The enigma of the equations of fluid motion: A survey of existence and regularity results

Transcription

1 The enigma of the equations of fluid motion: A survey of existence and regularity results RTG summer school: Analysis, PDEs and Mathematical Physics The University of Texas at Austin Lecture 1 1

2 The review presented here is far from being exhaustive and it is beyond the scope of this lecture to describe all the large body of work on the fluid equations. 2

3 Outline of the lecture 1. The equations of fluid motion 2. Methods for obtaining solutions to a deterministic nonlinear PDE 3. A review of existence and regularity results for Euler and Navier-Stokes equations 3

4 1 The equations of fluid motion The equations that describe the most fundamental properties of viscous, incompressible fluids are the Navier-Stokes equations: u t + (u )u = p + ν u + f, (1.1) u = 0, (1.2) with initial conditions u(x, 0) = u 0 (x), (1.3) for an unknown velocity vector u(x, t) = (u i (x, t)) 1 i n R n and pressure p = p(x, t) R, for position x R n and time t [0, ). Here = P n i=1 2 x 2 i is the Laplacian in the space variable (u )u = ( P n j=1 u j u i x j ) 1 i n u = div u = P n i=1 u i x i 4

5 Figure 1: Bust of Claude-Louis Navier ( ) at the École Nationale des Ponts et Chaussées; Photo is a courtesy of Wikipedia 5

6 Figure 2: The photo appears in the article by Susan Friedlander and Marco Cannone Navier: Blow-up and Collapse, Notices of the AMS, January

7 Figure 3: Sir George Gabriel Stokes ( ); Photo is a courtesy of Wikipedia 7

8 OUTSTANDING OPEN PROBLEM: Global existence of physically reasonable a solutions to the Navier-Stokes equations in 3D In 2D the positive answer has been known for a long time (Leray s thesis in 1933 and the work of Ladyzhenskaya in 1960 s). a A solution is considered to be physically reasonable if u, p C (R n [0, )) and R R n u(x, t) 2 dx < C, for all t 0. 8

9 Why is the global existence problem for 3D Navier-Stokes difficult? Some answers to the above question: turbulence (the behavior of 3D Navier-Stokes at fine scales is much more unstable than at coarse scales) super-critical equation (the available globally controlled quantities for the Navier-Stokes are super-critical with respect to scaling) 9

10 Figure 4: Turbulence in the tip vortex from an airplane wing; Photo is a courtesy of Wikipedia 10

11 The Euler equations are derived for an incompressible, inviscid fluid with constant density (known as an ideal fluid). They are given by: u t + (u )u = p + f (1.4) u = 0, (1.5) with initial conditions u(x, 0) = u 0 (x). 11

12 Figure 5: A portrait of Leonhard Euler ( ) by Emanuel Handmann; Photo is a courtesy of Wikipedia 12

13 OUTSTANDING OPEN PROBLEM: Global existence of physically reasonable solutions to the Euler equations in 3D In 2D the positive answer have been known for a long time (see e.g. the work of Wolibner 1933, Yudovich 1963). 13

14 What is a solution to a nonlinear PDE? Answer varies depending on what we require from a solution. Some properties that one desires from a solution: 1. Existence: for any choice of initial data u 0 Y, there exists a positive time T = T (u 0 ), such that a solution to the initial value problem exists on the time interval [0, T ] 2. Uniqueness: the solution is unique in a certain solution class, for example in C 0 t Y x ([0, T ] R n ) 3. Continuous dependence on the initial data: the solution map from Y x to C 0 t Y x ([0, T ] R n ) depends continuously on the initial data. In other words, small perturbations in the initial data lead to small changes in the solution. 4. Persistence of regularity: the solution is always as smooth as the initial data. 5. Approximability by smooth solutions: the solutions can be written as the limit (in some topology) of smoother solutions. If T can be taken arbitrarily large, we call such a solution global in time. 14

15 A classical solution: a solution which has so much differentiability and decay to justify all the formal manipulations with the equations. More precisely, a classical solution to the Navier-Stokes equations is a pair of function (u, p) satisfying the system (1.1) - (1.3) for which all the terms appearing in the equations are continuous functions of their arguments. 15

16 Conserved quantities We recall two properties of the Euler and Navier-Stokes equations: (a) The Divergence free condition u = 0 (b) The skew-symmetry property (u )u, u L 2 (R n ) = 0 As a consequence it is easy to see that classical solutions to the Euler equations u + (u )u = p t satisfy conservation of energy: u(, T ) L 2 = u 0 L 2, while classical solutions to the Navier-Stokes equations u t satisfy decay of energy: + (u )u + p = ν( )u u(, T ) 2 L 2 = u 0 2 L 2 2 Z T 0 ν ( )u, u. 16

17 Scaling invariance If the pair (u(x, t), p(x, t) solves the Navier-Stokes equations (1.1) in R n then (u λ (x, t), p λ (x, t)) with u λ (x, t) = 1 λ u( x λ, t λ 2 ), p λ (x, t) = 1 λ 2 p( x λ, t λ 2 ) is a solution to the system (1.1) for the initial data u 0,λ = 1 λ u 0( x λ ). Moreover u λ 0 Ḣs = λ n 2 1 s u 0 Ḣs. Hence such a scaling leaves the Ḣs c invariant. We refer to regularities with s c = n 2 1 norm of a solution s > s c s = s c s < s c as subcritical (as λ, u λ 0 Ḣs 0; the problem is a small perturbation of the linear ) as critical (large date global existence open) as super-critical (large data global existence completely open!) 17

18 Why do we say that the 3D Navier-Stokes is a super-critical equation? The energy for the 3D Navier-Stokes u(, t) L 2 is super-critical with respect to the scaling invariant regularity a Ḣ 1/2. a Note, H s space is equipped with the norm f H s x = (1 + ξ 2 ) s 2 ˆf L 2 ξ, while Ḣ s space is equipped with the norm f Ḣs x = ξ s ˆf L 2 ξ. 18

19 2 Methods for obtaining solutions to a deterministic nonlinear PDE (A) Construct explicit solutions (B) Energy methods (C) Iterative methods 19

20 Energy methods: one integrates by parts the IVP to obtain an a priori bound of the type sup 0 t T u(, t) Y C(T, u 0 ). (2.1) Then one uses a weaker notion of a solution by, for example, weakening the nonlinear term strengthening the linear term (introducing hyper-dissipation) performing a discretization of spatial scales to obtain a sequence for which (2.1) is valid and takes a limit (in some topology) to obtain a solution. Bad news: in order to ensure that the limit is smooth we need convergence in a strong topology (critical or sub-critical). Hence one obtains weak solutions whose uniqueness and regularity are open questions. 20

21 Iterative methods: treat the nonlinear term N(u) of the IVP Lu = N(u), u(x, 0) = u 0 as a perturbation. More precisely, one starts with the solution u (0) to the linear problem Lu = 0, u(x, 0) = u 0, and then forms successive approximations u (1), u (2),... by solving the inhomogeneous linear problems: Lu (j+1) = N(u (j) ), u (j+1) (0) = u 0. Goal: establish convergence (in a suitable space) of u (j) to a solution u of the original IVP via the fixed point method. Bad news: the fixed point method works when the problem is a small perturbation of the linear one, for example, short times or small data. 21

22 3 A review of existence and regularity results for Euler and Navier-Stokes equations 3.1 Weak solutions Idea behind weak solutions: multiply the equation by the test function, then integrate and use the integration by parts to move the derivatives to fall on the test function. Strategy: establish global existence of weak solutions, and prove their uniqueness and regularity. 22

23 Definition (Weak solution) A Leray-Hopf weak solution of the Navier-Stokes equations in R 3 (0, T ) is a vector u(x, t) such that a u L ((0, T ); PL 2 ) L 2 ((0, T ); PH 1 ), and Z T 0 ( u, t φ + (u )u, φ + u, φ ds = 0 for all φ Ċ 0, Z 1 t 2 u(, t) 2 L 2 + u(, t) 2 L u(, 0) 2 L2, for all t [0, T ]. Leray [1934] and Hopf [1951] proved global existence of weak solutions in R 3 (0, ) under assumption that u(x, 0) PL 2. a Here P is a projection into divergence-free vector fields. 23

24 Open questions: Uniqueness and regularity of Leray-Hopf weak solutions. Some known results: due to Prodi [1959], Serrin [1963], Ladyzhenskaya [1967] and Escauriaza-Seregin-Šverak [2003] could be summarized in the following way: If u 1 and u 2 are two Leray-Hopf solutions to the Navier-Stokes equations with u 1 (x, 0) = u 2 (x, 0) PL 2 and if for some T > 0 the velocity field u i, i = 1, 2 satisfies u i L l tl s x(r 3 (0, T )) with 3 s + 2 l = 1, s [3, + ] then u i is a smooth function in R 3 (0, T ) and u 1 = u 2 in R 3 (0, T ). Bad news in 3D: it is known that Leray-Hopf solutions are in L l tl s x(r 3 (0, T )) with 3 s + 2 l = 3, s [2, 6]. 2 Good news in 2D: Leray-Hopf solutions are unique and regular in 2D. 24

25 Uniqueness of weak solution for the Euler equations is false! Scheffer [1993] and Shnirelman [1997] constructed weak solutions of the Euler equations on R 2 R with compact support in space-time. In particular, the weak solution corresponding to the zero initial velocity is not unique. 25

26 Definition (Suitable weak solutions) Let Ω be an open set in R 3. We say that a pair (u, p) is a suitable weak solution to the Navier-Stokes equations in Ω ( T 1, T ) if: u L t L 2 x(ω ( T 1, T )) L 2 t Hx(Ω 1 ( T 1, T )), p L 2 3 (Ω ( T1, T ), u and p satisfy the Navier-Stokes equations in the sense of distributions, and u and p satisfy local energy inequality: Z Z φ u(x, t) 2 dx + 2 φ u 2 dxdt Ω Z Ω ( T 1,t) Ω ( T 1,t) ( u 2 ( φ + t φ) + u φ( u 2 + 2p))dxdt, for a.a. t ( T 1, T ) and for all nonnegative functions φ Ċ 0. 26

27 Partial regularity results for suitable weak solutions are obtained by Scheffer [ ], Caffarelli-Kohn-Nirenberg [1982], Lin [1998], Vasseur [2005]. By a partial regularity result, one means an estimate for the dimension of the singular set. Here, the singular set of a suitable weak solution u is the set of all points (x, t) R 3 R such that u is unbounded in every neighborhood. of (x, t). Caffarelli, Kohn, and Nirenberg showed that the singular set of a suitable weak solution to the Navier-Stokes equations has parabolic Hausdorff dimension at most 1. 27

28 The parabolic analogue of Hausdorff dimension is defined using parabolic cylinders Q r = B r I r, where B r is a ball in R n of radius r, while I r is an interval in R of length r 2. Given any set A R n R, the k-dimensional parabolic Hausdorff measure of A, P k (A), is given by: where P k (A) = lim δ 0 P k δ (A), X Pδ k (A) = inf{ i=1 r k i Q r1, Q r2,... cover A, and each r i < δ}. Having defined Hausdorff measure we can speak about the Hausdorff dimension which is given by: inf k. P k (A)=0 28

29 3.2 Mild solutions Idea behind mild solutions: solutions to the corresponding integral equation. Strategy: establish local existence of solutions and use some a priori control to prove that they are global in time. 29

30 Definition A mild solution to the Navier-Stokes equations is a solution to the integral equation u = e t u 0 Z t 0 e (t s) P (u u) ds, u 0 = 0, such that u(x, t) C([0, T ]; PX), where X is a Banach space of distributions and P is a projection into divergence-free vector fields. The notion of mild solutions was introduced by Browder [1964] and Kato-Fujita [1962] in a more abstract context. Existence of mild solutions is obtained through the Picard fixed point argument. 30

31 Sub-critical spaces: L p with p > n, H s with s > n 2 1 Existence results in the period due to Kato-Fujita, Fabes-Jones-Riviere, Kato, Giga, Kato-Ponce, Cannonne-Meyer Local existence of smooth solutions of the Navier-Stokes equations Global existence of smooth solutions of the Navier-Stokes equations for small initial data in Sobolev spaces Long time behavior results that give decay as t of solutions to the Navier-Stokes in sub-critical spaces due to Kato [ Lebesgue spaces], Kato [ Sobolev spaces] 31

32 Critical spaces: Ḣ n 2 1 L n Ḃ 1+ n p, p p< BMO 1. (3.1) Existence results in the period from due to Kato, Giga-Miyakawa, C.P. Calderon, Taylor, Kato-Ponce, Planchon, Cannone, Koch-Tataru local existence of smooth solutions of the Navier-Stokes equations (BMO 1 replaced with V MO 1 ) global existence of smooth solutions of the Navier-Stokes equations for small initial data Higher regularity + long time behavior results that give decay as t of solutions to the Navier-Stokes in critical spaces obtained in last few years by Schonbeck, Giga-Sawada, Dong-Du, Sawada, Gallagher-Iftimie-Planchon, Auscher-Dubois-Tchamitchian, Miura-Sawada, Germanin-P.-Staffilani. 32

33 3.3 Blow-up criteria Blow-up criteria: results that identify certain norms that control the breakdown of smooth solutions to Euler and Navier-Stokes equations. Analytical and numerical results suggest the connection between the accumulation of vorticity ω = u and development of finite time singularities for the 3D Euler equations. This was made rigorous by Beale, Kato, Majda and Constantin, Fefferman, Majda. 33

34 A version of the local existence theorem for Euler equations can be stated as follows: Theorem 3.1. Suppose an initial velocity field u 0 is specified in H s (R 3 ), s 3, with u 0 H 3 (R 3 ) N 0, for some N 0 > 0. Then there exists T 0 > 0, depending only on N 0, so that Euler equations have a solution in the class u C([0, T ]; H s (R 3 )) C 1 ([0, T ]; H s 1 (R 3 )) (3.2) at least for T = T 0 (N 0 ). This theorem does not say if solutions actually lose their regularity. 34

35 In 1984 Beale, Kato and Majda made the connection between the accumulation of vorticity and development of finite time singularities rigorous by proving: Theorem 3.2. Let u be a solution of Euler equations and suppose there is a time T such that the solution cannot be continued in the class (3.2) to T = T. Assume that T is the first such time. Then and in particular Z T 0 ω(t) L dt =, lim sup t T ω(t) L =. Constantin, Fefferman, Majda [1996] proved a criteria in which the direction of vorticity ω ω has an important role. 35

36 2D Euler equations from the perspective of Beale-Kato-Majda result In 2D the global solvability of Euler equations is guaranteed since L norm of vorticity is conserved. As in 3D, the proof of Beale, Kato, Majda gives a double exponential growth in time of the Sobolev norm of velocity. Open question: Is this growth estimate sharp in 2D? 36

37 3.4 Other strategies: Analysis of the surface 2D quasi-geostrophic equation (Caffarelli-Vasseur, Kiselev-Nazarov-Volberg) Analysis of complex valued Navier-Stokes equations (Li-Sinai) Shell models for the equations of fluid motion (Katz-P., Nazarov, Kiselev-Zlatoš, Cheskidov-Friedlander-P.) 37