ON LANDAU’S SOLUTIONS OF THE NAVIER-STOKES EQUATIONS

Vladimír Šverák

University of Minnesota

1. Introduction. In this note we will study a special class of solutions of the three-dimensional steady-state Navier-Stokes equations

(NSE) |

The equations have a non-trivial scaling symmetry and it is natural to try to find solutions which are invariant under this scaling. Explicit examples of such solutions were first calculated by L.D.Landau in 1944 ([L]) and can be found in standard textbooks. (See, for example, [LL], p. 82, or [B], p. 206.) The main idea of Landau’s calculation is that if we impose an additional symmetry requirement, namely that the solutions are axi-symmetric, the system (NSE) reduces to a system of ODEs which, surprisingly, can be solved explicitly in terms of elementary functions. The solutions were also independently found by H.B.Squire in 1951 ([Sq]). More recently, the topic has been re-visited in [TX] and [CK], where issues concerning Landau’s solutions are addressed from a slightly different viewpoint.

In this note we prove that even if we drop the requirement of axi-symmetry, Landau’s solutions are still the only solutions of (NSE) which are invariant under the natural scaling. More precisely, we will prove the following:

Theorem 1. Assume that is a non-trivial smooth solution of (NSE) satisfying for each . Then is a Landau solution. In other words, is axi-symmetric and, in a suitable coordinate frame, is described by formulae (E7) in Section 4. The proof of the theorem shows a connection between the scale-invariant solutions of (NSE) and the conformal geometry of the two-dimensional sphere. In fact, once the connection is understood, the formulae for Landau’s solutions can be derived without much calculation, using just the geometrical properties of the two-dimensional sphere.

Some implications of Theorem 1 are considered in the next two sections.

2. Implications for regularity of very weak solutions. By a very weak solution of the steady-state Navier-Stokes system (NSE) in a domain we mean a divergence-free vector field which satisfies

for each smooth, compactly supported, divergence-free vector field in .

It seems to be an open problem whether very weak solutions of (NSE) are regular. Standard regularity theory can be used to show that very weak solutions are regular under the additional requirement that . The equations (NSE) are usually considered with the assumption that , in which case regularity follows by a straightforward bootstrapping argument. The assumption is of course very natural when considering solutions describing real physical flows. However, one can speculate that very weak solutions might arise from a blow-up procedure of the usual weak solutions of the time-dependent Navier-Stokes equations at a possible singularity (if a singularity exists). The time-dependent 3-dimensional Navier-Stokes equations are supercritical with respect to the natural energy estimates, and in a blow-up procedure the information about energy can be lost.

A natural first step in understanding the regularity of the very weak solutions above is to study the scale-invariant solutions in which are smooth in . A calculation (which can be found in [B], p. 209, and also in [T], and [CK]) shows that Landau’s solutions are not very weak solutions of (NSE) across the origin. Therefore Theorem 1 implies the following.

Corollary. Let be a -homogeneous very weak solution of the Navier-Stokes equations in , which is smooth away from the origin. Then .

This result rules out only the simplest conceivable singularity of a very weak solution. For example, the question if one can have a non-trivial very weak solution smooth away from the origin and satisfying in is not answered by Theorem 1 and – as far as I know – remains open.

Remark: The question about the existence of -homogeneous solutions of the system (NSE) in (smooth away from the origin) can also be posed for . It turns out that for there are no such solutions. This was proved independently by several authors ([St], [Sv], [T]) and it also follows from results in [FR].

3. Asymptotic behavior of solutions in exterior domains. Theorem 1 has some relevance for the problem of long-range behavior of solutions of the Navier-Stokes equations in exterior domains. (See for example [G] for an overview of this topic.) For simplicity, we consider here only the following special case. Let be a compactly supported vector field in and consider the equations

together with a “boundary condition” at , which might take the form at and , for example. The existence of such solution was proved in a classic paper by Leray ([Le]), but there are many open questions about the behavior of these solutions for large , see [G]. Theorem 1 implies, roughly speaking, the following: If a solution of the above exterior problem is asymptotically -homogeneous, then the terms of order must be given by a Landau solution. To give this a more precise meaning, let us consider the scaled functions and defined by and . The functions and satisfy the same equations as and . Moreover the functions converge to a distribution , given by , where and is the Dirac function. Assume now that converges to a limit in, say, as . Our notion of “asymptotically - homogeneous” used above can be defined by requiring that this is really the case. (Of course, it is a difficult open problem to decide whether, for general large data, this might always be the case.) The limit functions and will again satisfy the same equations (in the sense of distributions). Under our assumptions the function is smooth away from the origin, satisfies for each , and, by Theorem 1, must therefore be a Landau solution or vanish identically. (The direction of the vector will be the axis of symmetry of the solution.) For we will have , which means that, under the above assumptions, the solution decays faster than .

4. Proof of Theorem 1. Let be a -homogeneous vector field in , smooth away from the origin. Clearly is determined by its restriction to the unit sphere . For we decompose as , where is the outer unit normal to , and is tangent to at , i. e. . We now write down the Navier-Stokes equations for and as a system of PDEs on . If satisfies the Navier-Stokes equation in in the very weak sense defined above, it is easy to see that there exists a suitable pressure function in which is -homogeneous and smooth away from the origin. The function is also determined by its values on , and the system (NSE) can be written down as a system of PDEs on for and . The differential operators in what follows will all be differential operators on , defined by the usual conventions of Riemannian geometry. The differential forms on will be identified with vector fields and vice-versa, as is usual on Riemannian manifolds. The Hodge Laplacian on 1-forms will be denoted by . (The reason for writing it as , with the minus sign, is to keep the equations on in a form which resembles the standard euclidean form of the equations as much as possible.) The Navier-Stokes equations (NSE) for written in terms of and as equations on are as follows:

(S) |

A straightforward (although perhaps not the most illuminating) way to derive these equations is to write the system (NSE) in spherical coordinates (see, for example, [B], p. 601) and check that for -homogeneous vector fields it reduces to the system (S). We remark that the spherical coordinates version of (NSE) in the second edition of the book [LL] (p. 49) contains a misprint in the right-hand side of the first equation, where an incorrect expression appears instead of the correct .

We will denote by the function on given by , where is the canonical volume form of . This corresponds to the formula used in .

By taking of the first equation of the system (S) we obtain

(E1) |

Lemma 1. With the notation introduced above, we have .

Proof. Let be the differential operator defined by . The adjoint operator is given by . The kernel of consists of constant functions, as can be seen from the strong maximum principle. The kernel of must therefore also be one dimensional. Let us denote by a non-trivial function in the kernel of . If changed sign on , we could find a strictly positive smooth function on with . But this would mean that the equation has a solution. However, the last equation cannot be satisfied at points where attains its minimum. From this we see that the function cannot change sign. At the same time, the definition of immediately implies that , and we see that must vanish.

Remark. I assume the above argument is known in one form or another, but I was not able to find a good reference for it.

Once we know that , the first equation of (S) simplifies. Indeed, when we have , and we also have . Using this, the first equation of (S) implies

where is a constant. The second equation of (S) now gives

(E2) |

Integrating (E2) over and using the third equation of (S) we see that . Since we can write for a suitable smooth function on . The equation (E2), together with the third equation of (S) and the fact that now gives

(E3) |

Letting , the last equation can be re-written as

The solutions of this equation are well-known: They are functions of the form , where is a constant. (An easy way to verify this is for example the following: Write in the form . We get an equation for for which the strong maximum principle implies that the solutions are exactly ) Integrating over the sphere we see that . Hence we have

for a constant . Changing by a constant, if necessary, we can assume without loss of generality, and we end up with

(E4) |

The interpretation of equation (E4) is well-known (see, for example, [CY]): Let be the canonical metric on and let be the metric on defined by . Equation (E4) says exactly that the Gauss curvature of the metric is , i. e. the metric is isometric to the metric . In other words, we have (pullback of by ) for a suitable diffeomorphism of . From the definitions we also see that has to be conformal or anti-conformal. Anti-conformal maps can be obtained from conformal maps by a composition with an isometry, and hence we can only consider the case when is conformal. For a given conformal , the function is given by

(E5) |

where denotes the (complex) derivative of at . It is well-known (see e. g. [DFN]) that all conformal diffeomorphisms of can be produced as follows. Let be the standard stereographic projection, and let be defined by . Let . Then any conformal diffeomorphism of can be produced by composing a suitable (with ) with isometries of . If is given by (E5) and we compose with and isometry, then the function either does not change or only changes by being shifted by the isometry. Therefore in a suitable coordinate frame all solutions of (E4) look like the solutions generated by the special above. We now consider the standard spherical coordinates on , given by

(SC) |

We will use the usual notation for the tangent vector field on corresponding to . Letting , calculating the maps above in these coordinates, and using the formula (E5), we obtain

(E6) |

This gives

(E7) |

which agrees with the formulae in [B], p. 207 if we set and with the formulae in [LL], p. 82, if we set . The proof of Theorem 1 is finished.

Remarks:

1. As we already mentioned in Section 1, the Landau solutions (given by (E7)) do not satisfy the Navier-Stokes equations (NSE) across the origin. A calculation in [B], p. 209, shows that for Landau’s solutions we have, in distributions,

where is the Dirac function and is a non-zero vector in depending in a non-trivial way on the parameter which parametrizes the solutions in the above coordinate frame. The exact formula for can be found in [B], p. 209, and was also calculated in [CK].

2. If is a non-trivial holomorphic map (which is not necessarily a diffeomorphisms) the formula (E5) gives a function which is regular away from a finite set where vanishes. The function will generate a -homogeneous solution of the Navier-Stokes equations in the region , where . However, the vector field will not be locally square integrable in , except for the case of Landau’s solutions, when does not vanish at any point.

5. Open problems. An interesting problem is to try to repeat the above analysis when is replaced by the half-space and the boundary condition is imposed on . A simple calculation shows that there are no non-trivial axi-symmetric -homogeneous solutions in that case. However, it seems to be an open problem if this conclusion is still true without assuming the rotational symmetry. We refer the reader to the very interesting paper [Se], where a related situation is studied in a different context.

Another interesting question is the following: Among smooth vector fields in satisfying for some , are the Landau solutions the only ones which satisfy the Navier-Stokes equations (NSE) in ? Such questions are relevant for the problem of asymptotic behavior of steady-state solutions in exterior domains mentioned in Section 3. A first natural step in addressing this question is to look at possible infinitesimal deformations of Landau solutions in the above class. This leads to linear equations which can be reduced to ODEs by classical methods of separation of variables, due to the symmetries of Landau’s solutions. Based on numerical experiments with these ODEs, the author conjectures that the Landau solutions are rigid with respect to infinitesimal deformations, i. e. it seems that there are no new solutions bifurcating from Landau’s solutions.

Acknowledgement

The research was supported in part by a grant from the National Science Foundation.

REFERENCES

[B] Batchelor, G. K. An Introduction to Fluid Dynamics, Cambridge University Press, 1974 paperback edition.

[CK] Cannone, M. and Karch, G. Smooth or singular solutions to the Navier-Stokes system? J. Differential Equations 197 (2004), no. 2, 247–274.

[CY] Chang, S.Y.A. and Yang, P.C.The inequality of Moser and Trudinger and applications to conformal geometry, Communications on Pure and Applied Mathematics, Vol. LVI, 1135–1150, 2003.

[DFN] Dubrovin, B.A., Fomenko, A.T. and Novikov, S.P. Modern Geometry – methods and applications, Springer 1984–1990.

[FR] Frehse, J, and Ružička, M. Existence of regular solutions to the steady Navier-Stokes equations in bounded six-dimensional domains, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 23 (1996), no. 4, 701–719 (1997).

[G] Galdi, G.P. Introduction to the Mathematical Theory of the Navier-Stokes Equations, Volume II, Springer 1994.

[L] Landau, L.D. A new exact solution of the Navier-Stokes equations, Dokl. Akad. Nauk SSSR, 43, 299, 1944.

[LL] Landau, L.D. and Lifschitz, E.M. Fluid Mechanics, second edition, Butterworth-Heinemann, 2000 paperback reprinting.

[Le] Leray, J. Etude de Diverses Équations Intégrales non Linéaires et de Quelques Problèmes que Pose l’ Hydrodynamique, J. Math. Pures Appl., 12, 1-82, 1933.

[Se] Serrin, J. The swirling vortex, Philosophical Transactions of the Royal Society of London, Volume 271, 325–360, 1972.

[Sq] Squire, H.B. The round laminar jet, Quart. J. Mech. Appl. Math. 4, (1951). 321–329.

[St] Struwe, M. personal communication, 1997.

[Sv] Šverák, V. unpublished note

[T] Tsai, T.P. Thesis, University of Minnesota, 1998.

[TX] Tian, G, and Xin, Z. One-point singular solutions to the Navier-Stokes equations, Topol. Methods Nonlinear Anal. 11 (1998), no. 1, 135–145.