A new approach to the study of the 3D-Navier-Stokes System

Size: px

Start display at page:

Download "A new approach to the study of the 3D-Navier-Stokes System"

Marsha Kelly Wilkinson
5 years ago
Views:

1 Contemporary Mathematics new approach to the study of the 3D-Navier-Stokes System Yakov Sinai Dedicated to M Feigenbaum on the occasion of his 6 th birthday bstract In this paper we study the Fourier transform of the 3D-Navier- Stokes-System without external forcing on the whole space The properties of solutions depend very much on the space in which the system is considered In this paper we deal with the space Φ(α, α of functions v(k = c(k/ k α where α = 2 + ɛ, ɛ > and c(k is bounded, sup k \ c(k < We construct the power series which converges for small t and gives solutions of the system for bounded intervals of time These solutions can be estimated at infinity (in k-space by exp{ const t k } The spaces Φ(α, α and the ruling parameter for the Navier-Stokes system in Φ(α, α Consider 3D-Navier-Stokes System (NSS on for incompressible free fluids fter Fourier transform and an elementary transformation it becomes a non-linear non-local equation for an unknown function v(k, t with values in having the form t ( v(k, t = e t k 2 v(k, + i e k 2 (t s ds k, v(k k, s P k v(k, s dk The function v(k, t must satisfy the condition v(k, t k for any k, k and t ; P k is the orthogonal projection to the subspace orthogonal to k; the viscosity is taken to be one, ie, ν = Classical solutions of ( on [, t ] are functions v(k, t, t t, for which all integrals in ( converge absolutely and ( becomes the identity There are several reasons by which it is natural to consider ( in the spaces of functions having singularities near k = or k = In this paper, we restrict ourselves to the spaces of functions v(k = c(k/ k α where α = 2 + ɛ, ɛ > and sufficiently small, c(k is continuous everywhere outside k = and uniformly bounded, ie, sup k \ c(k = c < (see [7], [] If the solution of ( belongs to Φ(α, α then v(k, t = c(k, t/ k α, t t, c(k, t k for any k \, t, and c(k, t satisfies the equation which is equivalent to (: t R3 (2 c(k, t = e k 2t c(k, +i k α e k 2 (t s k, c(k k, s P k c(k, s dk ds It is easy to check that for typical c Φ(α, α the initial condition has infinite energy and enstrophy c (copyright holder

2 2 YKOV SINI ssume that c(k, = and take a one-parameter family of initial conditions c (k, = c(k,, where is a parameter taking positive values In [] the local existence theorem for solution of (2 was proven Below we outline this proof and show that if λ = t ɛ/2 λ, where λ is an absolute constant which may depend on α, then there exists the unique solution of (2 in the corresponding time interval Usual arguments are based on the classical iteration scheme Put c ( (k, t = e k 2t c(k, and (3 c (n (k, t = c ( (k, t + i k α t e k 2 (t s ds (n k, c (k k, s P k c (n (k, s dk The first step in the proof of the convergence of the iterations c (n (k, t as n is to show that all c (n (k, t remain close to c( (k, t in the sense of the norm in Φ(α, α If c (n that c (n (4 c (n 2c( = sup s t, k \ c (n (k, s then we would like to show = 2 for all n By induction and with the use of (3 we can write c( + sup k \ s t k α ( e k 2t (c (n 2 The last integral satisfies the inequality R3 dk (5 B k 2α 3 dk Here and below the letter B with indices is used for various absolute constants which appear during the proofs These constants may depend on α Now, we have to show that sup k 2 α ( e k 2t (c (n 2 B c ( k By induction c (n 2c ( Therefore we have to show that sup k 2 α ( e k 2t 4 c ( k Consider two cases ( k 2 /t Then (2 k 2 /t Then Thus we can write k 2 α ( e k 2t k 4 α t t 4 α 2 + = t ɛ/2 k 2 α ( e k 2t k 2 α t α 2 2 = t ɛ/2 c (n +42 (c ( 2 t ɛ/2 B = ( + 4 t ɛ/2 B = ( + 4B λ We used the fact that c ( = c ( If λ < /(4B then c (n argument shows how the parameter λ arises 2c( This The next step in the proof of the existence of solutions is to show that the converge to a limit We have from (3 iterations c (n c (n (k, t c(n (k, t = i k α (n k, c [ t e k 2 (t s ds (k k, s c (n 2 (k k, s P k c (n (k, s dk

3 and (6 NEW PPROCH TO THE STUDY OF THE 3D-NVIER-STOKES SYSTEM 3 c (n + k, c (n 2 (k, t c(n (k, t 2 c (n From (5 and (6, (n c c(n 2 B (n c (k k, s ( (n P k c (k, s c (n 2 (k, s dk ] c (n 2 k α ( e k 2 t dk R 3 c (n 2 sup k α ( e k 2t k \ The same arguments as before give that sup k α ( e k 2t k \ k 2α 3 B 2 t ɛ/2 Therefore, for some constant B 3, (n c c(n B3 λ (n c c (n 2 k 2α 3 From the last inequality it follows that if λ is less than some absolute constant then the iteration scheme converges and gives the desired solution Thus λ is really a ruling parameter in the current situation The main purpose of this paper is to construct a general power series in λ which provides the solution of (2 with the given initial condition Write down the solution of (2 with the initial condition c(k,, c(k,, in the form ( t c (k, t = c(k, e t k 2 + e (t s k 2 λ p h p (k, s ds (7 ( = c(k, e t k 2 + p p t p e (t s k 2 s pɛ/2 h p (k, s ds where now λ = s ɛ/2 Substituting this expression into (2 we get the system of recurrent relations for h p Below we give the explicit formulas for h, h 2 and then the general formula for h p, p 3 We have (8 2 s ɛ/2 h (k, s = i 2 k α k, c(k k, P k c(k, e s k k 2 s k 2 dk R, 3 (9 3 s ɛ h 2 (k, s = and i 3 k α [ s s + s R3 s ɛ/2 ds k, h (k k, s P k c(k, e (s s k k 2 s k 2 dk 2 (s s 2 k 2 dk ] R3 s ɛ/2 2 ds k, c(k k, P k h (k, s 2 e s k k 2 ( p+ s pɛ/2 h p (k, s = i p+ k α [ s s (p /2 ds + s (p ɛ/2 2 ds 2 k, h p (k k, s P k c(k, e (s s k k 2 s k 2 dk k, c(k k, P k h p (k, e s k k 2 (s s 2 k 2 dk

4 4 YKOV SINI s s pɛ/2 p,p 2 p +p 2=p + ds s s p2ɛ/2 2 ds 2 ] R3 k, h p (k k, s P k h p2 (k, s 2 e (s s k k 2 (s s 2 k 2 dk Use the following nsatz: h p (k, s = s ɛ/2 k α g p (k s, s and in all integrals above make the change of variables: s = s s, s 2 = s s 2, k s = k, k s = k Thus h p (k, s = s ɛ/2 k α g p ( k, s Instead of (8, (9, ( we shall get the system of recurrent relations for the functions g p ( k, s: ( ( k k 2 s ɛ k α g ( k, s = i 2 k α s ɛ s, k P kc s, e k k 2 k 2 d k k k α k α or ( g ( k, s = i ( (, k P kc s, e k k 2 k 2 d k k k, α k α or 3 s 3ɛ/2 k α g 2 ( k, s = i 3 k α s 3ɛ/2 [ ( k, g ( k k s (, s s P kc k s, e ( s k k 2 k 2 d k s ɛ d s (2 g 2 ( k, s = + s ɛ s ɛ d s (, k α g k k 2 ( s 2 k 2 d k ] k k α ( k, g ( k k s (, s s k P kc s, e ( s k k 2 k 2 d k + s ɛ For general g p ( k, s, p 3, k ( k, α c, g k k 2 ( s 2 k 2 d k k k α p+ s p+ 2 ɛ k α g p ( k, s = i p+ s p+ 2 ɛ k α [ ( k, gp ( k k s (, s s P kc s pɛ/2 k s, d s or + s pɛ/2 + p, p 2 p +p 2=p e ( s k k 2 k 2 d k k α (, P kg p k k 2 ( s 2 k 2 d k s (p+ɛ/2 d s s (p2+ɛ/2 k k α ( k, gp ( k k s, s s P kg p2 ( k s ] 2, s s 2 e ( s k k 2 ( s 2 k 2 d k

5 NEW PPROCH TO THE STUDY OF THE 3D-NVIER-STOKES SYSTEM 5 (3 g p ( k, s = [ i s pɛ/2 d s + s pɛ/2 + p,p 2 p +p 2=p ( k, gp ( k k s (, s s k P kc s, e ( s k k 2 k 2 d k k α (, P k g p k k 2 ( s 2 k 2 d k s (p+ɛ/2 d s s (p2+ɛ/2 k k α ( k, gp ( k k s, s s P kg p2 ( k s ] ( s k k 2 ( s 2 k 2 d k These recurrent relations allow to express each g p (k s, s through the initial conditions c(k, It is easy to see that this expression will be the sum of not more than b p 4p-dimensional integrals containing products of c(, with different values of the arguments where b is some constant We shall discuss the related questions in another paper Write down the inequality where g p ( k, s C p f ( k e k 2 s/(p+ = C p f ( k e k 2 /(p+ f(x = { x x, x x The main result of this paper is the following theorem Main Theorem: The numbers C p can be chosen in such a way that (4 C p = B C p C p2 (p + (p 2 + (p + We prove the main theorem in the next section First we analyze p =, 2, 3 and then the general case p > 3 We use the identity (5 a k k 2 + a 2 k 2 = a a 2 k 2 + (a + a 2 a + a 2 k a 2 k a + a 2 valid for arbitrary k, k It follows easily from (4 that the C p grow no faster than exponentially (see 2, C p b b p 2 for some constants b, b 2 < depending on α Corollary: If t ɛ/2 < b 2 then the series (7 converges for every k \ It is interesting to remark that all but one of the terms of the series (7 have finite energy and enstrophy Other expansions for the Navier-Stokes system which are formal can be found in the monographs [4], [5] General approach to the existence problem for the Navier-Stokes System is discussed in [3] 2 The discussion of the results First we show that the constants C p grow not faster than exponentially Denote C p = C p (p + The numbers C p satisfy the relation C p = B p,p 2 p +p 2=p C p C p 2

6 6 YKOV SINI Using the induction, let us prove that C p ( B p (p + 3/2 for any p and some constant B For p = we can choose B ssuming that the last inequality is valid for all q < p we can write C p p B ( B (p + 3/2 (p 2 + B B 3/2 ( B p (p + 3/2 p,p 2 p +p 2=p Take B = B B This gives the result Now we see that the series (7 converges if λ = t ɛ/2 < ( B In many estimates done for the NSS system, people assumed that solutions v(k, t at infinity in k satisfy the inequality v(k, t e f(t k which is different from the diffusion-like asymptotics If this asymptotics represents the true decay of solutions, it is an interesting question how does it appear The series (7 sheds some light on this question We can write v(k, t Const B p λ p 2 e t k /(p+ (p + 3/2 p The usual asymptotical method shows that the largest term in this last sum is when t k 2 /(p max + 2 = ln(b 2 λ, ie, p max = t k / ln(b 2 λ and the whole sum behaves as e 2pmax ln(b2λ = e 2 t ln(b 2λ k This is the asymptotics which was mentioned above It also shows that in the domain of convergence of the series the enstrophy of the solution is finite for t > This type of decay of solutions in various situations was obtained earlier in the works [2], [6] I would like to thank C Fefferman and V Yakhot for many useful discussions and remarks I thank also S Friedlander, N Pavlovic and E Siggia for their interest in this work My deep gratitude goes to G Pecht for her excellent typing of the manuscript The financial support from NSF, Grant DMS is highly appreciated References [] Yu Yu Bakhtin, E I Dinaburg, Ya G Sinai, bout solutions with infinite energy and enstrophy of Navier-Stokes system, Russian Mathematical Surveys 59(6 (24 [2] R Bhattacharya, L Chen, S Dobson, R Guenther, C Orum, M Ossiander, E Thomann, E Waymire, Majorizing kernels and stochastic cascades with applications to incompressible Navier-Stokes equations, TMS 355(2 (23, [3] M Cannone, Harmonic analysis tools for solving the incompressible Navier-Stokes equations, Handbook of Mathematical Fluid Dynamics, vol 3, 22 [4] U Frisch, Turbulence, The Legacy of N Kolmogorov, Cambridge University Press, 996 [5] S Monin, M Yaglom, Statistical Fluid Dynamics, vols and 2, MIT Press, Cambridge, M, 97, 975 [6] M Oliver, E Titi, Remark on the rate of decay of higher order derivatives for solutions to the Navier-Stokes equations in R n, Journal of Functional nalysis 72 (2, 8 [7] Ya G Sinai, On local and global existence and uniqueness of solutions of the 3D-Navier- Stokes Systems on, in Perspective in nalysis, Springer-Verlag, 25 (to appear Mathematics Department of Princeton University & Landau Institute of Theoretical Physics

Mathematical Hydrodynamics

Mathematical Hydrodynamics Ya G. Sinai 1. Introduction Mathematical hydrodynamics deals basically with Navier-Stokes and Euler systems. In the d-dimensional case and incompressible fluids these are the