ′′ − ( r + a ( r )) f ′ = ffi cult even in the bounded, axially symmetriccase, especially without the extra assumption Γ = rv θ ∈ L ∞ . The reason is that it con-tains another long standing question for the stationary NS as a special case. Namely, is ahomogeneous D solution zero? We will discuss the latter in detail in the following section.At the end of this section, we introduce a local representation formula for smooth solu-tions of NS, which is a useful tool in proving convergence results in NS during and afterblow up or scaling, including ancient solutions. It is largely due to O’leary [86] and hasbeen used in several papers including [121]. However, as pointed out by H. J. Dong, thereis a missing term in that formula. A corrected one is given in the addendum of [121], whichis given below. As one application, we can prove that for a local smooth solution v of NS, v and ∇ v is controlled by a local integral of v or |∇ v | without any assumption on the pressure.The smoothness assumption can be relaxed. We refer to [121] addendum for details.Given r > x , t ) in the space time. Let Q r ( x , t ) = { ( y , s ) | | x − y | < r , t − r < s < t } be a standard parabolic cube. Denote by Γ = Γ ( x , y ) = / (4 π | x − y | ) the Green’s function on R , G = G ( x , t , y ) the standard heat kernel, K = K ( x , t , y ) the Oseen kernel given in (3.1).Fixing ( x , t ), we construct a standard cut-o ff function η such that η ( y , s ) = Q r / ( x , t ), η ( y , s ) = Q r ( x , t ), 0 ≤ η ≤
1. Denote by K j the j − th column of the Oseenkernel. The following vector was introduced in [86]. Φ j = Φ j ( x , t − s , y ) = η ( y , s ) K j ( x , t − s , y ) + π ∇ η ( y , s ) × Z R curlK j ( x , t − s , z ) | z − y | dz ≡ η K j ( x , t − s , y ) + −→ Z j ( x , t − s , y ) . Proposition 3.8 ([121], Addendum) . Suppose v = ( v , v , v ) is a smooth solution of theNS (1.1) in Q r ( x , t ) and K j be the j − th column of the Oseen kernel K. Thenv j ( x , t ) = Z Q r ( x , t ) v ( y , s ) · h K j ( x , t − s , y )( ∆ η + ∂ s η )( y , s ) + ∇ η ( y , s ) ∇ y K j ( x , t − s , y ) i dyds + Z Q r ( x , t ) v ( y , s ) · (cid:20) ( ∆ y −→ Z j + ∂ s −→ Z j )( x , t − s , y ) + v ( y , s ) ∇ y Φ j ( x , t − s , y ) (cid:21) dyds + Z B r ( x ) h ( ∇ η · ∇ y Γ ) v j − ∂ y j Γ ( ∇ η · v ) − ∂ y j η ( ∇ y Γ · v ) i ( y , t ) dy . | {z } (3.8)In the above, if X , Y are two vector fields. Then X ∇ Y ≡ P j X j ∂ j Y . The last term in (3.8)was the missing one. Since all integrands vanish on the lateral boundary of Q r ( x , t ), onecan apply ∇ x freely into the integrals on the right sides of (3.8). Also it is not hard to seethat the formula also works if one replaces ( x , t ) by any points in Q r / ( x , t ) while keepingthe cube Q r ( x , t ) unchanged.Using the explicit formula above, one deduces Proposition 3.9 (mean value inequalities for (NS), [121], Addendum) . Let v be a smoothsolution of the NS (1.1) in Q r ( x , t ) . Then there exists an absolute constant λ > such that EVIEW ON AXIALLY SYMMETRIC NS 17 (a). | v ( x , t ) | ≤ λ r Z Q r ( x , t ) − Q r / ( x , t ) | v ( y , s ) | dyds + λ r Z B r ( x ) − B r / ( x ) | v ( y , t ) | dy + λ Z Q r ( x , t ) K ( x , t ; y , s ) | v ( y , s ) | dyds , where K ( x , t ; y , s ) = / ( | x − y | + √ t − s ) , if t ≥ s and if t < s.(b). |∇ v ( x , t ) | ≤ λ r Z Q r ( x , t ) − Q r / ( x , t ) |∇ v ( y , s ) | dyds + λ r Z Q r ( x , t ) − Q r / ( x , t ) | v ( y , s ) | dyds + λ r Z B r ( x ) − B r / ( x ) | v ( y , t ) | dy + λ Z Q r ( x , t ) K ( x , t ; y , s ) | v ( y , s ) | |∇ v ( y , s ) | dyds . Using an iteration, one can also remove the kernel function K from the above meanvalue inequality, assuming v is in a parabolic type Kato class. The later contains the stan-dard L px L qt regularity class alluded in Section 1 for q < ∞ . Definition 3.10 ([121]) . A vector valued function b = b ( x , t ) in L loc ( R n + ) is in class K ifit satisfies the following condition:lim h → sup ( x , t ) ∈ R n + Z tt − h Z R n [ K ( x , t ; y , s ) + K ( x , s ; y , t − h )] | b ( y , s ) | dyds = . The parabolic Kato norm of b on a time interval [ t , t ] and scale h is B ( b , t , t , h ) ≡ sup ( x , t ) ∈ R n × [ t , t ] Z tt − h Z R n [ K ( x , t ; y , s ) + K ( x , s ; y , t − h )] | b ( y , s ) | dyds . Proposition 3.11 ( [121], Addendum) . Let v be a local solution of the NS (1.1) in Q r ( x , t ) ⊂ R × R , satisfying the energy inequality (1.2) localized in Q r ( x , t ) . Suppose also thatv | Q r ( x , t ) is in class K . Then both v and |∇ v | are bounded functions in Q r = Q r ( x , t ) .Moreover, for some positive constants C = C ( v ) and r , depending only on the size ofthe Kato norm of v in the time interval [ t − r , t ] with scale r such that the followinghold. When < r < r , | v ( x , t ) | ≤ Cr sup s ∈ [ t − (2 r ) , t ] Z B r ( x ) | v ( y , s ) | dy , |∇ v ( x , t ) | ≤ Cr k∇ v k L ( Q r ) + Cr sup s ∈ [ t − (2 r ) , t ] Z B r ( x ) | v ( y , s ) | dy .
4. S tationary S olutions D solutions.
In this section we discuss the stationary Navier-Stokes equations v ∇ v + ∇ p − ∆ v = f , in Ω ⊂ R , ∇ · v = . (4.1)Here Ω is an open domain and f is a forcing term. Boundary conditions will vary case bycase.We will focus on the decay and vanishing properties of the so-called homogeneous Dsolutions to (4.1) in certain unbounded domain Ω ⊂ R with various boundary conditions and requirements of the behavior of v at infinity. Here the name “D solutions” arises fromthe condition that solutions v have finite Dirichlet integrals (energy) Z Ω |∇ v ( x ) | dx < + ∞ . (4.2) Definition 4.1.
A smooth solution to (4.1) is called a D solution on Ω if (4.2) holds. It isreferred to as DS.A homogenous D solution on Ω is a D solution v on Ω such that v = ∂ Ω , f = | v ( x ) | → | x | → ∞ . It is referred to as HDS.An axially symmetric homogenous D solution on Ω is referred to as ASHDS.If no confusion arises, we will drop the reference to the domain Ω from D solutions.Existence of D solutions with several boundary conditions were studied in the pioneerwork of Leray [61] (p24) by variational method. The following uniqueness problem hasbeen open since then: Is a homogeneous D solution equal to ? This is also part of the very di ffi cult uniqueness problem for the steady Navier-Stokesequation, which states that if the right hand of (4.1) is a nontrivial smooth function, de-caying su ffi ciently fast, do we have the uniqueness of D solutions. For axially symmetriccompact domains with a hole at the axis, the uniqueness fails, as pointed out by Yudovich[119] over 50 years ago. However for general domains, the problem is wide open. Let usexplain Yudovich’s construction of 2 solutions. Let Ω be the above compact domain. Fora divergence free vector field A , find a nontrivial solution φ of the following ”eigenvalueproblem” for vector fields. ( ∆ φ − A ∇ φ − φ ∇ A − ∇ τ = , in Ω , ∇ · φ = Ω ; φ = , on ∂ Ω . (4.3)Then A + φ and A − φ will be two distinct solutions of the stationary NS with the sameforcing term, the same boundary value, but di ff erent pressure terms. But there is a problem.How to find a nontrivial solution to (4.3). Yudovich found a special A so that (4.3) has avariational structure and found a nontrivial solution by solving a constrained maximumproblem. His choice of A is A = − r − e θ in the polar coordinates.Now let us focus on the decay and vanishing problems for DS. In the 2 dimensions,the corresponding vanishing property in the full space case is solved by Gilbarg and Wein-berger [41], using the line integral method. Theorem 4.2. ( [41] Let velocity v and pressure p be a solution of the Navier-Stokes equa-tion defined over the entire R and assume R R |∇ v ( x ) | dx < + ∞ . Then v and p are constant.
They also proved a number of asymptotic properties of 2 dimensional D solutions ingeneral. For example, they proved that the pressure p from Leray’s D solution in 2 Dexterior domains with f = R . The situation is akin to the regularity problem. This time the scaling gaphappens at infinity. Namely, to prove vanishing, one always needs to impose a conditionthat the solution decays in some sense near infinity su ffi ciently fast. However, no decay ratehas been proven a priori for HDS (homogeneous D solutions) in R , making the situation EVIEW ON AXIALLY SYMMETRIC NS 19 looks bleaker than the regularity problem. Another eerie similarity with the regularityproblem occurs in that some partial a priori decay can be proven quickly outside a smallset. In fact R. Finn [35] (page229), already observed the following partial decay propertyfor any 3 dimensional vector field v having a finite Dirichlet integral and tending to aconstant vector v ∞ as | x | → ∞ . For any δ >
0, there exists a measurable set E δ ⊂ S , suchthat | E δ | ≤ δ and | v − v ∞ | ≤ C δ / | x | / , ∀ x = | x | ω, ω ∈ S \ E δ . (4.4)So the current situation is, in order to prove vanishing, one needs to establish a priori decayof su ffi ciently fast order, such as | v ( x ) | ≤ c / | x | / , c.f. Theorem 4.5. However the only apriori decay is a slow and partial one, except for solutions which are small in suitable sense.Perhaps it is also not a surprise that just the decay and convergence property itselffor general D solutions is an intricate matter in both 2 and 3 dimensions. In [34, 35],Finn proved that any D solution in 3 dimensional exterior domains converges pointwiseuniformly to a constant vector v ∞ at infinity. Furthermore, in case v ∞ ,
0, he showed thatif | v ( x ) − v ∞ | ≤ C | x | − α for some α > / x → ∞ , then α can be replaced by 1. He alsointroduced a concept called ”physically reasonable (PR) solutions” to the stationary NS in3 dimensional exterior domains, which are those satisfying v ( x ) = O ( | x | − ) if v ∞ = | v ( x ) − v ∞ | = O ( | x | − α ) for some α > /
2, if v ∞ ,
0. Finn [36] then proved the existence anduniqueness of a PR solution in a 3 dimensional exterior domain when the boundary dataare small enough. It is straight forward to show a PR solution is a D solution. However,in case v ∞ =
0, whether the converse is true has remained open till now. As mentioned, if v ∞ =
0, we can not prove any decay rate at infinity for v so far, not to mention the expecteddecay rate of | x | − . One exception is when the solution is small in a suitable sense. Thenthe decay can be proven as a linear Stokes problem in exterior domains. See Galdi [38]where the viscosity constant ν is large and therefore a fixed Dirichlet integral is relativelysmall. Note that one will not run into the Stokes paradox which happens when v ∞ , v = Ω . In the case of v ∞ ,
0, Babenko [3]showed that every D solution is a PR solution if the forcing term is of bounded support.One would be tempted to think that the 2 dimensional situation is simpler, which is thecase for the regularity problem. But this is not true. Finiteness of the Dirichlet integral(4.2) does not entail any decay of v in 2 dimensions. Only recently, Korobkov, Pileckas,and Russo [54] managed to prove the following convergence result. Theorem 4.3.
Let v be a D-solution to NS in an exterior domain Ω ⊂ R without forcingterm. Then v converges pointwise uniformly at infinity to a constant vector v ∞ ∈ R . The proof builds on earlier work in Gilbarg and Weinberger [41], Amick [1, 2]. Fineanalysis of the level sets of the vorticity plays an important role. These are curves in R . There is still a problem on whether v ∞ agrees with the one from Leray’s originalconstruction [61]. The interested reader may consult the introduction of [54] for a detailedaccount of related and expanded results.Let’s recall some of vanishing results with extra integral or decay assumptions for thesolution v in 3 dimensions. If the domain Ω = R , Galdi [39] Theorem X.9.5 proved thatif v ∈ L / ( R ) is a homogeneous D-solution, then v =
0. A log factor improvement wasshown in Chae and Wolf [26]. In [25], Chae proved that homogeneous D solutions are 0by assuming ∆ v ∈ L / ( R ), which scales the same as k∇ v k . Theorem 4.4. ( [25] ) Let v be a homogeneous D solution in R . Suppose k ∆ v k L / ( R ) < ∞ ,then v = . Chae’s proof uses the property that the head pressure Q = | v | + p is non-positive if p ( x ) → v ( Q − ǫ ) δ − as a test function in the stationary NS. Letting ǫ, δ →
0, one concludes that R R |∇ v | dx = v = v ∈ L ( R ) ∩ BMO − . In a recent paper [55], Kozono-Terasawa-Wakasugi showed vanishingof homogeneous D solutions if either the vorticity ω = ω ( x ) decays faster than c / | x | / atinfinity, or the velocity v decays like c / | x | / with c being a small number. i.e. They proved Theorem 4.5. ( [55] ) Let v be a homogeneous D solution in R and ω = ∇ × v be thevorticity. If either | ω ( x ) | = o ( | x | − / ) or | v ( x ) | ≤ c | x | − / for a small positive constant andall large | x | , then v = . Afterwards, W. D. Wang [114] and N. Zhao [122] proved a similar result for the axiallysymmetric D solutions independently. Note that no decay condition is imposed in the e direction on one hand, but no improvement on the decay exponent is made on the other.However the result is still encouraging since we now have a priori estimates on v and ω ,c.f. Theorems 4.9 and 4.10. Theorem 4.6. ( [114] and [122] ) Let v be an axially symmetric homogeneous D solutionin R and ω = ∇ × v be the vorticity. If either | ω ( x ) | = o ( r − / ) or | v ( x ) | = o ( r − / ) for alllarge | x | , then v = . The results in [55] has also been extended to D solutions in some Lebesgue and Morreyspaces in [20].Under certain smallness assumption, vanishing result for homogeneous 3 dimensionalsolutions in a slab R × [0 ,
1] was also obtained in the book [39], Chapter XII.As mentioned, D solutions in general noncompact domains have been studied in [61].Besides the whole space, the next simplest noncompact domains are the half space andslabs. Let us present a vanishing result for D solutions in a slab in R . Theorem 4.7 ([24] vanishing of HDS in a slab) . Let v be a smooth solution to the problem ( v · ∇ ) v + ∇ p − ∆ v = , ∇ · v = in R × [0 , , v ( x ) | x = = v ( x ) | x = = , (4.5) such that the Dirichlet integral satisfies the condition: Z Z R |∇ v ( x ) | dx < ∞ . (4.6) Then, v ≡ . Comparing with the full space case, one can show by Poincar´e inequality, with the helpof Dirichlet boundary and the finite integral condition (4.6), that the velocity v belongsto L , which indicates that the decay rate of v = v ( x ) is like 1 / | x | in the integral sense.However one does not have a good knowledge of the pressure p . The Dirichlet boundarycondition is known to induce complications on the vorticity and pressure. The main di ffi -culty is to deal with the pressure term. Earlier, Pileckas and Specovius-Neugebauer [92]studied the asymptotic decay of solutions of the Navier-Stokes equation in a slab. Theyprove, under certain weighted integral assumption on the velocity v = v ( x ) and 3rd orderderivatives, v = v ( x ) decays like 1 / | x | or 1 / | x | . See Theorem 3.1 in [92]. Then the van-ishing of v in the homogeneous case follows easily. However, these authors required that EVIEW ON AXIALLY SYMMETRIC NS 21 (1 + | x | ) + β | v ( x ) | + (1 + | x | ) + β | ∂ x v ( x ) | with β ∈ ( − , −
1) is in L in addition to further inte-gral decay conditions of the first, second and third order derivatives of v , and consequentlyrestriction on the pressure. These conditions are not available to us. Furthermore in theperiodic case, which will be dealt with later, it is not even known that v is L .Besides if the Dirichlet energy is infinite, the vanishing property may be false. Anexample is v = ( x (1 − x ) , , , p = − x .In the paper [24], there is an extra assumption that v is bounded, which turns out to beunnecessary. Also it seems interesting to study the decay property of nonhomogeneoussolutions in a slab.Now if the domain is the whole R , one can show that if the positive part of the radialcomponent of D solutions decays at order − Theorem 4.8 ([24]) . Let v ρ = v ρ ( x ) be the radial component of 3 dimensional D-solutionsin spherical coordinates. If v ρ ( x ) ≤ C | x | , x ∈ R , (4.7) for some positive constant C, then v ≡ . We should compare with the result in [55] where the authors prove, if the weak L / norm of v is small, then v vanishes. This includes the case | v ( x ) | ≤ c | x | − / for certain smallconstant c . In contrast, our assumption here is worse on the order of the distance function.However we only impose the condition on the positive part of the radial component of thesolution and there is no restriction on the other two components.Next we concentrate on axially symmetric homogeneous D solutions, for which thevanishing problem is also wide open. However as we shall see, a priori decay of thesolutions v and vorticity ω in the e r direction is available, albeit insu ffi cient for vanishingresult in the whole space case. The set up of equations have been given in (1.5) for thevelocity and (2.5) for the vorticity, except that we are happy to drop the time variable alltogether. The relation between v and ω are given in (2.6).For the decay of v and ω , the combined results of Chae-Jin and S. K. Weng spanning 8years can be condensed into: Theorem 4.9. ( [19] and [116] ) Let v be a homogeneous D solution of ASNS in R . Forx ∈ R , | v ( x ) | ≤ C (cid:0) log r / r (cid:1) / , | ω θ ( x ) | ≤ Cr − (19 / − , | ω r ( x ) | + | ω ( x ) | ≤ Cr − (67 / − . (4.8) Here C is a positive constant and for a positive number a, a − represents a number whichis smaller than but close to a. Their proof is based on line integral techniques from Gilbarg and Weinberger [41]. Inour recent work [23], the decay estimate on the vorticity ω is improved and a short prooffor the decay of the velocity v is found under a slightly more general condition. Let usmention that in both the previous and following theorems, one can add a fast decayingforcing term and still obtain the decay estimates. The proof is the same. Theorem 4.10 ([23] a priori decay v and ω ) . Let u be a smooth axially symmetric solutionto the problem ( v · ∇ ) v + ∇ p − ∆ v = , ∇ · v = , in R , lim | x |→∞ v ( x ) = , (4.9) such that the Dirichlet integral satisfies the condition: for a constant C, and all R ≥ , Z ∞−∞ Z R ≤| x ′ |≤ R (cid:0) |∇ v ( x ) | dx + | v ( x ) | (cid:1) dx ′ dx < C < ∞ . (4.10) Then the velocity and vorticity satisfy the following a priori bound. For a constant C > ,depending only on the constant C in (4.10) such that | v ( x ) | ≤ C (ln r ) / √ r ; | ω θ ( x ) | ≤ C (ln r ) / r / , | ω r ( x ) | + | ω ( x ) | ≤ C (ln r ) / r / , r ≥ e . Now we outline the proof of the decay result in Theorem 4.10 briefly. We start with theobservation that in a large dyadic ball, far away from the x axis, after scaling, the axiallysymmetric Navier-Stokes equation resembles a 2 dimensional one. Then the 2 dimensional Brezis-Gallouet inequality introduced in [8] implies that a smooth vector field with finiteDirichlet energy is almost bounded. After returning to the original scale, one can show that v is bounded by C (cid:0) ln rr (cid:1) / for large r . It is curious that the ASNS hardly plays a role in theproof, except for guaranteeing ∆ v is bounded.Next by combining the energy estimates of the equations for ω in the stationary (2.5) , Brezis-Gallouet inequality and scaling technique, we will show that, with x taken as 0 forconvenience, | ω θ ( r , | ≤ Cr − (ln r ) / k ( v r , v θ , v ) k / L ∞ ([ r , r ] × [ − r , r ]) , (4.11)and | ω r ( r , | + | ω z ( r , | ≤ Cr − (ln r ) / k ( v r , v ) k / L ∞ ([ r , r ] × [ − r , r ]) + Cr − / (ln r ) / k ( ∇ v r , ∇ v ) k / L ∞ ([ r , r ] × [ − r , r ]) . (4.12)The details can be founded in [23]. Then using the decay of v and (4.11), we can deducethat the decay rate of ω θ is r − / (ln r ) / .In order to obtain decay of ω r and ω from (4.12), we need the decay of ∇ v r , ∇ v whichcan be connected with ω θ by the Biot-Savart law − ∆ ( v r e r + v e ) = ∇ × ( ω θ e θ ) . Then ∇ v r , ∇ v can be written as integral representations of ω θ in the form of R R K ( x , y ) ω θ ( y ) dy ,where K ( x , y ) are Calderon-Zygmund kernels. The decay relations between ∇ v r , ∇ v and ω θ are shown in Lemma 3.2 of [23]. At last, a combination of (4.12), decay of v and ∇ v r , ∇ v imply the decay of ω r , ω in Theorem 4.10.Clearly, if the Dirichlet integral is finite i.e. k∇ v k L ( R ) < ∞ , then (4.10) is satisfied. In[23], we also proved a vanishing result when D solutions are periodic in x variable underthe additional assumption that v θ and v have zero mean in the x direction. With someextra work, one can also reach the vanishing result assuming the integral in (4.10) growsat a certain positive power of R .In a subsequent paper [24], the extra condition that v θ , v have zero mean in the x direction has been removed, under the stronger assumption that the Dirichlet integral isfinite. We state it as the following theorem, with ASHDS standing for axially symmetrichomogeneous D solutions. EVIEW ON AXIALLY SYMMETRIC NS 23
Theorem 4.11. (Vanishing of Periodic ASHDS) Let v be a smooth axially symmetric solu-tion to the problem ( v · ∇ ) v + ∇ p − ∆ v = , ∇ · v = , in R × S = R × [ − π, π ] , v ( x , x , x ) = v ( x , x , x + π ) , lim | x ′ |→∞ v ( x ) = , (4.13) with finite Dirichlet integral: R π − π R R |∇ v ( x ) | dx < + ∞ . Then v = . We outline the proof of the above results briefly. We start with the observation thatin x − periodic case with Ω = R × S , the horizontal radial component of the solution v r satisfies R π − π v r dx =
0. Poincar´e inequality and the finite Dirichlet integral conditionindicate that v r ∈ L ( Ω ). Then using of x -periodicity again, we can actually prove thatthe L ∞ oscillation of the pressure p is bounded in a dyadic annulus. The method is similarto the line integral technique in [41] except it is carried out in a 3 dimensional domain.At last by testing the vector equation (4.13) with v φ ( | x ′ | ), where φ ( x ′ ) is supported in { x ′ | | x ′ | < R } and equal to 1 in { x ′ | | x ′ | < R } , and making R approach ∞ , we can prove that v ≡
0. This result seems to add an extra weight in the belief that ASHDS in R is 0. Thereason is that in the periodic case, finiteness of the Dirichlet integral of v does not implyany decay of v . For example the function f = ln ln(2 + r ) has finite Dirichlet integral. Yet v still vanishes.The next theorem treats the case with Dirichlet boundary condition (DBC) in a slab,even allowing the Dirichlet integral to be log divergent. Recall from Theorem 4.7 that 3dimensional D solutions in a slab with DBC is 0. Why do we come back to the slab case?One reason is that infinite Dirichlet energy may induce non-uniqueness / non-vanishing asshown in the example after Theorem 4.7. So one would like to know, under what rate ofdivergence of the energy, vanishing of D solutions and especially ASHDS is preserved.In addition, solutions with infinite Dirichlet energy is also of interest in turbulence theory.These include the study of Kolmogorov flows such as flows in a channel with periodicforcing terms. See [37] e.g. Theorem 4.12. ( [24] ) Let u be a smooth, axially symmetric solution to the problem ( u · ∇ ) v + ∇ p − ∆ v = , ∇ · v = , in R × [0 , , lim | x ′ |→∞ v = , v ( x ) | x = = v ( x ) | x = = , (4.14) such that the Dirichlet integral satisfies the condition: for a constant C, and all R ≥ , Z Z R ≤| x ′ |≤ R |∇ v ( x ) | dx < C < ∞ . (4.15) Then v θ = . Moreover, there exists a positive constant C , depending only on the constantC in (4.15) such that | v r ( x ) | + | v ( x ) | ≤ C ln rr ! / . (4.16) Since v θ = ( v r ∂ r + v ∂ x ) v r + ∂ r p = ( ∆ − r − ) v r , ( v r ∂ r + v ∂ x ) v + ∂ x p = ∆ v ,∂ r v r + r − v r + ∂ x v = , lim r →∞ ( v r , v ) = , ( v r , v )( r , x ) | x = , = . (4.17) (cid:3) So our vanishing problem now is much like a two dimensional problem. But we do notknow any vanishing result for swirl free case in a slab with Dirichlet boundary conditionand (4.15).The decay estimate in the e r direction still holds if there is an inhomogeneous term ofsu ffi ciently fast decay. However, we are not aware any decay estimate in the e direction,except in the swirl free case, c.f. Theorem 1.2 [116]. If one works a little harder, thenone can reach the same conclusion as the theorem assuming the integral in (4.15) grows atcertain power of R .Let us give a description of the proof of Theorem 4.12. From (2.2), the quantity Γ : = rv θ satisfies ( v r ∂ r + v ∂ x ) Γ − ( ∆ − r ∂ r ) Γ = . (4.18)So Γ enjoys maximum principle, which means, for bounded open sets Ω ⊂ R ,sup x ∈ Ω | Γ | ≤ sup x ∈ ∂ Ω | Γ | . (4.19)We will show that the decay rate of v θ is at least r − ( ) − for large r . Take this decay forgranted at the moment. By using the above maximum principle and a sliding argumentalong the x axis together with the fact that Γ ( x ) → r → ∞ , one can prove v θ ≡ v θ , which take three steps.In step one: the Laplace Green’s function G on R × [0 ,
1] with homogeneous Dirichletboundary condition will be introduced and a number of properties of G explained. Thepoint is that G has fast decay near infinity. See [81] e.g.In step two: we obtain the following decay of ω r , ω by using a refined Brezis-Gallouet inequality, energy methods and scaling techniques: | ( ω r ( x ) , ω ( x )) | . r − ln r , for large r . Furthermore, by the same procedure, we can show that | ( ∂ x ω r ( x ) , ∂ r ω ( x )) | . r − / (ln r ) / . (4.20)In step three, we use the Biot-Savart law to get the representation of v θ by integralsinvolving G and ( ∂ x ω r , ∂ r ω ) which implies that v θ decays in the same rate as ∂ x ω r and ∂ r ω . Now that we know v θ decays faster than order 1, as mentioned one can apply themaximum principle on the function Γ = rv θ to conclude v θ = H extension domains. An analogous inequality also holds in higher dimensionsby Brezis-Wainger [11]. Recall that a domain Ω ⊂ R n is called a H extension domain if EVIEW ON AXIALLY SYMMETRIC NS 25 the following properties hold. There exists an extension operator P : H ( Ω ) → H ( R n )such that P is a bounded operator from H i ( Ω ) to H i ( R n ), i = , P f | Ω = f for all f ∈ H ( Ω ). For instance, Lipschitz domains are H extension domains. See [13] e.g. Lemma 4.13 ([8]) . Let Ω ⊂ R be a bounded open domain with H extension property, itscomplement or R . Let f ∈ H ( Ω ) . Then there exists a constant C Ω , depending only on Ω ,such that k f k L ∞ ( Ω ) ≤ C Ω k f k H ( Ω ) log / (cid:0) e + k ∆ f k L ( Ω ) k f k H ( Ω ) (cid:1) . (4.21)The proof uses extension properties of H functions and Fourier transform. A variantof it can be found in [44] with a proof using the Green’s function.It is easy to see that the above inequality implies the next one, which was used in[23, 24]. k f k L ∞ ( Ω ) ≤ C Ω (1 + k f k H ( Ω ) ) log / (cid:0) e + k ∆ f k L ( Ω ) (cid:1) . (4.22)The constant C Ω depends on the domain in an implicit way. In applications, it is convenientto have an estimate on C Ω . Next we prove the following refined Brezis − Gallouet inequalityfor a class of domains in the ( r , x ) plane, whose constant is independent of the thinness ofthe domains. A price to pay is that the functions need to have zero boundary value in the x direction or mean zero in the x direction. Lemma 4.14.
For R ≫ and ≤ α ≤ , set ¯ D = (cid:8) ( r , x ) : 1 − R α − ≤ r ≤ + R α − , | x | ≤ R α − r α (cid:9) . Then if f ∈ H ( ¯ D ) satisfies f | | x | = R α − r α = , (4.23) we have k f k L ∞ ( ¯ D ) ≤ C (1 + k∇ f k L ( ¯ D ) ) log / ( e + R α − k ∆ f k L ( ¯ D ) ) , (4.24) where C is independent of R. Here ∇ = e r ∂ r + e ∂ x and ∆ = ∂ r + ∂ x are the twodimensional gradient and Laplacian in the ( r , x ) plane, respectively. Proof.
Note that we can not simply make zero extension for f outside of the domain andapply the regular Brezis-Gallouet inequality. The reason is that the extended function maynot be in H .Define with scaled function ˜ f (˜ r , ˜ x ) = f ( R α − ˜ r , R α − ˜ x ) where (˜ r , ˜ x ) ∈ ˜ U and˜ U = (cid:8) (˜ r , ˜ x ) : R − α − / < ˜ r < R − α + / , | ˜ x | ≤ | R α − ˜ r | α (cid:9) . Observe that ˜ U is almost a square for large R .Using (4.22), we know k f ( r , x ) k L ∞ ( ¯ D ) = k ˜ f (˜ r , ˜ x ) k L ∞ ( ˜ U ) ≤ C (1 + k ˜ f k H ( ˜ U ) ) log / (cid:0) e + k ∆ ˜ f k L ( ˜ U ) (cid:1) = C (1 + k∇ ˜ f k L ( ˜ U ) + k ˜ f k L ( ˜ U ) ) log / (cid:0) e + k ∆ ˜ f k L ( ˜ U ) (cid:1) . (4.25)The point is that the constant C is independent of R . This is because we can first extendthe function ˜ f to be a H function in the whole (˜ r , ˜ x ) plane. From the proof of the originalBrezis-Gallouet inequality, we know the constant relies only on the H extension propertyof functions in a domain. The extension property only depends on the thickness of theoriginal domain, which is scaled to 1. By the change of variables and relationship between f and ˜ f , we deduce k∇ ˜ f k L ( ˜ U ) = k∇ f k L ( ¯ D ) , k ˜ f k L ( ˜ U ) = R − α k f k L ( ¯ D ) , k ∆ ˜ f k L ( ˜ U ) = R α − k ∆ f k L ( ¯ D ) . Inserting the above equalities into (4.25), we find k f ( r , x ) k L ∞ ( ¯ D ) ≤ C (1 + k∇ f k L ( ¯ D ) + R − α k f k L ( ¯ D ) ) log / (cid:0) e + R α − k ∆ f k L ( ¯ D ) (cid:1) . (4.26)Now if f satisfies (4.23), the Poincar´e inequality implies R − α k f k L ( ¯ D ) ≤ C k∇ f k L ( ¯ D ) , where C is independent of R . At last, combination of the above inequality and (4.26) infers(4.24). (cid:3) Now let us introduce Bogovski˘i’s [4, 5] work on solving the divergence equation ona bounded domain with W , p functions (4.27). We only present a special case written asLemma III.3.1 of [39]. More general results can be found in Chapter III of the same book.See also a later paper by Brezis and Bourgain [6] for further results in the special case p = n on torus. Lemma 4.15.
Let
O ⊂ R n , n ≥ , be a bounded domain which is star shaped with respectto every point in a ball B ( x , R ) ⊂ O . Then for any f ∈ L ( O ) , satisfyingf ∈ L p ( O ) , < p < ∞ , with Z O f = , there exists a constant C = C ( O , p , n ) and at least one vector field V : O → R n such that ∇ · V = f , V ∈ W , p ( O ) , k∇ V k L p ≤ C k f k L p . (4.27) Furthermore, let diam ( O ) be the diameter of O , there is a positive constant c ( n , p ) , de-pending only on n , p such that the following estimate holds:C ≤ c ( n , p ) [ diam ( O ) / R ] n (1 + diam ( O ) / R ) . The proof is based on an explicit integral formula found by Bogovski˘i. V ( x ) = Z O N ( x , y ) f ( y ) dy , N ( x , y ) = x − y | x − y | n Z ∞| x − y | φ ( y + r x − y | x − y | ) r n − dr , (4.28)where φ ∈ C ∞ ( B R ( x )) with R φ = C in (4.27) can be improved for some domains, as indicated inthe following: Proposition 4.16.
Consider the domains O R = (cid:8) x | R ≤ | x h | ≤ R , | x | ≤ | x h | α (cid:9) ⊂ R , where α ∈ [0 , and R ≥ . For any f ∈ L ( O R ) with R O R f = , Problem (4.27) with p = has asolution such that k∇ V k L ( O R ) ≤ C α R − α k f k L ( O R ) , (4.29) where C α is independent of R. Proof.
The existence of V is already known by Lemma 4.15. So we just need to prove(4.29). For ¯ x = ( ¯ x , ¯ x , ¯ x ) ∈ O , define¯ f ( ¯ x , ¯ x , ¯ x ) : = f ( R ¯ x , R ¯ x , R α ¯ x ) = f ( x , x , x ) . Note x = R ¯ x , x = R ¯ x but x = R α ¯ x . EVIEW ON AXIALLY SYMMETRIC NS 27
It is easy to see that ¯ f satisfies the assumption in Lemma 4.15. So by Lemma 4.15, thereexists a vector function ¯ V : O → R satisfying (4.27). Then for x ∈ O R , define V ( x , x , x ) = ( V ( x , x , x ) , V ( x , x , x ) , V ( x , x , x )) = ( R ¯ V ( x R , x R , x R α ) , R ¯ V ( x R , x R , x R α ) , R α ¯ V ( x R , x R , x R α )) = ( R ¯ V ( ¯ x , ¯ x , ¯ x ) , R ¯ V ( ¯ x , ¯ x , ¯ x ) , R α ¯ V ( ¯ x , ¯ x , ¯ x )) . (4.30)By a direct computation, we have ∇ · V = f , V ∈ W , ( O R ) , in x variables ∇ · ¯ V = ¯ f , ¯ V ∈ W , ( O ) , in ¯ x variables , where ¯ V = ( ¯ V ( ¯ x ) , ¯ V ( ¯ x ) , ¯ V ( ¯ x )). Now we estimate the L norm of ∇ V . We use α, β to takevalues only on 1 , i , j to take values on 1 , ,
3. So we have k∇ V k L ( O R ) = X i , j = Z | x |≤| x h | α Z R ≤| x h |≤ R | ∂ V j ∂ x i | dx h dx = Z | x |≤| x h | α Z R ≤| x h |≤ R (cid:0) X α,β = | ∂ V β ∂ x α | + X β = | ∂ V β ∂ x | + X α = | ∂ V ∂ x α | + | ∂ V ∂ x | (cid:1) dx h dx = Z | x |≤| x h | α Z R ≤| x h |≤ R (cid:16) X α,β = | ∂ ¯ V β ∂ ¯ x α | ( x h R , x R α ) + R − α X β = | ∂ ¯ V β ∂ ¯ x | ( x h R , x R α ) + R − α ) 2 X α = | ∂ ¯ V ∂ ¯ x α | ( x h R , x R α ) + | ∂ ¯ V ∂ ¯ x | ( x h R , x R α ) (cid:17) dx h dx . Therefore k∇ V k L ( O R ) = R + α Z | ¯ x |≤| ¯ x h | α Z ≤| ¯ x h |≤ (cid:16) X α,β = | ∂ ¯ V β ∂ ¯ x α | ( ¯ x h , ¯ x ) + R − α ) 2 X β = | ∂ ¯ V β ∂ ¯ x | ( ¯ x h , ¯ x ) + R − α ) 2 X α = | ∂ ¯ V ∂ ¯ x α | ( ¯ x h , ¯ x ) + | ∂ ¯ V ∂ ¯ x | ( ¯ x h , ¯ x ) (cid:17) d ¯ x h d ¯ x ≤ CR − α k∇ ¯ V k L ( O ) . (4.31)Also it is easy to see k f k L ( O R ) = R + α k ¯ f k L ( O ) . (4.32)Combining (4.31), (4.32) and (4.27), we have k∇ V k L ( O R ) ≤ CR − α k∇ ¯ V k L ( O ) ≤ CR − α k ¯ f k L ( O ) = CR − α ) k f k L ( O R ) . This finishes the proof of Proposition 4.16. (cid:3)
In Section 4.3, we will see some applications of Lemma 4.15 and Proposition 4.16 toflows in a slab (channels of fixed finite depth) or aperture domains.4.2.
Self-similar solutions.
Another useful class of special solutions to the NS are self-similar solutions and their variants such as discretely self-similar solutions etc. A solution v to the NS is called self-similar if it is invariant under the natural scaling (1.3) for allparameters λ >
0, namely v ( x , t ) = v λ ( x , t ) ≡ λ v ( λ x , λ t ) , ∀ λ > , x ∈ R and all t > t <
0. In general, if v = v λ for one particular λ , then v is called a discretely self-similar solution with factor λ . Given a self-similar v , if it is independent of time, it is called stationary self-similar; otherwise, if v is defined for t <
0, then it is called backward self-similar; if v is defined for t >
0, then it is called forward self-similar. Stationary self-similarsolutions can serve as one model of stationary solutions of NS near a singularity or spatialinfinity. A backward one serves as a model of possible type I singularity of solutions of NS;and a forward self-similar solution can be used to describe long time behavior of solutionsof the NS. Detailed study can be found in [110] Chapter 8.Self-similar solutions are determined by their profile U . In the stationary case v ( x ) = | x | − U ( x ) , U ( x ) ≡ v ( x / | x | ) . Here U can be regarded as a vector defined on the unit sphere or a homogeneous vectorfield on R − { } of degree 0. In the time dependent case v ( x , t ) = λ ( t ) U ( λ ( t ) x ) , U ( x ) = v ( x , sgn t ) , λ ( t ) = ( t sgn t ) − / . Let the number m = , , − U satisfy the following stationary equations in R if m , R −{ } if m = − ∆ U + U ∇ U + ∇ P − m U − m x ∇ U = , div U = , (4.33)where, as usual x = ( x , x , x ), U = ( U , U , U ) and x ∇ U = P i = x i ∂ i U .There is a family of stationary self-similar solutions, called Landau solutions [60],which, in the spherical coordinates, are explicitly given by U b = curl ( L e θ ) = ρ sin φ ∂ φ ( L sin φ ) e ρ − ρ ∂ ρ ( ρ L ) e φ , L = φ ( a − cos φ ) , b = π a + a a − a + + a a − ! e , a > . (4.34)With the parameter b , U b actually solves the nonhomogeneous problem in R : − ∆ U b + U b ∇ U b + ∇ P b = b δ, div U b = , where δ = δ ( x ) is the Dirac delta function centered at 0. The spherical system we take isthe standard one x = ( x , x , x ) = ( ρ sin φ cos θ, ρ sin φ sin θ, ρ cos φ ) . Here ρ = | x | , φ is the polar angle between x and e , θ is the azimuthal angle. This systemis a convenient one for studying self-similar solutions.The Landau solutions are axially symmetric without swirl. The following result due toSverak [103] shows that they are the only (-1)-homogenous ones. Earlier Tian and Xinproved in [111] that all (-1)-homogeneous, axially symmetric solutions in C ( R \{ } ) areLandau solutions. Here we are using the notion of ”(- m )-homogeneous” interchangeablywith ”homogeneous of degree − m ”. Theorem 4.17.
If a homogeneous vector field v of degree − is a solution to the stationaryNS in R \ , then v = U b for some b ∈ R . Extension of the theorem to (-1)-homogenous solutions with singularities at the polesof S can be found in recent papers by Li, Li and Yan [66, 67].Backward self-similar solutions of the form v ( x , t ) = √ T − t U x √ T − t ! , t ∈ [ t , T ) , x ∈ R , (4.35)was proposed by Leray as a candidate for possible type I singularity of the time dependentNS at time T . Here t is the initial time. Leray asked whether there is a self-similar solution EVIEW ON AXIALLY SYMMETRIC NS 29 of the NS in above form with finite energy, i.e. (1.2) holds. For such solutions U ∈ L ( R ).In 1996, Necas, Ruzicka and Sverak [85] proved that the only such solution is 0. In [109],T. P. Tsai generalized the result to very weak self-similar solutions satisfying the local finiteenergy condition. For a definition of very weak solutions, see [110] p53 e.g. Theorem 4.18.
Suppose v is a self-similar very weak solution of the NS with zero force inthe cylinder B × ( − , ⊂ R × ( − , . It is zero if it has finite local energy sup − < t < Z B | v ( x , t ) | dx + Z − Z |∇ v ( x , t ) | dxdt < ∞ , or if the profile U ∈ L q ( R ) for some q ∈ [3 , ∞ ) . Also U is constant if U ∈ L ∞ ( R ) . The proof is based on the fact that the total head pressure Q = | U ( x ) | + P ( x ) + x · U ( x )satisfies the equation − ∆ Q + ( U ( x ) + x ) · ∇ Q ( x ) = −| curl U ( x ) | ≤ . If one can prove that Q behaves well at infinity, then the maximum principle and finiteenergy assumption will force U = C α ( R \{ } ). Theorem 4.19 ([48]) . Assume v is scale-invariant and locally H¨older continuous in R \{ } with div v = in R . Then the Cauchy problem (1.1) has at least one scale-invariant solution v which is smooth in R × (0 , ∞ ) and locally H¨older continuous in ( R × ([0 , ∞ )) \{ (0 , } . The proof uses Leray-Schrauder fixed point theorem to solve equation (4.33) with m = µ e ∆ v = µ R R G ( x , , y ) v ( y ) dy . Here µ ∈ (0 ,
1] is aparameter and G is the standard heat kernel on R . More specifically, one looks for U µ solving (4.33) such that (cid:12)(cid:12)(cid:12)(cid:12) ∇ α (cid:16) U µ ( x ) − µ e ∆ v ( x ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( α, v )(1 + | x | ) + α , α = , . One can convert this to a fixed point problem of an integral operator involving the Stokeskernel, which is easy to solve if µ is small. If one has compactness for the integral operator,then Leray Schauder theorem ensures the solution exists for µ =
1. The compactness isbased on a priori H¨older estimates near the initial time for the local Leray solutions fromthe book [63]. Then v = t − / U ( t − / x ) is a solution in the above theorem.These solutions potentially have application in searching non-unique Leray Hopf solu-tions of NS. Denote by v µ = t − / U µ ( t − / x ) the unique scale-invariant solution with initialdata µ v and µ su ffi ciently close to 0. Note for large µ the uniqueness is unknown. In thepaper [49], Jia and Sverak consider solutions of NS in the form v µ ( x , t ) + t − / φ ( t − / x , t ).The linearization of the equation for φ takes the form t ∂ t φ = L µ φ . For small enough µ theeigenvalues of L µ are away from the imaginary axis with real part less than zero. Theysuggest two potential scenarios under which the solution curve can be continued as a reg-ular function of µ and the eigenvalues of L µ can cross the imaginary axis. Under thesehypotheses two solutions of the NS are obtained with initial data µ v ; Although these haveinfinite energy but by analysis on certain critical singularities in some lower order terms, the authors are then able to localize these solutions to obtain possible non-uniqueness forLeray-Hopf solutions.4.3. Addendum: Extra results on D solutions, by Xinghong Pan and Q. S. Zhang.
In this subsection, we present some new vanishing results on D solutions which extendthose in [24]. The improvements are on wider domains and relaxation of symmetry orgrowth condition on the solutions. These results seem to be new and are not presentedelsewhere in the literature. The general idea of the proof is similar to that of the previousresults.For Theorem 4.11, we can weaken the axially symmetric assumption on the three com-ponents of v r , v θ , v to the case that only v θ is axially symmetric. Denote x h = ( x , x ) and S the 1 dimensional periodic domain with period being 1. We have the following result. Theorem 4.20 (vanishing of periodic HDS with just axial symmetry of v θ ) . Let ( v , P ) be abounded smooth solution to the problem ( v · ∇ v + ∇ P − ∆ v = , ∇ · v = , in R × S , v ( x h , x ) = v ( x h , x + , in R × S , (4.36) with finite Dirichlet integral Z R × S |∇ v ( x ) | dx < + ∞ . (4.37) If just v θ is axially symmetric (independent of θ ), then we have v ≡ ce . The next theorem deals with ASHDS in an aperture domain Ω : = { x | x h ∈ R , | x | ≤ (max(1 , r )) α } for some α ≥
0. Study of flows in aperture domains is also of usefulness, asmentioned in [39] Chapter XIII.
Theorem 4.21 (ASHDS on aperture domains) . Let ( v , P ) be an axially symmetric boundedsmooth solution to the problemv · ∇ v + ∇ P − ∆ v = , ∇ · v = in Ω ; v ( x ) = on ∂ Ω (4.38) with finite Dirichlet integral (4.2) . Then we have (i) v ≡ if ≤ α < / ; (ii) v θ ≡ if ≤ α < / . When α =
0, we can remove the axial symmetry assumption of the solution, which goesback to Theorem 4.7.4.3.1.
Proof of Theorem 4.20, one component axially symmetric periodic solutions.
The proof is divided into two steps.
Step 1. L boundedness of v r and L mean oscillation of P. Lemma 4.22.
For n ∈ N / { } , denote C n = { x h | n ≤ | x h | ≤ n } . Let ( v , P ) be the solution of (4.36) . Under the assumptions of Theorem 4.20, we have k v r k L ( R × S ) < C ∗ , (4.39) and k P − P n k L ( C n × S ) ≤ C ∗ n , (4.40) where C ∗ = C ( k v k L ∞ , k∇ v k L ) and P n : = |C n × S | R C n × S Pdx is the average of P on C n × S . EVIEW ON AXIALLY SYMMETRIC NS 31
Proof.
In cylindrical coordinates, if v θ is independent of θ , the divergence free condition(4.36) is translated as ∇ · v = r ∂ r ( rv r ) + r ∂ θ v θ + ∂ x v = r ∂ r ( rv r ) + ∂ x v = . Integrating the above inequality in S about x and by using the periodic boundary condition,we can deduce R S v r dx = . Then the one dimensional Poincar´e inequality indicates that Z R × S | v r | dx = Z R × S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v r − | S | Z v r dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dx dx h . Z R Z | ∂ x v r | dx dx h < ∞ . This proves (4.39).In Proposition 4.16, if α = O R is replaced by Ω R = (cid:8) x | R ≤ | x h | ≤ R , | x | ≤ R (cid:9) , it is not hard to see the proof is still valid. Now set R = n with n ∈ N \{ } .From Proposition 4.16, there exists a V ∈ H ( Ω n ) satisfying ∇ · V = f and (4.29) with f = P − P n and O R replaced by Ω n .Now multiplying (4.36) with V and integration on Ω n , we get Z Ω n ∇ ( P − P n ) · Vdx = Z Ω n ( ∆ v − v · ∇ v ) · Vdx . Integration by parts and (4.29) indicate that Z Ω n ( P − P n ) dx = Z Ω n ( P − P n ) ∇ · Vdx = − Z Ω n (cid:0) ∆ v − v · ∇ v (cid:1) · Vdx = Z Ω n X i , j = ∂ i v j ∂ i V j + ∂ i ( v i v j ) V j dx = Z Ω n X i , j = (cid:0) ∂ i v j − v i v j (cid:1) ∂ i V j dx ≤k∇ V k L ( Ω n ) (cid:0) k∇ v k L ( Ω n ) + k v k L ∞ k k L ( Ω n ) (cid:1) ≤ C k P − P n k L ( Ω n ) (cid:0) k∇ v k L ( Ω n ) + k v k L ∞ k k L ( Ω n ) (cid:1) (4.41)Then we can obtain k P − P n k L ( Ω n ) ≤ C (cid:0) k∇ v k L ( Ω n ) + k v k L ∞ ( Ω n ) n / (cid:1) . Remembering that ( v , P ) is periodic in the x direction, the above inequality can be rewrit-ten as n / k P − P n k L ( C n × S ) ≤ C (cid:0) n / k∇ v k L ( C n × S ) + k v k L ∞ ( C n × S ) n / (cid:1) . This implies (4.40). (cid:3)
Step 2. Trivialness of v
Let φ ( s ) be a smooth cut-o ff function satisfying φ ( s ) = s ∈ [0 , φ ( s ) = s ≥ , (4.42)with the usual property that φ , φ ′ and φ ′′ are bounded. Set φ n ( y h ) = φ ( | y h | n ). Testing the NSin (4.36) with v φ n , we find that Z Ω × S |∇ v | φ n dx = − X i = Z Ω × S v i ∇ v i · ∇ φ n dx + Z Ω × S | v | + ( P − P n ) ! v · ∇ φ n dx . Here Ω is the 2 dimensional ball { x h | | x h | < n } . It follows that Z Ω × S |∇ v | φ n dx . Z C n × S | v ||∇ v ||∇ φ n | dx + Z C n × S | v · ∇ φ n | | v | dx + Z C n × S | P − P n | | v · ∇ φ n | dx : = I + I + I . (4.43)Observe that, as n → ∞ , I . k v k L ∞ ( C n × S ) n k∇ v k L ( C n × S ) k k L (( C n × S ) . k v k L ∞ ( C n × S ) k∇ v k L ( C n × S ) → I . k v k L ∞ ( C n × S ) Z C n × S | v r ∂ r φ n | dx . k v k L ∞ ( C n × S ) n k v r k L ( C n × S ) k k L ( C n × S ) . k v k L ∞ ( C n × S ) k v r k L ( C n × S ) → I . Z C n × S | P − P n | | v r ∂ r φ n | dx . n k P − P n k L ( C n × S ) k v r k L ( C n × S ) . C k v r k L ( C n × S ) → . Here we have applied Cauchy-Schwarz inequality, the boundedness of the oscillation of P in dyadic annulus from Lemma 4.22 and v r ∈ L ( Ω × S ). Combining those estimates of I , I and I , (4.43) yields R Ω × S |∇ v | dx = , which means v is a constant vector. The v r = v θ = v ≡ ce . (cid:3) Proof of Theorem 4.21, axially symmetric solutions on aperture domains.
First we show an a priori decay of v . Proposition 4.23.
Under the assumption of Theorem 4.21, we havev = O (cid:16) ln / rr / (cid:17) as r → + ∞ . (4.44) Proof.
The proof is based on
Brezis − Gallouet inequality [8] and its refinement, together withscaling and dimension reduction techniques, which is similar to [24, part 5.2, p. 1403-1406].Fixing x ∈ Ω : = { x || x h | ∈ Ω , | x | ≤ max { , r α } such that | x ′ | = r is large. Withoutloss of generality, we can assume, in the cylindrical coordinates, that x = ( r , , x ) = θ =
0. Consider the scaled solution ˜ v ( ˜ x ) = r v ( r ˜ x ) which is also axiallysymmetric. Hence ˜ v can be regarded as a two variable function of the scaled variables˜ r , ˜ x . Consider the two dimensional domain¯ D = (cid:8) (˜ r , ˜ x ) : 1 − r α − ≤ ˜ r ≤ + r α − , | ˜ x | ≤ r α − ˜ r α (cid:9) . Then for ˜ v = ˜ v (˜ r , ˜ x ), we have ˜ v (1 , = r v ( x ). Recall that ˜ v satisfies the Dirichletboundary condition. Applying the refined Brezis-Gallouet inequality (Lemma 4.14) on¯ D , after a simple adjustment on constants, we can find an absolute constant C such that | ˜ v (1 , | ≤ C Z ¯ D | ˜ ∇ ˜ v | d ˜ rd ˜ x ! / + × log / r α − Z ¯ D | ˜ ∆ ˜ v | d ˜ rd ˜ x ! / + e , EVIEW ON AXIALLY SYMMETRIC NS 33 where ˜ ∇ = ( ∂ ˜ r , ∂ ˜ x ) and ˜ ∆ = ∂ r + ∂ x . From this and the assumption that 1 / ≤ ˜ r ≤
2, wesee that | ˜ v (1 , | ≤ C Z ¯ D | ˜ ∇ ˜ v | ˜ rd ˜ rd ˜ x ! / + × log / r α − Z ¯ D | ˜ ∆ ˜ v | ˜ rd ˜ rd ˜ x ! / + e . Now we can scale this inequality back to the original solution u and variables r = r ˜ r and z = r ˜ x to get r | v ( x ) | ≤ C √ r Z D |∇ v | rdrdx ! / + × log / r α + / Z D ( | ∂ r v | + | ∂ x v | ) rdrdx ! / + e , where D = (cid:8) ( r , x ) : r − r α ≤ r ≤ r + r α , | x | ≤ r α (cid:9) . By condition (4.2), this provesthe claimed decay of velocity. Note that by our assumption in the theorem, the solution u is globally bounded and then it is not hard to prove that the first and second derivative of v are also bounded. (cid:3) The rest of the proof of Theorem 4.21 is divided into two subsections.4.3.3.
Proof of case (i): ≤ α < / . First we will give a L estimate of v and meanoscillation of the pressure P by using the preceding Proposition 4.16. Lemma 4.24.
Let ( v , P ) be the solution of (4.38) and O R = (cid:8) x | R ≤ | x h | ≤ R , | x | ≤ | x h | α (cid:9) ,then we have Z O R | v | dx ≤ o ( R α ) , (4.45) k P − P R k L ( O R ) ≤ o (1) R (1 − α ) (1 + R α − / ln / R ) , (4.46) as R → ∞ . Here P R : = |O R | R O R Pdx is the average of P on O R . Proof.
Since we have zero boundary, the one dimensional Poincar´e inequality indicatesthat Z O R | v | dx = Z R ≤| x h |≤ R Z | x |≤ r α | v | dx dx h . R α Z R ≤| x h |≤ R Z | x |≤ r α | ∂ x v | dx dx h . o ( R α ) . by the definition of D solutions. This proves (4.45).From Proposition 4.16, there exists a V ∈ H ( O R ) satisfying ∇ · V = f and (4.29) with f = P − P R . Now multiplying (4.38) with V and integrating on O R , we get Z O R ∇ ( P − P R ) · Vdx = Z O R ( ∆ v − v · ∇ v ) · Vdx . The same derivation as (4.41) indicates that Z O R ( P − P R ) dx = Z O R ( P − P R ) ∇ · Vdx ≤k∇ V k L ( O R ) (cid:0) k∇ v k L ( O R ) + k v k L ∞ ( O R ) k v k L ( O R ) (cid:1) ≤ CR − α k P − P R k L ( O R ) (cid:0) k∇ v k L ( O R ) + k v k L ∞ ( O R ) k v k L ( O R ) (cid:1) . Here to reach the last line, we used (4.29). Then we can obtain k p − p R k L ( O R ) ≤ CR (1 − α ) (cid:0) k∇ v k L ( Ω R ) + k v k L ∞ ( Ω R ) k v k L ( Ω R ) (cid:1) = o (1) R (1 − α ) (1 + R α − / ln / R ) , which is (4.46). (cid:3) Vanishing of v Now we are in a position to complete the proof of case (i) of Theorem 4.21. Let φ ( s )be the smooth cut-o ff function defined in (4.42). Set φ R ( y ′ ) = φ ( | y ′ | R ) where R is a largepositive number. Now testing the NS in (4.38) with v φ R , we obtain Z Ω − ∆ v ( v φ R ) dx = Z Ω − ( v · ∇ v + ∇ ( P − P R )) ( v φ R ) dx . Integration by parts yields that Z Ω |∇ v | φ R dx − Z Ω | v | ∆ φ R dx = Z Ω | v | v · ∇ φ R dx + Z Ω ( P − P R ) v · ∇ φ R dx . Then we have, since φ R depends only on r , that Z Ω |∇ v | φ R dx . R Z O R | v | dx + R Z O R | v r | | v | dx + R Z O R | P − P R | | v r | dx . R Z O R | v | dx + k v r k ∞ R Z O R | v | dx + R Z O R ( v r ) dx ! / Z O R | P − P R | dx ! / . (cid:0) R α − + R α − / ln / R + o (1) + R α − / ln / R (cid:1) . When 0 ≤ α < /
2, let R → + ∞ , we arrive at R Ω |∇ v | dx = , which shows that v ≡ c . Besides, recall v = v ≡ . This completes theproof of case (i) of Theorem 4.21.4.3.4.
Proof of case (ii): / ≤ α < / . The proof is divided into 2 steps.
Step 1. Caccioppoli inequality for
Γ = rv θ First for R >
2, pick the domains D ( R ) = Ω ∩ ( B R × R ) , and D ( R , R ) = D ( R ) \ D ( R ) , R > R > . Here B R is the 2 dimensional ball centered at the origin in R , with radius R . Wefirst derive a Caccioppoli type energy estimates of Γ by equation (2.2). Consider a standardtest function ψ ( r ) satisfying supp ψ ⊂ B σ , ψ = B σ , ≤ ψ ≤
1; 0 < σ < σ ≤ |∇ k ψ | ≤ C ( σ − σ ) k for k = , . Lemma 4.25.
Let ≤ σ < σ ≤ , R ≥ and ψ R ( x ) = ψ ( rR ) . Denote f : = | Γ | q for q > .Then we have Z D ( σ R ) |∇ ( f ψ R ) | . (1 + k v r k L ∞ R α ) ( σ − σ ) R Z D ( σ R ,σ R ) f , here and in the proof k v r k L ∞ = k v r k L ∞ ( D ( σ R ,σ R )) .Proof. Recall (2.2) without time variable is: ∆Γ − b ∇ Γ − r ∂ r Γ = . with b = v r e r + v e . Testing this by q | Γ | q − Γ ψ R in Ω gives12 Z Ω ( b · ∇ f + r ∂ r f ) ψ R dx = q Z Ω ∆Γ | Γ | q − Γ ψ R dx . (4.47) EVIEW ON AXIALLY SYMMETRIC NS 35
Later the integral variables dx will not be written out for simplicity unless there is confu-sion. Using Cauchy-Schwartz’s inequality and integration by parts, we have q Z Ω ∆Γ | Γ | q − Γ ψ R = q Z D ( σ R ) ∆ | Γ || Γ | q − ψ R = − q Z D ( σ R ) (2 q − |∇| Γ || Γ q − ψ R + ∇| Γ || Γ | q − · ∇ ψ R = − Z D ( σ R ) (2 − q ) |∇ ( f ψ R ) | − (2 − q ) f ∇ ψ R · ∇ ( f ψ R ) − q f |∇ ψ R | . − Z D ( σ R ) |∇ ( f ψ R ) | + C Z D ( σ R ) f |∇ ψ R | . (4.48)Also we have − Z Ω r ∂ r f ψ R = − π Z σ R Z | x |≤ (max { , r } ) α ∂ r f ψ R dx dr ≤ π Z σ R Z | x |≤ (max { , r } ) α f ∂ r ψ R dx dr . Z D ( σ R ,σ R ) f ( |∇ ψ R | ) (4.49)Consequently, using (4.47) and combining (4.48) and (4.49), we get Z D ( σ R ) |∇ ( f ψ R ) | ≤ C Z D ( σ R ,σ R ) f |∇ ψ R | − Z D ( σ R ) b · ∇ f ψ R ≤ C ( σ − σ ) R Z D ( σ R ,σ R ) f − Z D ( σ R ) b · ∇ f ψ R . (4.50)By the divergence-free property of the drift term b , we have − Z Ω b · ∇ f ψ R = Z D ( σ R ) f ψ R b · ∇ ψ R = Z D ( σ R ,σ R ) ( f ψ R ) f v r ∂ r ψ R ≤ C k v r k L ∞ ( σ − σ ) R k f ψ R k L ( D ( σ R ,σ R )) k f k L ( D ( σ R ,σ R )) ≤ C ε k v r k L ∞ R α ( σ − σ ) R k f k L ( D ( σ R ,σ R )) + ε R − α k f ψ R k L ( D ( σ R ,σ R )) ≤ C ε k v r k L ∞ R α ( σ − σ ) R k f k L ( D ( σ R ,σ R )) + C ε k∇ ( f ψ R ) k L ( D ( σ R ,σ R )) ; (4.51)where at the third line of (4.51), we have used the fact f = Ω , andthe following 1-dimensional Poincar´e inequality. k f ψ R k L ( D ( σ R ,σ R )) = Z π Z σ R σ R Z r α − r α | f ψ R | dx rdrd θ ≤ CR α Z π Z σ R Z r α − r α | ∂ x ( f ψ R ) | dx rdrd θ ≤ CR α k∇ ( f ψ R ) k L ( D ( σ R )) . (4.52)Combining (4.50) and (4.51), by choosing small ε , we get Z D ( σ R ) |∇ ( f ψ R ) | . (1 + k v r k L ∞ R α ) ( σ − σ ) R Z D ( σ R ,σ R ) f . (4.53) (cid:3) Step 2. Vanishing of v θ We will show that Γ decays to 0 as r → ∞ . Then the maximum principle implies Γ andhence v θ =
0. We use Moser’s iteration.By H¨older inequality and Sobolev imbedding theorem, for any 0 < β <
4, we have k f ψ R k + β L + β ( D ( σ R )) = Z D ( σ R ) | f ψ R | + β = Z D ( σ R ) | f ψ R | − β | f ψ R | β . (cid:0) Z D ( σ R ) | f ψ R | dy (cid:1) − β (cid:0) Z D ( σ R ) | f ψ R | dy (cid:1) β/ . k f ψ R k − β L ( D ( σ R )) k∇ ( f ψ R ) k β L ( D ( σ R )) . R α (2 − β ) k∇ ( f ψ R ) k − β L ( D ( σ R )) k∇ ( f ψ R ) k β L ( D ( σ R )) ≤ R α (2 − β ) k∇ ( f ψ R ) k + β L ( D ( σ R )) , (4.54)where at the last line we have used (4.52). Using (4.53) and (4.54), we deduce Z D ( σ R ) f + β . R α (2 − β ) h (1 + k v r k L ∞ R α ) ( σ − σ ) R k f k L ( D ( σ R ,σ R )) i + β . Remember f = | Γ | q and set κ = + β , then we obtain Z D ( σ R ) | Γ | q κ . R α (2 − β ) h + k v r k L ∞ R α ( σ − σ ) R i κ (cid:16) Z D ( σ R ,σ R ) | Γ | q (cid:17) κ . For integer j ≥ σ = , set σ = (1 + σ j + ) and σ = (1 + σ j ). Let q = κ j , then we arrive at Z D ( R (1 + σ j + )) Γ κ j + dy ( κ ) j + . R α (1 − β ) ( κ ) j + h + k v r k L ∞ R α σ j + R i ( κ ) j Z D ( R (1 + σ j ) , R (1 + σ j + )) Γ κ j dy ( κ ) j . By iterating j , the above inequality gives Z D ( R (1 + σ j + )) Γ κ j + dy ( κ ) j + . R α (1 − β ) j P i = ( κ ) i + (1 + k v r k L ∞ R α ) P ji = ( κ ) i σ P ji = ( i + κ ) i R P ji = ( κ ) i Z D ( R , R ) Γ dy . Letting j → ∞ yields thatsup x ∈ D ( R ) | Γ | . R α ( β − ) (1 + k v r k L ∞ R α ) + ββ R + ββ Z D ( R , R / Γ dy ! . (4.55)Next we use the 1-dimensional Poincar´e inequality to see that Z D ( R , R / Γ dy = Z π Z RR / Z r α − r α r ( v θ ) dx rdrd θ ≤ R Z π Z RR / Z r α − r α ( v θ ) dx rdrd θ ≤ CR + α Z π Z RR / Z r α − r α ( ∂ x v θ ) dx rdrd θ ≤ CR + α k∇ v k L ( Ω ∩{ R / ≤| x ′ |≤ R } ) . (4.56)Inserting (4.56) into (4.55), we can getsup x ∈ D ( R ) | Γ | . R ( α − β + α (1 + k v r k L ∞ R α ) + ββ k∇ v k L ( Ω ∩{ R / ≤| x ′ |≤ R } ) . Using the decay estimates for v r , we havesup x ∈ D ( R ) | Γ | . R ( α − β + α (1 + R α − / ln / R ) + ββ k∇ v k L ( Ω ∩{ R / ≤| x ′ |≤ R } ) . (4.57) EVIEW ON AXIALLY SYMMETRIC NS 37
When 1 / ≤ α < /
4, from (4.57), we findsup x ∈ D ( R ) | Γ | ≤ CR ( α − β + α ( R α − / ln / R ) + ββ ≤ CR (2 α − ) β + α − (ln R ) + β β → , as R → ∞ by choosing su ffi ciently small β such that (2 α − ) β + α − < , since 2 α − <
0. Thisindicates that Γ ≡ Ω .So at last we have proved that Γ ≡ ≤ α < /
4, which shows that v θ ≡ (cid:3) A cknowledgments We wish to thank the following people and organizations for their supports. Prof.Yanyan Li for the invitation to write the review paper and for his encouragement; Prof.Hongjie Dong and Prof. Zhen Lei for the discussion and collaboration over the years; Pro-fessors B. Carrillo, Zijin Li and Dr. Na Zhao for the recent collaboration ; Prof. Xin Yangand Mr. Chulan Zeng for going over the paper and making suggestions; Natural ScienceFoundation of Jiangsu Province for grant No. BK20180414; National Natural ScienceFoundation of China for grant No. 11801268; Simons Foundation for grant 710364.R eferences [1] Amick, C.J.,
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Methods Appl. Anal. 9, 563-578, (2002).(Xinghong Pan) D epartment of M athematics , N anjing U niversity of A eronautics and A stronautics , N anjing hina . Email address : xinghong [email protected] (Q. S. Zhang) D epartment of mathematics , U niversity of C alifornia , R iverside , CA 92521, USA Email address ::