A complete solution to the incompressible Navier-Stokes equation
AA complete solution to the incompressibleNavier-Stokes equation
Yanyou QiaoAerospace Information Research Institute, Chinese Academy of Sciences,Beijing 100101, [email protected]
Abstract:
In this paper, we solve the Navier-Stokes equation, one of the sevenMillennium Prize Problems suggested by Clay Mathematics Institute(CMI). We provethat, for any initial velocity ( , )u x t , there has been a force vector ( , )f x t , such thatthere exists no smooth solution to the respective Navier-Stokes equation. This resultalso holds for the Euler equation. For any n , similar results also hold for n dimensional Navier-Stokes equation or Euler equation. Key Word: the Navier-Stokes equation; the Euler Equation; global regularity;Millennium Prize Problems;Clay Mathematics Institute
1. The Equations
The Euler and Navier–Stokes equations [1] describe the motion of a fluid in R . Theseequations are to be solved for an unknown velocity vector
31 2 3 ( , ) ( ( , ), ( , ), ( , ))ux t u x t u x t u x t R and pressure ( , )p x t R , defined forposition
31 2 3 ( , , )x x x x R and time t . For the incompressible fluidsfilling all of R , the Navier–Stokes equations are given by: ( , ),( 1,2,3) i ij i ij j i u u pu u f x t it x x (1.1)
0u u ux x x (1.2) ( , ) | ( ) t ux t u x (1.3)Where ( ) ( ( ), ( ), ( ))u x u x u x u x is a given, C divergence–free vectorfield on R , it is the initial condition; x is the Laplacian in the spacevariables; ( ( , ), ( , ), ( , ))f f x t f x t f x t is a given externally applied force; is a positive coefficient (the viscosity), and the Euler equations are theabove equations with .For physically reasonable solutions, it is supposed that: ( , ) [ [0, )]p u C R (1.4)and ( , )ux t doesn’t grow large as | |x . Hence, the forces f and the initialcondition u are restricted to satisfy ( ) (1 | |) Kx K u x C x (1.5)on R for any multi-index and , , ( , ) 1 | | Kmx t m K f x t C x t (1.6)on [0, )R for any multi-index , nonegative integer m, and And there is a constant C such that: | ( , ) | R ux t dx C (1.7)If there is a ( , )p u satisfying (1.1)-(1.7), we say that the Navier-Stokes equation hasa smooth or regular solution; otherwise, there are blow-ups in the equation.n this paper, we take ( 1)(| | 1) (| | 1) 20,0
1( , ) [ (| | 1)]!( ) ,( 1,2,3) t x x n ni nninn f x t e e x tnf xt i (1.8)
2. Notations
The Schwartz space is define as { ( ) ( ) | ( ) (1 | |) , , 0} Kx K
S ux C R ux C x K (2.1)It can be seen that ( ) | | R ux S u dx .The divergence free Schwartz space is define as { ( ) | 0}u u uS ux S x x x (2.2)It can be seen that f is satisfied with (1.6) and
1, 2, 3, ( ( ), ( ), ( )) ,( 0,1,...) n n n f x f x f x S n .For any u S , there is a function ( )C R , so that ( ) ( ), ( ). ( ) 0 R u P u P u dx (2.3)Where ( )P u S is the projection of u onto S .
3. A crucial Lemma
Lemma 3.1 : If ( , )p u is a smooth solution and | ( , ) | R u x t dx , then: ( , ) 0,( 1,2,3) iR u x tdx i Proof:
Since ( , )p u is a smooth solution, we have: ( , , , ) ( , , , ) ( , , , )( , , , ) [ ] x x u x x t u x x t u x x tu x x x t d dx x x
In combination with | ( , ) | R u x t dx , we have: ( , , , )( , , , ) ( , , , )[ ( ) ] [ ( ) ] 0 R x xR R u x x x tdxdxdxu x x t u x x tdx d dxdx dx d dxdxx x
In the same way, ( , ) 0, ( , ) 0
R R u x tdx u x tdx .
4. The series form solution
Theorem 4.1 : If ( , )p u is a smooth solution, then they can be written as followingseries: ,0 ( , ) ( ) ,( 1,2,3) ni inn u x t u xt i (4.1) ( , ) ( ) nnn p x t p xt (4.2)Where
1, 2, 3, 0 ( , , ) ,( 0,1,2,...) i i i u u u S i , and ( ),( 0,1,2,...) i p x i areall known functions, and have , ( ) 0,( 1,2,3; 0,1,2,...) ijR u xdx i j (4.3) Proof:
The proof can be done through three steps:(i) . To get a local seriesSince ( , )p u is smooth, they can be locally expanded by Taylor expansion.There is
0, 0 , such that for ,| |t x , there are , , , , 1 2 3 ,, , , 0 0 ( , ) ( ) ,( 1,2,3) j k l n ni ij kln inj kln n u x t a x x xt u xt i (4.4) , , , 1 2 3, , , 0 0 ( , ) ( ) j k l n nj kln nj kln n p x t b x x xt p xt (4.5)Just as in (4.1) and (4.2).(ii). To compute the coefficientsSince ( ( ), ( ), ( ))u x u x u x are already known, we need to solve thefollowing coefficients: ),...(),(),,,(),(),(),(),(),( xuxuzyxuxpxuxuxuxp
For any n , if ))(),(),((,...,))(),(),(( SxuxuxuSxuxuxu nnn havealready been solved, we can then solve: )( xp n , )( ,1 xu n , )( ,2 xu n , )( ,3 xu n Lets combine (1.1), (4.1) and (4.2), and compare the coefficients of t in(1.1) and let:
1, 1 1, 11 1 11, 1, 1 1, 1 1, 11, 2, 3,0 0 01 2 3 n nn n nn n i n i n ii i ii i i u fv u u un u u ux x x (4.6)
2, 1 2, 11 1 12, 2, 1 2, 1 2, 11, 2, 3,0 0 01 2 3 n nn n nn n i n i n ii i ii i i u fv u u un u u ux x x (4.7)
3, 1 3, 11 1 13, 3, 1 3, 1 3, 11, 2, 3,0 0 01 2 3 n nn n nn n i n i n ii i ii i i u fv u u un u u ux x x (4.8)hen ,
1, 2, 3, ( , , ) n n n v v v S and
1, 2, 3, 1, 2, 3, 0 ( , , ) (( , , )) n n n n n n u u u P v v v S .By careful calculation, we can get: ( )( ) 4 | | nn R p x dx (4.9)Where
10 21,33,210 31,11,310 21,11,2 10 31,33,310 21,22,210 11,11,13 1,32 1,21 1,11
222 )( ni inini inini ini ni inini inini ininnnn xuxuxuxuxuxu xuxuxuxuxuxuxfxfxf x (iii) . To extend the local series into global seriesSince ( , )p u is smooth, so (4.1) and (4.2) can extend to ( , ) [0, )x t R .It’s easy to check that , ijR u dx i j
5. The main result
Theorem 5.1:
For any initial u satisfying (5) and f defined in (1.8), there existblow-ups in the Navier-Stokes equation. Proof:
It’s easy to check that txf i , so that R i dxtxf (5.1)f there is a smooth solution ),( up to the Navier-Stokes equation, then we have theseries form solution in (4.1). Since ,...)1,0(,),,( jSuuu jjj , we can see that ),( txu is integrable on R , so that integration can be done on (1.1) and got R iR i dxfdxut (5.2)By using Lemma 3.1, we have R i dxf (5.3)As (5.3) is contradicted to (5.1), there should exist blow-ups in the Navier-Stokesequation.From the process, we can see that the Euler equation has similar results.For any n , similar results also hold for n dimensional equations.
6. Conclusion
The result is corresponding to the case (C) in CMI’s official statement [1] , so we cansay that we solve the Navier-Stokes equation problem.