LIOUVILLE THEOREM FOR 2D NAVIER-STOKES EQUATIONS. We consider the standard Navier-Stokes equations (NSE) 1 for incompressible (Newtonian) fluids:

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1 LIOUVILLE THEOREM FOR 2D NAVIER-STOKES EQUATIONS GABRIEL KOCH We consider the standard Navier-Stokes equations (NSE) 1 for incompressible (Newtonian) fluids: } u (NSE) t ν u + u u + p = 0 in R u = 0 n (t 1, t 2 ). We are interested in the following problem: { Is it possible to describe all global solutions of NSE in R (L) n R which are bounded, i.e., u(x, t) C everywhere in R n R? (One may modify the question by putting various other restrictions on (L); for example, one can consider only steady-state solutions, or solutions with finite rate of dissipation or belonging to various other function spaces, etc.) We have proved a positive result for dimension n = 2 which we will discuss below, but let us begin by mentioning why the basic problem is interesting. Generally speaking, it is known that Liouville properties are related to regularity (see, e.g., [6] and [9]), and we expect that progress in 3D Liouville problems for NSE, even in the steady-state case, will lead to progress in regularity theory. The difficulties which come up in the 3D regularity theory for NSE appear in the Liouville problem in a slightly different light, so hopefully studying the 3D Liouville problem will motivate some new ideas for the 3D regularity problem. We note that our 2D Liouville theorem is closely related to the fact that in 2D we have regularity for NSE (see e.g., [7]). We will highlight now some particular instances in the literature relating Liouville theorems to regularity; one of particular interest is in the 1987 paper of Giga and Kohn ([5]). They are interested in characterizing blow-up for non-negative solutions to the non-linear heat equation, u t u u p 1 u = 0 in R n (0, T ). (Note that for p = 3, the equation has the same scaling symmetries as does NSE.) Their result limits the rate of blow up at time 1 We generally assume for simplicity that ν = 1, as this does not change anything from the point of view of regularity, or in any details of the proofs. 171

2 172 WOMEN IN MATHEMATICS: MAY 18 20, 2006 t = T, that is, not to exceed a rate of the order 1 (T t) 1 p 1 in the spatial supremum-norm, when 1 < p < n+2 n 2 or n 2. The argument is by contradiction, that if the supremum norm exceeds such a growth rate, it is possible (by various changes of variables and re-scalings) to construct a non-trivial, non-negative steady state solution as a limit of certain sequences obtained from the growth assumption; that is, there would exist a function v 0, v 0 satisfying v v p = 0 in R n. However, this contradicts a Liouville-type theorem of Gidas and Spruck (see [2], [3]), which asserts exactly the triviality of such solutions which are globally bounded from below. Another interesting and somewhat related instance is the recent results regarding a blow-up question which had been long left open in the study of NSE, dating back to the 1934 paper of Leray [8]. This concerns the (non-trivial) self-similar, singular solutions, which were supposed to quite potentially exist. These are solutions of the form ( ) 1 x u(x, t) = U 2a(T t) 2a(T t) for non-constant U, which then gives the equation ν U + a(u + (y )U) + (U )U + P = 0 for the divergence-free vector-field U. The triviality of such solutions was shown first in 1995 by Nečas, Růžička, and Šverák [10] under global energy assumptions, and in 1998 by Tsai [13] under only local energy assumptions. In [13], for example, Tsai proved a Liouville-type theorem for the equation for U, asserting the triviality of globally bounded U, which immediately denies the possibility of Leray s self-similar type of blow-up for u (except for the trivial cases). Incidentally, Tsai uses a similar method to that used in an early paper of Giga and Kohn in the same series of papers concerning the nonlinear heat equation (see [4]). Having established now our interest in the problem, we explicitly state the progress which we have made. 2 The result we obtained is the following theorem. Theorem. Let n = 2, and let u be a global bounded solution of NSE in R 2 R. Then u is of the form for some bounded u 0 : R R 2. u(x, t) = u 0 (t) 2 Collaboration of G. Koch, N. Nadirashvili, and V. Šverák, to be submitted

3 THE LEGACY OF LADYZHENSKAYA AND OLEINIK 173 (We note that there are no assumptions on the pressure.) In particular, in the steady-state case, globally bounded velocities are constant. The main idea of the proof is that, in two dimensions, the vorticity ω (= x1 u 2 x2 u 1 ) satisfies ω t ω + u ω = 0, a parabolic scalar equation with bounded first-order coefficients. As a first step, we can formally apply the parabolic Harnack inequality to this equation, together with some other properties of ω to show that ω must vanish everywhere. Once we know ω 0, the proof is finished easily. To make this rigorous, we need to prove that, under our assumptions, u is sufficiently regular. This is complicated by the presence of solutions of the form u(x, t) = a(t), 3 where a(t) may not have any regularity. Such solutions cannot be avoided if we do not want to make assumptions about the pressure. The necessary regularity for u is obtained by relatively standard bootstrapping techniques, but since we do not impose conditions on the pressure, the details of the bootstrapping arguments and regularity estimates are somewhat non-trivial. We also make some use of the well-known results (see, e.g., [1], [8], [12]) that for minimally regular solutions to NSE, the velocity field u is smooth for each fixed time t, and any spatial derivative of u has the regularity Dx α u L loc (Rn+1 ). The problem (L) for n = 2 becomes more difficult if the spatial domain R 2 is replaced by an exterior domain. It is open even in the steady-state case and when the domain is taken to be Ω = R 2 \B 1 (0). Even under the additional assumptions u B1 = 0 and u 0 at infinity, it is unknown if u 0. (The situation may be extended similarly to the case of a half-plane.) Our proof relies heavily on the fact that the region has infinite diameter, with no boundary. The problem (L) in three dimensions is, of course, much harder. In fact, it is open even in the steady-state case and under the additional assumptions u 2 < and u(x) 0 as x. In that case, the quantity u 2 + p = θ satisfies the equation 2 θ + u θ = ω 2. This implies a maximum principle for θ, and we hoped that this could be helpful in showing that under the above assumptions, u 0. However, even this special case appears to be difficult, and some new ideas seem to be necessary to complete the proof. The best known result for dimension n = 3 is under the assumption u L 9 2 (or L 9 2 ɛ, but in that case the proof is quite easily accomplished); 3 Note that these solutions present an unavoidable obstacle to improving our result to a constant-in-time result.

4 174 WOMEN IN MATHEMATICS: MAY 18 20, 2006 bounded 4, L 9 2 steady-state solutions of NSE for n = 3 are constant. In the context of this result, it is interesting to mention a result concerning a model equation mentioned in the paper of Plecháč and Šverák [11]. The equation they consider is for κ 0: u (M) t u κ div u+(u )u+1 (div u)u = 0. 2 For large κ, (M) can be considered to approximate NSE in some sense; moreover, it shares the same scaling properties as NSE (if u(x, t) is a solution, then so is λu(λx, λ 2 t)), as well as the same type of energy estimate. For the model equation (M), it is noted in [11] that there are non-trivial, radial steady-state solutions v with the decay property ( ) v(x) = O x 2 3 as x ; in relation to the Liouville property mentioned above for the 3D NSE, we note that if we were to have a solution ṽ to NSE with ṽ x ( 2 3 +ɛ) at infinity for any positive ɛ, then we would have ṽ L 9 2 (R 3 ), and hence is trivial. This shows that the method of proof based on energy estimates cannot be used to improve the result for NSE. References [1] L. Caffarelli, R. Kohn, and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math. 35 (1982), no. 6, [2] B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math. 34 (1981), no. 4, [3], A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations 6 (1981), no. 8, [4] Y. Giga and R. V. Kohn, Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math. 38 (1985), no. 3, [5], Characterizing blowup using similarity variables, Indiana Univ. Math. J. 36 (1987), no. 1, [6] E. Giusti, Minimal Surfaces and Functions of Bounded Variation. Monographs in Mathematics, 80. Basel: Birkhaüser-Verlag, [7] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow. 2nd English ed., revised and enlarged. Translated from the Russian by Richard A. Silverman and John Chu. New York: Gordon and Breach, Science Publishers, [8] J. Leray, Sur le mouvement d un liquide visqueux emplissant l espace, Acta Math. 63 (1934), [9] J. Nečas, A necessary and sufficient condition for the regularity of weak solutions to nonlinear elliptic systems of partial differential equations, in Nonlinear Analysis (Berlin, 1979). Abh. Akad. Wiss. DDR, Abt. Math. Naturwiss. Tech., 1981, 2. Berlin: Akademie-Verlag, In fact, the boundedness condition is unnecessary, but makes the proof slightly simpler and relates this to our Liouville-type class of results.

5 THE LEGACY OF LADYZHENSKAYA AND OLEINIK 175 [10] J. Nečas, M. Růžička, and V. Šverák, On Leray s self-similar solutions of the Navier- Stokes equations, Acta Math. 176 (1996), no. 2, [11] P. Plecháč and V. Šverák, Singular and regular solutions of a nonlinear parabolic system, Nonlinearity 16 (2003), no. 6, [12] J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal. 9 (1962), [13] T.-P. Tsai, On Leray s self-similar solutions of the Navier-Stokes equations satisfying local energy estimates, Arch. Rational Mech. Anal. 143 (1998), no. 1, School of Mathematics; University of Minnesota, Twin Cities; Minneapolis, MN USA current address: Department of Mathematics; University of Chicago; 5734 S. University Avenue; Chicago, IL USA address: koch@math.uchicago.edu