EXISTENCE AND SMOOTHNESS OF THE NAVIER–STOKES …

1 EXISTENCE AND SMOOTHNESS OF THENAVIER STOKES EQUATIONCHARLES L. FEFFERMANThe Euler and navier Stokes equations describe the motion of a fluid inRn(n= 2 or 3). These equations are to be solved for an unknown velocity vectoru(x, t) = (ui(x, t))1 i n Rnand pressurep(x, t) R, defined for positionx Rnand timet 0. We restrict attention here to incompressible fluids filling all Stokesequations are then given by tui+n j=1uj ui xj= ui p xi+fi(x, t)(x Rn, t 0),(1)divu=n i=1 ui xi= 0(x Rn, t 0)(2)with initial conditions(3)u(x,0) =u (x) (x Rn).Here,u (x) is a given,C divergence-free vector field onRn, fi(x, t) are the com-ponents of a given, externally applied force ( gravity), is a positive coefficient(the viscosity), and =n i=1 2 x2iis the Laplacian in the space variables. TheEulerequationsare equations (1), (2), (3) with set equal to (1) is just Newton s lawf=mafor a fluid element subject to the ex-ternal forcef= (fi(x, t))1 i nand to the forces arising from pressure and (2) just says that the fluid is incompressible.

2 For physically reasonablesolutions, we want to make sureu(x, t) does not grow large as|x| . Hence, wewill restrict attention to forcesfand initial conditionsu that satisfy(4)| xu (x)| C K(1 +|x|) KonRn,for any andKand(5)| x mtf(x, t)| C mK(1 +|x|+t) KonRn [0, ),for any , m, accept a solution of (1), (2), (3) as physically reasonable only if it satisfies(6)p, u C (Rn [0, ))and(7) Rn|u(x, t)|2dx < Cfor allt 0 (bounded energy).Alternatively, to rule out problems at infinity, we may look for spatially periodicsolutions of (1), (2), (3). Thus, we assume thatu (x), f(x, t) satisfy(8)u (x+ej) =u (x), f(x+ej, t) =f(x, t) for 1 j n12 CHARLES L. FEFFERMAN(ej=jthunit vector inRn).In place of (4) and (5), we assume thatu is smooth and that(9)| x mtf(x, t)| C mK(1 +|t|) KonR3 [0, ),for any , m, then accept a solution of (1), (2), (3) as physically reasonable if it satisfies(10)u(x, t) =u(x+ej, t) onR3 [0, ) for 1 j nand(11)p, u C (Rn [0, )).A fundamental problem in analysis is to decide whether such smooth, physicallyreasonable solutions exist for the navier Stokes equations.]]]]]

3 To give reasonable lee-way to solvers while retaining the heart of the problem, we ask for a proof of oneof the following four statements.(A) EXISTENCE and SMOOTHNESS of navier Stokes solutions >0 andn= 3. Letu (x) be any smooth, divergence-free vector field satisfying (4).Takef(x, t) to be identically zero. Then there exist smooth functionsp(x, t), ui(x, t)onR3 [0, ) that satisfy (1), (2), (3), (6), (7).(B) EXISTENCE and SMOOTHNESS of navier Stokes solutions inR3 >0 andn= 3. Letu (x) be any smooth, divergence-free vector field satisfying(8); we takef(x, t) to be identically zero. Then there exist smooth functionsp(x, t),ui(x, t) onR3 [0, ) that satisfy (1), (2), (3), (10), (11).(C) Breakdown of navier Stokes solutions >0 andn= there exist a smooth, divergence-free vector fieldu (x) onR3and a smoothf(x, t) onR3 [0, ), satisfying (4), (5), for which there exist no solutions (p, u)of (1), (2), (3), (6), (7) onR3 [0, ).(D) Breakdown of navier Stokes Solutions onR3 >0 andn= 3.]]]]

4 Then there exist a smooth, divergence-free vector fieldu (x) onR3and asmoothf(x, t) onR3 [0, ), satisfying (8), (9), for which there exist no solutions(p, u) of (1), (2), (3), (10), (11) onR3 [0, ).These problems are also open and very important for the Euler equations ( = 0),although the Euler equation is not on the Clay Institute s list of prize me sketch the main partial results known regarding the Euler and navier Stokes equations, and conclude with a few remarks on the importance of the two dimensions, the analogues of assertions (A) and (B) have been knownfor a long time (Ladyzhenskaya [4]), also for the more difficult case of the Eulerequations. This gives no hint about the three-dimensional case, since the maindifficulties are absent in two dimensions. In three dimensions, it is known that (A)and (B) hold provided the initial velocityu satisfies a smallness condition. Forinitial datau (x) not assumed to be small, it is known that (A) and (B) hold (alsofor = 0) if the time interval [0, ) is replaced by a small time interval [0, T),withTdepending on the initial data.]]]]

5 For a given initialu (x), the maximumallowableTis called the blowup time. Either (A) and (B) hold, or else there isa smooth, divergence-freeu (x) for which (1), (2), (3) have a solution with a finiteblowup time. For the navier Stokes equations ( >0), if there is a solution withEXISTENCE AND SMOOTHNESS OF THE navier STOKES EQUATION3a finite blowup timeT, then the velocity (ui(x, t))1 i 3becomes unbounded nearthe blowup unpleasant things are known to happen at the blowup timeT, ifT < .For the Euler equations ( = 0), if there is a solution (withf 0, say) with finiteblowup timeT, then the vorticity (x, t) = curlxu(x, t) satisfies T0{supx R3| (x, t)|}dt= (Beale Kato Majda),so that the vorticity blows up numerical computations appear to exhibit blowup for solutions of theEuler equations, but the extreme numerical instability of the equations makes itvery hard to draw reliable above results are covered very well in the book of Bertozzi and Majda [1].

6 Starting with Leray [5], important progress has been made in understandingweaksolutionsof the navier Stokes equations. To arrive at the idea of a weak solution ofa PDE, one integrates the equation against a test function, and then integrates byparts (formally) to make the derivatives fall on the test function. For instance, if (1)and (2) hold, then, for any smooth vector field (x, t) = ( i(x, t))1 i ncompactlysupported inR3 (0, ), a formal integration by parts yields(12) R3 Ru tdxdt ij R3 Ruiuj i xjdxdt= R3 Ru dxdt+ R3 Rf dxdt R3 Rp (div ) that (12) makes sense foru L2,f L1,p L1, whereas (1) makes senseonly ifu(x, t) is twice differentiable inx. Similarly, if (x, t) is a smooth function,compactly supported inR3 (0, ), then a formal integration by parts and (2)imply(13) R3 Ru 5x dxdt= solution of (12), (13) is called aweak solutionof the navier Stokes long-established idea in analysis is to prove EXISTENCE and regularity of solutionsof a PDE by first constructing a weak solution, then showing that any weak solutionis smooth.

7 This program has been tried for navier Stokes with partial in [5] showed that the navier Stokes equations (1), (2), (3) in three spacedimensions always have a weak solution (p, u) with suitable growth of weak solutions of the navier Stokes equation isnotknown. For theEuler equation, uniqueness of weak solutions is strikingly false. Scheffer [8], and,later, Schnirelman [9] exhibited weak solutions of the Euler equations onR2 Rwith compact support in spacetime. This corresponds to a fluid that starts fromrest at timet= 0, begins to move at timet= 1 with no outside stimulus, andreturns to rest at timet= 2, with its motion always confined to a ballB [7] applied ideas from geometric measure theory to prove a partialregularity theorem for suitable weak solutions of the navier Stokes L. FEFFERMANC affarelli Kohn Nirenberg [2] improved Scheffer s results, and Lin [6] sim-plified the proofs of the results in Caffarelli Kohn Nirenberg [2]. The partial regu-larity theorem of [2], [6] concerns a parabolic analogue of the Hausdorff dimensionof the singular set of a suitable weak solution of navier Stokes.

8 Here, thesingu-lar setof a weak solutionuconsists of all points (x , t ) R3 Rsuch thatuis unbounded in every neighborhood of (x , t ). (If the forcefis smooth, and if(x , t ) doesn t belong to the singular set, then it s not hard to show thatucan becorrected on a set of measure zero to become smooth in a neighborhood of (x , t ).)To define the parabolic analogue of Hausdorff dimension, we useparabolic cylin-dersQr=Br Ir R3 R, whereBr R3is a ball of radiusr, andIr Ris aninterval of lengthr2. GivenE R3 Rand >0, we setPK, (E) = inf{ i=1rKi:Qr1, Qr2, coverE,and eachri< }and then definePK(E) = lim 0+PK, (E).The main results of [2], [6] may be stated roughly as (A) Letube a weak solution of the navier Stokes equations, satisfyingsuitable growth conditions. LetEbe the singular set ofu. ThenP1(E) = 0.(B) Given a divergence-free vector fieldu (x)and a forcef(x, t)satisfying (4)and (5), there exists a weak solution of navier Stokes (1), (2), (3) satisfying thegrowth conditions in (A).

9 In particular, the singular set ofucannot contain a spacetime curve of the form{(x, t) R3 R:x= (t)}. This is the best partial regularity theorem known sofar for the navier Stokes equation. It appears to be very hard to go me end with a few words about the significance of the problems posed are important and hard to understand. There are many fascinating prob-lems and conjectures about the behavior of solutions of the Euler and navier Stokesequations. (See, for instance, Bertozzi Majda [1] or Constantin [3].) Since we don teven know whether these solutions exist, our understanding is at a very primitivelevel. Standard methods from PDE appear inadequate to settle the problem. In-stead, we probably need some deep, new [1] A. Bertozzi and A. Majda, vorticity and Incompressible Flows, Cambridge U. Press, Cam-bridge, 2002.[2] L. Caffarelli, R. Kohn, and L. Nirenberg,Partial regularity of suitable weak solutions of theNavier Stokes equations, Comm. Pure & Appl.

10 (1982), 771 831.[3] P. Constantin,Some open problems and research directions in the mathematical study offluid dynamics, in Mathematics Unlimited 2001 and Beyond, Springer Verlag, Berlin, 2001,353 360.[4] O. Ladyzhenskaya,The Mathematical Theory of Viscous Incompressible Flows(2nd edition),Gordon and Breach, New York, 1969.[5] J. Leray,Sur le mouvement d un liquide visquex emplissent l espace, Acta Math. (1934),193 248.[6] Lin,A new proof of the Caffarelli Kohn Nirenberg theorem, Comm. Pure. & (1998), 241 257.[7] V. Scheffer,Turbulence and Hausdorff dimension, in Turbulence and the navier Stokes Equa-tions, Lecture Notes in , Springer Verlag, Berlin, 1976, 94 AND SMOOTHNESS OF THE navier STOKES EQUATION5[8] V. Scheffer,An inviscid flow with compact support in spacetime, J. Geom. Analysis3(1993),343 401.[9] A. Shnirelman,On the nonuniqueness of weak solutions of the Euler equation, Comm. Pure& Appl. (1997), 1260 further conditionp(x+ej,t) =p(x,t) should be made explicit in Eqn (8).