Existence and smoothness of the solution to the Navier-Stokes
aa r X i v : . [ m a t h . G M ] F e b Existence and smoothness of the solution to theNavier-Stokes equation
Dr. Bazarbekov Argyngazy B.
Kazakh National State University, EKSU. e-mail : [email protected]
Abstract .
A fundamental problem in analysis is to decide whether a smooth solution exists for the Navier-Stokes equations in three dimensions . In this paper we shall study this problem. The Navier-Stokes equations are givenby: u it ( x, t ) − ρ △ u i ( x, t ) − u j ( x, t ) u ix j ( x, t ) + p x i ( x, t ) = f i ( x, t ) , div u ( x, t ) = 0 , i = 1 , , withinitial conditions u | ( t =0) S ∂ Ω = 0 . We introduce the unknown vector-function: (cid:16) w i ( x, t ) (cid:17) i =1 , , : u it ( x, t ) − ρ △ u i ( x, t ) − dp ( x,t ) dx i = w i ( x, t ) with initial conditions: u i ( x,
0) = 0 , u i ( x, t ) | ∂ Ω = 0 . The solution u i ( x, t ) of this problem isgiven by: u i ( x, t ) = R t R Ω G ( x, t ; ξ, τ ) (cid:18) w i ( ξ, τ ) + dp ( ξ,τ ) dξ i (cid:19) dξdτ where G ( x, t ; ξ, τ ) is the Green function. We consider thefollowing Navier- Stokes -2 problem : find a solution w ( x, t ) ∈ L ( Q t ) , p ( x, t ) : p x i ( x, t ) ∈ L ( Q t ) of the system ofequations: w i ( x, t ) − G (cid:18) w j ( x, t ) + dp ( x,t ) dx j (cid:19) · G x j (cid:18) w i ( x, t ) + dp ( x,t ) dx i (cid:19) = f i ( x, t ) , i = 1 , , satisfying almost every where on Q t . Where the vector-function (cid:16) p x i ( x, t ) (cid:17) i =1 , , is defined by the vector-function (cid:16) w i ( x, t ) (cid:17) i =1 , , . Using the following estimatesfor the Green function: (cid:12)(cid:12)(cid:12) G ( x, t ; ξ, τ ) (cid:12)(cid:12)(cid:12) ≤ c/ ( t − τ ) µ · | x − ξ | − µ ; (cid:12)(cid:12)(cid:12) G x ( x, t ; ξ, τ ) (cid:12)(cid:12)(cid:12) ≤ c/ ( t − τ ) µ · | x − ξ | − (2 µ − (1 / < µ < , from this system of equations we obtain: w ( t ) < f ( t ) + b (cid:18) R t w ( τ )+ p ( τ )( t − τ ) µ dτ (cid:19) where µ : 5 / < µ < , b = const ; w ( τ ) = k w ( x, τ ) k L (Ω) ; f ( t ) = k f ( x, t ) k L (Ω) , p ( τ ) = P k ∂p ( x,τ∂x i k L (Ω) . Using the estimate: p ( t ) < c w ( t ) from this inequality weinfer: w ( t ) < f ( t ) + b · (cid:18) R t w ( τ )( t − τ ) µ dτ (cid:19) where b is real number. After the replacements of the functions R t w ( τ ) dτ ( t − τ ) µ = w ( t ) and z ( t ) = z (0) e − k R t w ( τ ) dτ this inequality will accept the following form: k R t z ( τ ) d z ( τ ) dτ ( t − τ ) − µ dτ + f ( t ) > where µ : 5 / < µ < is a real number. This is analogue of the replacement of function by Riccati : z ( t ) = − b · u ′ ( t ) u ( t ) for the solution of the followingordinary nonlinear equation : dz ( t ) dt = f ( t ) + bz ( t ); z (0) = 0[10 p. . From the last inequality we obtain the a priori estimate: k w ( x, t ) k L ( Q t ) < √ k f ( x, t ) k L ( Q t ) where Q t = Ω × [0 , t ] , t > 0 is an arbitrary real number. By the well known Leray-Schauder’s method and this a priori estimate the existence and uniqueness of the solution u ( x, t ) : u ( x, t ) ∈ W , ( Q t ) T H ( Q t ) is proved. We used the nine known classical theorems.2000 Mathematics Subject Classification . Primary : K ; Secondary : E ; Keywords : The Holder’s inequalities,Theorems: of Volterra V., of Hardy-Littlewood , of Sobolev S.L., of Lerau-Schauder, of Weyl H., of Abel-Carleman , of Riccati,Gronwall’ Lemma. Estimates for the Green function , Gamma function , projection operators , a priori estimate.
1. Introduction . The Navier-Stokes equations are given by ∂u i ( x, t ) ∂t − ρ △ u i ( x, t ) + n X j =1 u j ( x, t ) ∂u i ( x, t ) ∂x j + ∂p ( x, t ) ∂x i = f i ( x, t ) div u ( x, t ) = 0 ; i = 1 , ..., n with the initial condition : u ( x,
0) = u ( x ) (1 . where u ( x, t ) = ( u i ( x, t )) i =1 , , and p ( x, t ) ∈ R are the unknown velocity vector andpressure defined for position x ∈ R and time t ≥ . Here, u ( x ) is a given divergence-free vector field on R n , f i ( x, t ) are the components of a given externally force , ρ > is a positive coefficient and △ = P i =1 d dx i is the Laplacian in the space variables .Starting with Lerau [1] , important progress has been made in understanding weaksolutions of the Navier-Stokes equations.For instance, if (1.1) and (1.2) hold, then for anysmooth vector field ϕ ( x, t ) = ( ϕ i ( x, t )) i =1 , , , compactly supported in R n × (0 , ∞ ) , a formalintegration by parts yields Z Z R n × R u ( x, t ) ∂ϕ∂t − X i,j Z Z R n × R u i ( x, t ) u j ( x, t ) ∂ϕ i ∂x j dxdt == − ρ Z Z R n × R u ( x, t ) △ ϕdxdt + Z Z R × R f ( x, t ) ϕdxdt − Z Z R n × R p ( x, t )( divϕ ) dxdt (1 . Note that (1.3) makes sense for u ∈ L , f ∈ L , p ∈ L whereas (1.1) makes sense onlyif u(x,t) is twice differentiable in x . Similarly, if ϕ ( x, t ) is a smooth function, compactlysupported in R n × (0 , ∞ ) , then a formal integration by parts and (1.2) imply: Z Z R n × R u ( x, t ) ▽ x ϕ ( x, t ) dxdt = 0 (1 , . A solution (1.3),(1.4) is called a weak solution of the Navier-Stokes equations. Leray in [1]showed that the Navier-Stokes equation (1.1), (1.2), (1.3) in three space dimensions alwayshave a weak solution (u(x,t) , p(x,t)). The uniqueness of weak solutions of the Navier-Stokesequation is not known. In two dimensions the existence, uniqueness and smoothness of weaksolutions have been known for a long time (R.Temam [2],O. Ladyzhenskaya [3],I. Lions[4]).In three dimensions, this questions studied for the initial velocity u ( x ) satisfying asmallness condition . For the initial data u ( x ) not assumed to be small , it is known thatthe existence of smooth weak solutions holds if the time interval [0 , ∞ ) is replaced by asmall time interval [0 , T ) depending on the initial data .A fundamental problem in analysis is to decide whether a smooth solution exists for theNavier-Stokes equations in three dimensions.
2. Results
Let Ω ⊂ R be a finite domain bounded by the Lipchitz surface ð Ω . Q t = Ω × [0 , t ] , x =( x , x , x ) and u ( x, t ) = ( u i ( x, t ) i =1 , , , f ( x, t ) = ( f i ( x, t ) i =1 , , are vector-functions. Here t> 0 is an arbitrary real number. The Navier-Stokes equations are given by ∂u i ( x, t ) ∂t − ρ △ u i ( x, t ) − X j =1 u j ( x, t ) ∂u i ( x, t ) ∂x j + ∂p ( x, t ) ∂x i = f i ( x, t ) (2 . ,div u ( x, t ) = X i =1 ∂u i ( x, t ) ∂x i = 0 , i = 1 , , The Navier-Stokes problem 1.
Find a vector-function u ( x, t ) = ( u i ( x, t )) i =1 , , : Ω × [0 , t ] → R and a scalar function p ( x, t ) : Ω × [0 , t ] → R satisfying the equation (2.1) andthe following initial condition u ( x,
0) = 0 , u ( x, t ) | ∂ Ω × [0 ,t ] = 0 (2 . Let p > , r > be real numbers. We shall use the following functional spaces. L p,r ( Q t ) is the Banach space with the norm [3 p.33] k u ( x, t ) k L p,r ( Q t ) = h Z t (cid:16) Z Ω | u ( x, t ) | p dx (cid:17) r/p dt i /r , L p,p ( Q t ) = L p ( Q t ) .W , p ( Q t ) is the Banach space supplied by the norm k u ( x, t ) k W , p ( Q t ) = h k u k pL p ( Q t ) + k u t k pL p ( Q t ) + k u x k pL p ( Q t ) + k u xx k pL p ( Q t ) i /p L p,r ( Q t ) is the Banach vector-space with the norm k u ( x, t ) k L p,r ( Q t ) = X i =1 k u i ( x, t ) k L p,r ( Q t ) L ( Q t ) is the Hilbert vector-space with the inner product ( u ( x, t ) , v ( x, t )) L ( Q t ) = X i =1 ( u i ( x, t ) , v i ( x, t )) L ( Q t ) V ( Q t ) ( V ( Q t )) are the vector-spaces of smooth functions V ( Q t ) = { u ( x, t ) ∈ C ( Q t ) , div u ( x, t ) = 0 , u · n | ∂ Ω = X i =1 u i ( x, t ) cos ( n , x i ) | ∂ Ω = 0 } ,V ( Q t ) = ( { u ( x, t ) ∈ C ( Q t ) : div u ( x, t ) = 0 } ) H ( Q t ) is the closure of V ( Q t ) in the norm of L ( Q t ) . [2 p.13] I.e. H ( Q t ) = { u ( x, t ) : u ( x, t ) ∈ L ( Q T ) , div u ( x, t ) = 0 , u · n | ∂ Ω = 0 } E ( Q t ) is the closure of V ( Q t ) in the norm of L ( Q t ) . [2 p. 13] I.e. E ( Q t ) = { u ( x, t ) : u ( x, t ) ∈ L ( Q t ) , div u ( x, t ) = 0 } It is obvious that H ( Q t ) ⊆ E ( Q t ) . Further, we shall denote the vector-functions andvector-spaces by bold type. The following is principal result.
Theorem 2.1.
For any right-hand side f ( x, t ) ∈ L ( Q t ) in equation (2.1) and for anyreal numbers ρ > , t > , the Navier-Stokes problem-1 has a unique smooth solution u ( x, t ) : u ( x, t ) ∈ W , ( Q t ) ∩ H ( Q t ) and a scalar function p ( x, t ) : p x i ( x, t ) ∈ L ( Q t ) satisfying (2.1) almost everywhere on Q t , and the following estimates are valid: k u ( x, t ) k W , ( Q t ) ≤ c k f k L ( Q t ) , (cid:13)(cid:13)(cid:13) ∂p ( x, t ) ∂x i (cid:13)(cid:13)(cid:13) L ( Q t ) ≤ c k f k L ( Q t ) (2 . Here and bellow by symbol c, we denote a generic constant , independent on the solutionand right-hand side whose value is inessential to our aims, and it may change from line toline. Remark 2.1.
The case when the right-hand side f ( x, t ) has a small norm or a time t ≪ or ρ ≫ is well-known and so not interesting. But in Theorem 1 f ( x, t ) ∈ L ( Q t ) isan arbitrary vector-function and t > 0 , ρ > are arbitrary real numbers . In recentpaper [6] Ladyzhenskaja formulates the Navier-Stokes problem as in the formulas (2.1) -(2.2) and in Theorem 1 . For simplicity, we consider the Navier-Stokes problem for thehomogeneous case (i.e. u ( x,
0) = 0 , u ( x, t ) | ∂ Ω = 0 . ). We consider the inhomogeneous case(i.e. u ( x,
0) = u ( x ) , u ( x, t ) | ∂ Ω = 0 ) in Section 4. Definition 2.1.
A vector-function u ( x, t ) : u ( x, t ) | ( t =0) ∪ ∂ Ω = 0 and a scalar functionp(x,t) are called a smooth solution to the Navier-Stokes problem-1, if u ( x, t ) ∈ W , ( Q t ) ∩ H ( Q t ) and p x i ( x, t ) ∈ L ( Q t ) . We adduce the well-known definition of the Hopf solution to the Navier-Stokes equation.
Definition 2.2 (the Hopf’s solution). Let a right-hand side f ( x, t ) ∈ L ( Q t ) . Avector-function u ( x, t ) ∈ L ([0 , t ]; H (Ω)) ∩ L ∞ ([0 , t ]; L (Ω)) is called the Hopf’s solution, ifthe following equality [2 p.225]. ∂ ( u ( x, t ) , v ( x )) ∂t + ρ ( u x ( x, t ) , v x ( x )) − X i =1 u j ( x, t )( u x j ( x, t ) , v ( x )) = Z Ω f ( x, t ) · v ( x ) dx is fulfilled for all vector-functions v ( x ) ∈ H (Ω) = { u ( x ) : div u ( x ) = 0 , u ( x ) | ∂ Ω =0 , u ( x ) , u x i ( x ) ∈ L (Ω)[2 p. } . Remark 2.2.
By Theorem 2.1 it follows that the Hopf’s solution is the smoothsolution. ◭ For the proof of Theorem 2.1 we shall use the following known propositions.Theorem of Weyl H.
In the book [2 p.22] the following equalities are proved: L ( Q t ) = H ( Q t ) ⊕ G ( Q t ) where H ( Q t ) = { u ( x, t ) : u ( x, t ) ∈ L ( Q t ) , div u ( x, t ) =0 , u · n | ∂ Ω × [0 ,t ] = 0 } . G ( Q t ) = { u ( x, t ) : u ( x, t ) ∈ L ( Q t ) , u ( x, t ) = grad p ( x, t ) : p x i ( x, t ) ∈ L ( Q t ) } . I.e. for any f ( x, t ) ∈ L ( Q t ) , the following equality: f ( x, t ) = H ( f ( x, t )) + G ( f ( x, t )) is valid where H : L ( Q t ) ⇒ H ( Q t ) , G : L ( Q t ) ⇒ G ( Q t ) - are the projection operators. Proposition 1. (The Holder inequality). Let p > , p > r > , r > be areal numbers. Then, the following Holder inequality is valid [5 p.75]. k u ( x, t ) v ( x, t ) k L p p p p , r r r r ( Q t ) ≤ k u ( x, t ) k L p ,r ( Q t ) k v ( x, t ) k L p ,r ( Q t ) (2 . Proposition 2. (The system equations of Volterra V.) On the space of vector-functions u ( x, t ) = ( u i ( x, t )) i =1 , , ∈ L ( Q t ) we shall consider the following system of nonlinear integralequations of Volterra: [7 p.59 , p.62] u l ( x, t ) − X s =1 Z t Z Ω K l,s ( x, t ; ξ, τ ; u ( ξ, τ )) u s ( ξ, τ ) dξ dτ = f l ( x, t ) (2 , l = 1 ,2 ,3 . Or, in the vector form u ( x, t ) − K u ( x, t ) = f ( x, t ) ∈ L ( Q T ) (2 , This system of equations under some conditions to the nonlinear kernel K ( x, t ; ξ, τ ; u ( ξ, τ )) has been studied in the book [7 p.61]. We shall study this nonlinear system of equations byusing the theorem of Leray J., Schauder J. Proposition 3. (Theorem of Hardy G.H., Littlewood J.E.) Let µ : 0 < µ < be a real number. We shall consider the following operator of the fractionalintegration J µ u ( t ) = R t u ( τ ) dτ ( t − τ ) µ . Then: a) If < p < − µ , then the operator J µ is bounded from the space L p (0 , t ) intothe space L q (0 , t ) where q = p − p · (1 − µ ) and k J µ u ( t ) k L q (0 ,t ) ≤ c k u ( t ) k L p (0 ,t ) . [8 p.64]. Proposition 4. (Theorem of Sobolev S.L.) Let a function u(x) be represented as thepotential of a function f(x) , i.e. u ( x ) = R Ω f ( ξ ) d ξ | x − ξ | − λ λ > . [3 p.32]. Then a) If < λ < /p and f ( x ) ∈ L p (Ω) , then u ( x ) ∈ L q (Ω) where q ≤ p − p λ and k u ( x ) k L q (Ω) ≤ c k f ( x ) k L p (Ω) . b) If λ = 3 /p and f ( x ) ∈ L p (Ω) , then u ( x ) ∈ L ∞ (Ω) and k u ( x ) k L ∞ (Ω) ≤ c k f ( x ) k L p (Ω) . Proposition 5.
We shall consider the following problem on the domain Q t : u t ( x, t ) − ρ △ u ( x, t ) = g ( x, t ) , u ( x,
0) = 0 , u ( x, t ) | ∂ Ω × [0 ,t ] = 0 (2 , The solution u ( x, t ) of this problem is given by u ( x, t ) = Gg ( ξ, τ ) = Z t Z Ω G ( x, t ; ξ, τ ) g ( ξ, τ ) dξ dτ (2 , where G ( x, t ; ξ, τ ) is the Green function for Q t . The construction of the Green function isresulted in book [9 p.111]. The following estimates are valid.[9 p.170] | G ( x, t ; ξ, τ ) | ≤ const ( t − τ ) µ | x − ξ | − µ , < µ < , | ∂∂ x G ( x, t ; ξ, τ ) | ≤ const ( t − τ ) µ | x − ξ | − (2 µ − , / < µ < , From estimates in Propositions 3 , 4 and the estimates (2,9),(2,10) follow that: a) If g ( x, t ) ∈ L ( Q t ) and µ = 5 / , then Gg ( x, t ) ∈ L , ( Q t ) , G x g ( x, t ) = ∂Gg ( x, t ) ∂x ∈ L / , ( Q t ) , (2 , k Gg ( x, t ) G x g ( x, τ ) k L , ( Q t ) ≤ k Gg ( x, t ) k L , ( Q t ) k G x g ( x, τ ) k L / , ( Q t ) ≤ c k g ( x, t ) k L ( Q t ) b) If g ( x, t ) , g ( x, t ) ∈ L ( Q t ) and g i ( t ) = k g i ( x, t k L (Ω) i = 1 , , then for any µ : 5 / < µ < the following estimates are valid: (cid:13)(cid:13)(cid:13) Gg i ( x, t ) (cid:13)(cid:13)(cid:13) L − µ (Ω) ≤ c Z t g i ( τ ) dτ ( t − τ ) µ ; ; (cid:13)(cid:13)(cid:13) G x g i ( x, t ) (cid:13)(cid:13)(cid:13) L − µ (Ω) ≤ c Z t g i ( τ ) dτ ( t − τ ) µ (cid:13)(cid:13)(cid:13) Gg ( x, t ) G x g ( x, τ ) (cid:13)(cid:13)(cid:13) L − µ ) (Ω) ≤ c Z t g ( τ ) dτ ( t − τ ) µ · Z t g ( τ ) dτ ( t − τ ) µ (2 . . and for µ : ≤ µ < follows that ≤ − µ ) . Therefore, (cid:13)(cid:13)(cid:13) Gg ( x, t ) G x g ( x, τ ) (cid:13)(cid:13)(cid:13) L (Ω) ≤ ≤ c (cid:13)(cid:13)(cid:13) Gg ( x, t ) G x g ( x, τ ) (cid:13)(cid:13)(cid:13) L − µ ) (Ω) ≤ c Z t g ( τ ) dτ ( t − τ ) µ · Z t g ( τ ) dτ ( t − τ ) µ (2 . . ◮ a) We shall prove the inequality (2.11). From Holder inequality (2,4) (with p = 2 , p =12 , r = 8) and ( p = 2 , p = 12 / , r = 8) , the first inequality of (2.11) follows. From theestimate (2.9) with ( µ = 5 / we have k Gg ( x, τ ) k L , ( Q t ) = (cid:13)(cid:13)(cid:13) Z τ Z Ω G ( x, τ ; ξ, τ ) g ( ξ, τ ) dξ dτ (cid:13)(cid:13)(cid:13) L , ( Q t ) ≤≤ c (cid:13)(cid:13)(cid:13) Z τ τ − τ ) / (cid:13)(cid:13)(cid:13) Z Ω g ( ξ, τ ) dξ | x − ξ | − / (cid:13)(cid:13)(cid:13) L (Ω) dτ (cid:13)(cid:13)(cid:13) L (0 t ) ≤≤ c (cid:13)(cid:13)(cid:13) Z τ k g ( x, τ ) k L (Ω) dτ ( τ − τ ) / (cid:13)(cid:13)(cid:13) L (0 t ) ≤ c k g ( x, t ) k L ( Q t ) Here we used the fact that from Proposition a ( with p = 2 , λ = 5 / q = − λ = 12) follows the inequality (cid:13)(cid:13)(cid:13) R Ω g ( ξ,τ ) dξ | x − ξ | − / (cid:13)(cid:13)(cid:13) L (Ω) ≤ c k g ( x, τ ) k L (Ω) and since k g ( x, τ ) k L (Ω) ∈ L (0 , t ) , then from Proposition (with p = 2; µ = 5 / q = p − p (1 − µ ) = 8 ) the followinginequality (cid:13)(cid:13)(cid:13) R τ k g ( x,τ ) k L dτ ( τ − τ ) / (cid:13)(cid:13)(cid:13) L (0 ,t ) ≤ c k g ( x, t ) k L ( Q t ) follows.Using the estimate (2,10) (with µ : ≤ µ < ) , the Proposition a ( with λ = 1 / q = − λ = ) and Proposition (with p = 2; µ = 5 / q = p − p (1 − µ ) = 8 ) the followingestimate k ∂Gg ( x,t ) ∂x k L / , ( Q t ) ≤ c k g ( x, t ) k L ( Q t ) is proved similarly. By these estimates thesecond estimate of (2.11) follows. The inequality (2.11) is proved. b) The proofs of the inequalities in (2.12.1), (2.12.2) follow from the estimates of theGreen function (2,9), (2,10), Proposition a and is similar to the proof of a ). The parametersin the second inequality of (2,12.1) satisfy all conditions of parameters p , p in (2,4) ofProposition 1. ◭ Proposition 6. ( Theorem of Leray J., Schauder J.) Let А be a compact nonlinearoperator on L ( Q t ) . If every possible solution to the following equation [3 p.42] w ( x, t ) + Aw ( x, t ) = f ( x, t ) do not fall outside the bounds of some sphere | w ( x, t ) | L ( Q t ) ≤ c, then for any right-handside f ( x, t ) ∈ L ( Q t ) the equation has at least one solution in this sphere. Proposition 7. (The equation of Abel-Carleman.) The equation of Abel-Carleman isset by the following formulas [8 p.39]: Z t u ( τ )( t − τ ) µ dτ = f ( t ); u ( t ) = sinπµπ ddt Z t f ( τ )( t − τ ) − µ dτ (2 . Let −∞ < µ < −∞ < µ < . Then Z t dτ ( t − τ ) µ dτ Z τ g ( τ ) dτ ( τ − τ ) µ = Z t g ( τ ) dτ Z tτ dτ ( t − τ ) µ ( τ − τ ) µ == Γ µ µ Z t g ( τ ) dτ ( t − τ ) µ + µ − ; ; Γ µ µ = Γ(1 − µ )Γ(1 − µ )Γ(2 − µ − µ ) (2 . Proposition 8. (The linear Navier-Stokes equation) Let t > 0 be an arbitrary realnumber. We consider the following linear Navier-Stokes problem on the domain Q t : [3, p.95] ∂u i ( x, t ) ∂t − ρ △ u i ( x, t ) − ∂p ( x, t ) ∂x i = w i ( x, t ) (2 . ,div u ( x, t ) = 0 ; u ( x,
0) = 0 , u ( x, t ) | ∂ Ω × [0 ,t ] = 0 For this problem in the manuscript of author [12] is received the explicit expression to thepressure function p(x,t) , depending on the right-hand side w i ( x, t ) : p ( x, t ) = − T ∗ △ − ∗ Z t Z Ω 3 X dG ( x, t ; ξ, τ ) dx i w i ( ξ, τ ) dξdτ = (2 , − ddt △ − ∗ Z t Z Ω 3 X dG ( x, t ; ξ, τ ) dx i w i ( ξ, τ ) dξdτ + ρ · Z t Z Ω 3 X dG ( x, t ; ξ, τ ) dx i w i ( ξ, τ ) dξdτ where T ∗ u ( x, t ) = ddt u ( x, t ) − ρ △ x u ( x, t ) is the parabolic operator , △ − is the inverseoperator to Dirichlet problem for Laplase equation on the domain Ω . G ( x, t ; ξ, τ ) is theGreen function of Dirichlet problem for the parabolic equation on the domain Q t = Ω × [0 , t ] [9 p.106]. If w ( x, t ) ∈ L ( Q t ) , then R t R Ω G ( x, t ; ξ, τ ) w ( ξ, τ ) dξdτ ∈ W , ( Q t ) . It is obviousthat: (cid:12)(cid:12)(cid:12) Z t Z Ω 3 X dG ( x, t ; ξ, τ ) dx i w i ( ξ, τ ) dξdτ (cid:12)(cid:12)(cid:12) W , ( Q t ) < c · | w ( x, t ) | L ( Q t ) From this estimate follows the following estimate: Z t (cid:16)(cid:12)(cid:12)(cid:12) ddt △ − ∗ Z τ Z Ω 3 X dG ( x, t ; ξ, τ ) dx i w i ( ξ, τ ) dξdτ (cid:12)(cid:12)(cid:12) W (Ω) (cid:17) dτ < c | w ( x, τ ) | L ( Q t ) We consider the formula (2,16) in detail. It is known that the following classical problemfor the parabolic equations: u t − ρ △ u ( x, t ) = f ( x, t ) ∈ L ( Q t ) , u t =0 ∪ ∂ Ω = 0 has theunique solution u ( x, t ) ∈ W ( Q t ) and k u ( x, t k W ( Q t ) < c k f ( x, t ) k L ( Q t ) where Q t =Ω × [0 , t ] . Therefore, from the formula (2,16) follow the following estimates: (cid:13)(cid:13)(cid:13) ddx i ddt △ − ∗ Z t Z Ω 3 X dG ( x, t ; ξ, τ ) dx i w i ( ξ, τ ) dξdτ (cid:13)(cid:13)(cid:13) L ( Q t ) < c k w ( x, t ) k L ( Q t ) (cid:13)(cid:13)(cid:13) ddx i ρ · Z t Z Ω 3 X dG ( x, t ; ξ, τ ) dx i w i ( ξ, τ ) dξdτ (cid:13)(cid:13)(cid:13) L ( Q t ) < c k w ( x, t ) k L ( Q t ) Differentiating the function of pressure p(x,t) in formula (2,16) by x i , we find dp ( x,t ) dx i , i =1 , , , depending on the right-hand side w i ( x, t ) . And by these estimates, integrating on thedomain Q t , we obtain the following estimate: Z t X (cid:12)(cid:12)(cid:12) ∂p ( x, τ ) ∂x i (cid:12)(cid:12)(cid:12) L (Ω) dτ < c · Z t X | w i ( x, τ ) | L (Ω) dτ (2 , By the formula (2,16) and estimate (2,17) on the vector space w ( x, t ) = (cid:16) w i ( x, t ) (cid:17) i =1 , , ∈ L ( Q t ) we define the following linear and boundedoperator: P (cid:16) w i ( x, t ) (cid:17) i =1 , , = (cid:16) dp ( x, t ) dx i (cid:17) i =1 , , ∈ L ( x, t ) (2 , where the functions dp ( x,t ) dx i is defined by the functions w i ( x, t ) from the formula (2,16). ◭ Using the Green function G ( x, t ; ξ, τ ) , from the equation (2,15) we find : u i ( x, t ) = Z t Z Ω G ( x, t ; ξ, τ ) (cid:16) w i ( ξ, τ ) + ∂p ( ξ, τ ) ∂ξ i (cid:17) dξdτ (2 , And present the nonlinear Navier-Stokes equations (2,1) as: w i ( x, t ) − X j =1 G (cid:16) w j ( x, t ) + ∂p ( x, t ) ∂x j (cid:17) · G x j (cid:16) w i ( x, t ) + ∂p ( x, t ) ∂x i (cid:17) = f i ( x, t ) (2 . where G x j w i ( x, t ) = ∂Gw i ( x,t ) ∂x j . Differentiating the pressure function p(x,t) in the formula(2,16) by x i , we find dp ( x,t ) dx i , i = 1 , , , depending on the functions w i ( x, t ) , and substitute these functions to the equation (2,20). Then, for the definition of the three unknownfunctions ( w i ( x, t )) i =1 , , we obtain the three system of nonlinear equations ofVolterra (2,20) . Remark 2,3.
It is obvious that the vector function (cid:16) dp ( x,t ) dx i (cid:17) i =1 , , depends on the vectorfunction ( w i ( x, t )) i =1 , , linearly. But we shall not write the expressions of these depends, weshall use the estimate (2,17). ◭ Navier-Stokes problem 2.
Find the vector-function ( w i ( x, t )) i =1 , , ∈ L ( Q t ) ,satisfying the equation (2,20) almost every where on Q t . ◭ We will find the unknown vector-function w ( x, t ) = ( w i ( x, t )) i =1 , , ∈ L ( Q t ) . Theorem 2.2
For any right-side f ( x, t ) ∈ L ( Q T ) in the system of equations (2,20)there exists a unique vector-function w ( x, t ) ∈ L ( Q T ) , satisfying almost everywhere on Q T , the system equations (2,20) . And for any possible solution w ( x, t ) : k w ( x, t ) k L ( Q T ) = (cid:13)(cid:13)(cid:13) k w ( x, t ) k L (Ω) (cid:13)(cid:13)(cid:13) L (0 ,T ) = k w ( t ) k L (0 ,T ) < ∞ to the basis equation (2,20), the following apriori estimate is valid k w ( x, t ) k L ( Q T ) < √ · k f ( x, t ) k L ( Q T ) (2 . where Q T = Ω × [0 , T ] , ; T > 0 is an arbitrary real number.
3. Proof of Theorem 2.2.The following is the key Lemma for the proof of Theorem 2.2.Lemma 3.1.
Let the vector-function g ( x, t ) = ( g i ( x, t )) i =1 , , ∈ L ( Q t ) . I.e. | g ( x, τ ) | L ( Q t ) = P i =3 i =1 ( R Q t g i ( x, τ ) dxdτ ) / < ∞ . We define the following vector-function: Gg j ( x, t ) G x j g ( x, t ) = (cid:16) X j =1 Gg j ( x, t ) G x j g i ( x, t ) (cid:17) i =1 , , = (cid:16) X j =1 Gg j ( x, t ) ∂Gg i ( x, t ) ∂x j (cid:17) i =1 , , Then for any t > 0 there exists a constant b > 0 independent on g (x,t) such that thefollowing inequality is valid: (cid:13)(cid:13)(cid:13) Gg j ( x, t ) G x j g ( x, t ) (cid:13)(cid:13)(cid:13) L (Ω) ≤ b (cid:16) Z t g ( τ ) dτ ( t − τ ) µ (cid:17) (3 . where µ : ≤ µ < , g ( τ ) = k g ( x, τ ) k L (Ω) = P i =3 i =1 ( R Ω g i ( x, τ ) dx ) / . ◮ Let µ : ≤ µ < . By the estimate (2.12.2) in Proposition 5 and the followinginequality: P i,j =1 c i c j · a i · a j ≤ b ( P i =1 a i ) we have k Gg j ( x, t ) G x j g ( x, t ) k L (Ω) ≤ X i,j =1 k Gg j ( x, t ) G x j g i ( x, t ) k L (Ω) << X i,j =1 c i c j Z t g i ( τ ) dτ ( t − τ ) µ · Z t g j ( τ ) dτ ( t − τ ) µ < b (cid:16) Z t g ( τ ) dτ ( t − τ ) µ (cid:17) Lemma is proved. ◭ From the basis equation (2,20) and the inequality (3,1), integrating over the domain Ω ,and, using the Holder inequality, the estimates (2,9), (2,10) for Green function, we obtainthe following estimate: w ( t ) < f ( t ) + b (cid:16) Z t w ( τ ) + p ( τ )( t − τ ) µ dτ (cid:17) (3 . where w ( τ ) = k w ( x, τ ) k L (Ω) = i =3 X i =1 (cid:16) Z Ω w i ( x, τ ) dx (cid:17) / ≥ p ( τ ) = X (cid:13)(cid:13)(cid:13) ∂p ( x, τ∂x i (cid:13)(cid:13)(cid:13) L (Ω) ≥ f ( t ) = k f ( x, t ) k L (Ω) = i =3 X i =1 (cid:16) Z Ω f i ( x, τ ) dx (cid:17) / ≥ . ′ ) Remark 3-1.
Let T > 0 be an arbitrary number. We have proved thatfor all solutions w ( x, t ) , p ( x, t ) of the basis equation (2.20) the functions w ( τ ) = k w ( x, τ ) k L (Ω) , p ( τ ) = P (cid:13)(cid:13)(cid:13) ∂p ( x,τ∂x i (cid:13)(cid:13)(cid:13) L (Ω) satisfy to the inequality (3.2). The inequality(3.2) does not exclude the functions of the type w ( t ) + p ( t ) = tT − t * L (0 , T ) . I.e. thesefunctions satisfy the inequality (3.2). Note that µ : ≤ µ < and w ( t ) ∈ L (0 , T ) . ByProposition 3 and estimate (2,17) in Proposition 8 it follows that: (cid:13)(cid:13)(cid:13)(cid:16) Z t ( w ( τ ) + p ( τ )) dτ ( t − τ ) µ (cid:17) (cid:13)(cid:13)(cid:13) L (0 ,T ) < c · k w ( t ) + p ( t ) k L (0 ,T ) < c · k w ( t ) k L (0 ,T ) (3 . In the Theorem 2.2 we assumed that for all possible solutions w ( x, t ) of the basic equation(2,20) the function w ( t ) = k w ( x, t ) k L (Ω) ∈ L (0 , T ) . By the basis equation (2,20) we have proved the inequality (3,2). Below(see Theorem 3.1), we shall prove that for all functions w ( t ) , p ( t ) ∈ L (0 , T ) : k p ( t ) k L (0 ,T ) < c k w ( t ) k L (0 ,T ) < ∞ , satisfying the inequality (3,2), the following apriori estimate: k w ( t ) k L (0 ,T ) < √ k f ( t ) k L (0 ,T ) holds. ◭ Lemma 3.2.
From the estimates (2,17), (3,2) follows that: w ( t ) < f ( t ) + b · Z t w ( τ )( t − τ ) µ dτ (3 . where b = b + c is constant independent on function w(t). ◮ By Holder inequality and the inequality ( a + b ) < a + 2 b from the basic inequality(3,2) follows that: w ( t ) < f ( t ) + b (cid:16) Z t w ( τ ) + p ( t )( t − τ ) µ/ · t − τ ) µ/ dτ (cid:17) < f ( t )+ 2 b − µ · T − µ · Z t w ( τ ) + p ( τ )( t − τ ) µ dτ Using the estimate (2,17) in Proposition 8 and the inequality ( a + a + a ) < · ( a + a + a ) ,we infer: Z t p ( τ ) dτ = Z t (cid:16) X k p x i ( x, τ ) k L (Ω) (cid:17) dτ < · Z t X k p x i ( x, τ ) k L (Ω) dτ << c · Z t X k w i ( x, τ ) k L (Ω) dτ < c Z t (cid:16) X k w x i ( x, τ ) k L (Ω) (cid:17) dτ = 3 c · Z t w ( τ ) dτ I.e.: w ( t ) < f ( t ) + b Z t w ( τ ) + p ( τ )( t − τ ) µ dτ ; ; Z t p ( τ ) dτ < c · Z t w ( τ ) dτ (3 , ′ ) where b = b − µ · T − µ . We shall prove by the second inequality of (3,4’) that there existsa constant c > 0: Z t p ( τ )( t − τ ) µ dτ < c · Z t w ( τ )( t − τ ) µ dτ (3 , We shall prove this estimate by contradiction method and assumed that there exists aconstants c n → ∞ : Z t p ( τ )( t − τ ) µ dτ > c n · Z t w ( τ )( t − τ ) µ dτ Let us apply to this inequality the following operator: J − µ u ( t ) = R t u ( τ )( t − τ ) − µ dτ. Then: R t p ( τ ) dτ > c n · R t w ( τ ) dτ. But: R t p ( τ ) dτ < c · R t w ( τ ) dτ. And this contraction proves the estimate(3,4"). Lemma 3,2 is proved. ◭ Remark 3-2.
Below, using the
Riccati’s replacement of the function w(t), from theestimate (3,4) we derive the following estimate: k w ( t ) k L (0 ,T ) < √ k f ( t ) k L (0 ,T ) . For theseaims we consider the following equation. The equation of Riccati.
In 1715, Riccati has studied the following nonlinear equationon the segment [0,T] where T > 0 is an arbitrary real number [10 p.41]: dz ( t ) dt = f ( t ) + b z ( t ); z (0) = 0 By the replacement of the unknown function z ( t ) = − b · u ′ ( t ) u ( t ) , this nonlinear equation isreduced to the following linear equation of the second order: d u ( t ) dt + bf ( t ) u ( t ) = 0; du ( t ) dt | t =0 = 0 . Theorem 3.1.
For all functions w ( t ) ∈ L (0 , T ) satisfying the inequality (3,4) thefollowing estimate holds: k w ( t ) k L (0 ,T ) < √ · k f ( t ) k L (0 ,T ) (3 , This estimate does not depends on the number b in (3,4). ◮◮ In (3.4) we make the replacement of the function: w ( t ) = Z t w ( τ ))( t − τ ) µ dτ (3 , and, using the inequality ( a + b ) < a + 2 b , we rewrite the basis inequality (3,4) as: Z t dw ( τ ) dτ ( t − τ ) − µ dτ < f ( t ) + 2 b · w ( t ) (3 , Let k : 0 < k < ∞ and s : 0 < s < - are an arbitraries real numbers and for t > 1 wepresent the inequality (3,7) as: Z t dw ( τ ) dτ ( t − τ ) − µ dτ < f ( t ) + k Z t w ( τ ) dτ ( t − τ ) − µ + 2 b · t µ w ( t ) − k Z t w ( τ ) dτ ( t − τ ) − µ (3 , Applying the Mellin transformation, for s : 0 < s + µ < we obtain: Z ∞ t s − · (cid:16) b · t µ w ( t ) − k Z t w ( τ ) dτ ( t − τ ) − µ (cid:17) dt == (cid:16) b − k · Z dττ s + µ · (1 − τ ) − µ (cid:17) · Z ∞ τ s + µ − w ( τ ) dτ From this equality follows that: (cid:16) b · t − µ w ( t ) − k R t w ( τ ) dτ ( t − τ ) − µ (cid:17) < for all numbers k ≫ (cid:16) b − k · Z dττ s + µ · (1 − τ ) − µ (cid:17) < , where s : 0 < s + µ < . Using this inequality, we rewrite the inequality (3,8) as follows: Z t dw ( τ ) dτ ( t − τ ) − µ dτ < f ( t ) + k Z t w ( τ ) dτ ( t − τ ) − µ (3 , In this inequality , as Riccati , we shall make the replacement of the function w ( τ ) w ( τ ) = − k z ′ ( τ ) z ( τ ) ; ; dw ( τ ) dτ = − k · z ′′ ( τ ) z ( τ ) + 1 k (cid:16) z ′ ( τ ) z ( τ (cid:17) (3 , From the definition of the function w ( t ) by (3,6) follows that: z ( t ) = z (0) · e − k R t w ( τ ) dτ = z (0) · e − k − µ R t w ( τ )( t − τ ) − µ dτ (3 , ′ ) w (0) = 0 , then z ′ (0) = dz ( tdt | t =0 = 0 . From the inequality (3,10) we have − k Z t z ( τ ) d z ( τ ) dτ ( t − τ ) − µ dτ < f ( t ) (3 , Or − k Z t z ( τ · d z ( τ ) dτ dτ < µ − µ · Z t f ( τ )( t − τ ) µ dτ (3 , Let us denote: k Z t z ( τ ) · d z ( τ ) dτ dτ + 2Γ µ − µ · Z t f ( τ )( t − τ ) µ dτ = g ( t ) > Then k · d z ( t ) dt + 2Γ µ − µ · z ( t ) ddt Z t f ( τ )( t − τ ) µ dτ = z ( t ) dg ( t ) dt (3 , ′ ) Since g ( t ) > , g (0) = 0 , − dz ( τ ) dτ > , integrating by parts, we obtain: Z t z ( τ ) · ddτ g ( τ = z ( t ) g ( t ) + Z t (cid:16) − dz ( τ ) dτ (cid:17) · g ( τ ) dτ > Since dz ( t ) dt | t =0 = 0 , integrating the equation (3,13’) over [0,t] , we obtain the following important inequality : − dz ( t ) dt < k · Z t z ( τ ) dF µ ( τ ) dτ dτ (3 , where F µ ( t ) = 2Γ µ − µ · Z t f ( τ )( t − τ ) µ dτ ◭ (3 , Integrating by parts,from the basis inequality (3,14), we have: − k · dz ( t ) dt < z ( t ) · F µ ( t ) + Z t F µ ( τ ) (cid:16) − dz ( τ ) dτ (cid:17) · dτ (3 , In order to proof the a priori estimate , it is necessary that the function F µ ( t ) = µ − µ · R t f ( τ )( t − τ ) µ dτ is increasing on [0,T]. Otherwise, the proof of this estimate is difficult. Remark 3-3.
Not for all positive right-hand side f ( t ) : f ( t ) ∈ L (0 , T ) and realnumbers µ : 1 / < µ < the function f µ ( t ) = sinπδπ R t f ( τ )( t − τ ) µ dτ is increasing on [0,T]. Forexample, we consider the following function f µ ( t ) = sinπµπ R t − kτ ( t − τ ) µ dτ where (1 − kτ ) > on(0,t), i.e. < τ < /k. Then df µ ( t ) dt = sinπµπ · t µ · (cid:16) − k − µ · t (cid:17) , and for t : − µk < t < k thefunction f µ ( t ) is decreasing on ( − µk , k ) . ◭ Let f ( t ) : f ( t ) ∈ L (0 , T ) - is an arbitrary function. To define the right-hand side f ( t ) , for which the function F µ ( t ) is increasing on [0,T], we introduce the following functionalspaces: L +2 (0 , T ) = n f ( t ) : f ( t ) > , Z T f ( τ ) dτ < ∞ , o L +1 − µ (0 , T ) = n f − µ ( t ) : f − µ ( t ) = Z t g ( τ )( t − τ ) − µ dτ o (3 , where g ( t ) : g ( t ) ∈ L +2 (0 , T ) is an arbitrary function. Remark 3-4.
For all functions { f − µ ( t ) = R t g ( τ )( t − τ ) − µ dτ ∈ L +1 − µ (0 , T ) } the functions F µ ( t ) = 2Γ µ − µ · Z t f − µ ( τ )( t − τ ) µ dτ = 2Γ µ − µ · Z t dτ ( t − τ ) µ Z τ g ( τ ) dτ ( τ − τ ) − µ = 2 Z t g ( τ ) dτ (3 , are increasing on [0,T]. This remark is important for the proof the a priori estimate. ◭ Lemma 3.3
For any right-side f ( t ) = f − µ ( t ) = R t g ( τ )( t − τ ) − µ dτ ∈ L +1 − µ (0 , T ) in thebasis equations (3,14) and (3,15) the following a priori estimate holds: k w ( t ) k L (0 ,T ) < √ · k f − µ ( t ) k L (0 ,T ) (3 , ◮ Since (cid:16) − dz ( t ) d t (cid:17) > and for any function f − µ ( t ) = ∈ L +1 − µ (0 , T ) the function F µ ( t ) = 2 · R t g ( τ ) dτ is increasing (Remark 3.4), we rewrite the inequality (3,16) as: − k · dz ( t ) dt < z ( t ) · F µ ( t ) + F µ ( t ) · Z t (cid:16) − dz ( τ ) dτ (cid:17) · dτ (3 , Let us denote: R t (cid:16) − dz ( τ ) dτ (cid:17) dτ = z ( t ) . Then this equation will accept the following kind: dz ( t ) dt − k · F µ ( t ) z ( t ) < k · F µ ( t ) · z ( t ) (3 , ′ ) Since z (0) = 0 and k · F µ ( t ) · z ( t ) > , from this inequality and Gronwall’s Lemma we inferthat: z ( t ) = Z t (cid:16) − dz ( τ ) dτ (cid:17) dτ < k · (cid:16) Z t F µ ( τ ) z ( τ ) e − k R τ F µ ( τ ) dτ dτ (cid:17) · e k R t F µ ( τ ) dτ Since z ( t ) > and the function F µ ( t ) > is increasing, we rewrite this inequality: z (0) − z ( t ) < k · F µ ( t ) (cid:16) Z t z ( τ ) e − k R τ F µ ( τ ) dτ dτ (cid:17) · e k R t F µ ( τ ) dτ (3 , Let us denote: z ( t ) = Z t z ( τ ) e − k · R τ F µ ( τ ) dτ dτ (3 , and we present (3,21) in the following form: dz ( t ) dt + k · F µ ( t ) · z ( t ) > z (0) · e − k R t F µ ( τ ) dτ (3 , Using the Gronwall’s Lemma and z (0) = 0 , we infer that: z ( t ) > z (0) · t · e − k R t F µ ( τ ) dτ (3 , z ( t ) = R t z ( τ ) e − k R τ F µ ( τ ) dτ dτ < R t z ( τ ) dτ, it follows from (3,22),(3,24) that z (0) · t < (cid:16) Z t z ( τ ) dτ (cid:17) · e k R t F µ ( τ ) dτ (3 , where the function z(t) is defined by (3,11’) and (3,6): z ( t ) = z (0) · e − k R t w ( τ ) dτ = z (0) · e − k − µ R t w ( τ )( t − τ ) − µ dτ (3 , ′ ) Remark 3.5 . From (3,25’) we have: dz ( t ) dt = − z (0) · k · (cid:16) Z t w ( τ )( t − τ ) µ dτ (cid:17) · e − kµ R t w ( τ )( t − τ ) − µ dτ < . If for a some t ∈ (0 , ∞ ) dz ( tdt | t = t = 0 , then the following two cases are possible: R t w ( τ )( t − τ ) µ dτ = 0 , or R t w ( τ )( t − τ ) − µ dτ = ∞ . If R t w ( τ )( t − τ ) µ dτ = 0 , then w ( t ) ≡ on [0 , t ) , since w ( t ) ≥ on [0 , t ) . If R t w ( τ )( t − τ ) − µ dτ = ∞ , then below weshall prove that for all positive functions w ( t ) satisfying to the basis inequality (3,25): R t w ( τ )( t − τ ) − µ dτ < ∞ . ◭ The following is the key Lemma for the proof of Theorem 2.2.Lemma 3,5’.
Let t > 0 is an arbitrary real number. For all positive functions w ( τ ) satisfying to the basis inequality (3,25) follows that: Z t w ( τ )( t − τ ) − µ dτ < ∞ (3 , where µ : 1 / < µ < . Or, passing to limit µ → , we obtain: Z t w ( τ ) dτ < ∞ (3 , ′ ) ◮ We shall prove this Lemma by contradiction method and rewrite the basis inequality(3,25): z (0) · t < (cid:16) Z t z ( τ ) dτ (cid:17) · e k R t F µ ( τ ) dτ (3 , where the function z(t) is defined by (3,11’): z ( t ) = z (0) · e − k − µ R t w ( τ )( t − τ ) − µ dτ (3 , ′ ) Let for a some number t : R t w ( τ )( t − τ ) − µ dτ = ∞ . I.e. w ( τ ) ≈ c | t − τ | λ ·| t − τ | − µ where λ ≥ is a real number. Then for all t ≥ t : R t w ( τ )( t − τ ) − µ dτ = ∞ . And z ( t ) = z (0) · e − k − µ R t w ( τ )( t − τ ) − µ dτ ≡ for t ≥ t . Since e − k − µ R t w ( τ )( t − τ ) − µ τ · t (cid:12)(cid:12)(cid:12) t = 0 ,then, integrating by part, we have: z (0) Z t z ( τ ) dτ = + Z t t · e − k − µ R t w ( τ )( t − τ ) − µ dτ d (cid:16) k − µ Z t w ( τ )( t − τ ) − µ τ (cid:17) k − µ R t w ( τ )( t − τ ) − µ τ = t . Then: z (0) Z t z ( τ ) dτ < t · Z ∞ e − t d t = t From the definition of the function F µ ( t ) by (3,15) we obtain: Z t F µ ( τ ) dτ = 2Γ µ − µ · − µ · Z t f ( τ )( t − τ ) − µ dτ (3 , ′ ) Using the following limits: lim µ → Γ(1 − µ ) · (1 − µ ) = 1 and lim µ → ( t − τ ) − µ = 1 , weget: lim µ → R t F µ ( τ ) dτ = R t f ( τ ) dτ . Using these facts, from the basis inequality (2,25) wehave: t < t · e R T f ( τ ) dτ But, for t ≫ we have received the contradiction. Therefore R t w ( τ )( t − τ ) − µ dτ < ∞ . Lemma is proved. ◭ Remark 3.35’.
Hence, for any positive function w ( t ) ≥ satisfying to the basisinequality (3,25) the function z ( t ) > is continuous on [0 , ∞ ) and monotonously decreasingfrom z(0) to zero on [0 , ∞ ) . ◭ Let T > 0 be an arbitrary real number. We rewrite the basis inequality (3,25) for t =T as: T · z (0) < (cid:16) Z T z ( t ) dt + Z TT z ( t ) dt (cid:17) · e k R T F µ ( τ ) dτ (3 , Since the function z ( t ) = z (0) · e − k − µ R t w ( τ )( t − τ ) − µ dτ is continuous on [0 , ∞ ) andmonotonously decreasing from z(0) to zero, there exists the numbers t , t : t : 0 < t < T ; ; t : T < t < T (3 , ′ ) such that the following equalities holds: Z T e − k − µ R t w ( τ )( t − τ ) − µ dτ dτ = (cid:16) e − k − µ R t w ( τ )( t − τ ) − µ dτ (cid:17) · T (3 , Z TT e − k − µ R t w ( τ )( t − τ ) − µ dτ dτ = (cid:16) e − k − µ R t w ( τ )( t − τ ) − µ dτ (cid:17) · T (3 , ′ ) Using these equalities, we rewrite the inequality (3,27) as: < (cid:16) e − w ( t ) + e − w ( t ) (cid:17) · e k · R T F µ ( τ ) dτ (3 , where w ( t ) = k − µ Z t w ( τ )( t − τ ) − µ dτ ; ; w ( t ) = k − µ Z t w ( τ )( t − τ ) − µ dτ (3 , ′ ) and present the inequality (3,29) as: e w ( t ) < (cid:16) e w ( t ) e w ( t ) + 1 (cid:17) · e k · R T F µ ( τ ) dτ (3 , f ( t ) = f − µ ( t ) = Z t g ( τ )( t − τ ) − µ dτ ∈ L +1 − µ (0 , T ) (3 , where the g ( t ) : g ( t ) ∈ L (0 , T ) is an arbitrary function. Then, from (3,15) we have: F µ ( t ) = 2Γ µ − µ · Z t f − µ ( τ )( t − τ ) µ dτ = (3 , · µ − µ · Z t t − τ ) µ Z τ g ( τ ) dτ ( τ − τ ) − µ dτ = 2 · Z t g ( τ ) dτ From (3,31) and (3,32) follows that: g ( t ) = ddt Z t f − µ ( τ )( t − τ ) µ dτ ; ; F µ ( t ) = 2 Z t g ( τ ) dτ = 2 · Z t f − µ ( τ )( t − τ ) µ dτ (3 , ′ ) and from (3,32’) we have: Z t F µ ( τ ) dτ = 21 − µ · Z t ( t − τ ) − µ · f − µ ( τ ) dτ (3 , Since the function e − w ( t ) is decreasing on (0 , ∞ ) and t > t , we have: e w ( t ) /e w ( t ) < . And from (3,30) follows that e k − µ R t w ( τ )( t − τ ) − µ dτ < · e · k − µ R T f − µ ( τ )( t − τ ) − µ dτ or k Z t w ( τ )( t − τ ) − µ dτ < (1 − µ ) · ln k · Z T f − µ ( τ )( t − τ ) − µ dτ (3 , Passing to the limit µ → , from t > T, lim µ → ( t − τ ) − µ = 1 and this inequality weobtain: Z T w ( τ ) dτ < · Z T f − µ ( τ ) dτ (3 , ′ ) Lemma 3.3 is proved. ◭ The proof of Theorem 3.1. Lemma 3-4.
The space of functions L +1 − µ (0 , T ) isdense in the space of functions L +2 (0 , T ) in the norm of the space L (0 , T ) . I.e. for allfunctions f ( t ) ∈ L +2 (0 , T ) , there exists a sequence of functions: f n − µ ( t ) ∈ L +1 − µ (0 , T ) : lim n →∞ (cid:16) k f n − µ ( t ) k L (0 ,T ) −k f ( t ) k L (0 ,T ) (cid:17) < lim n →∞ k f n − µ ( t ) − f ( t ) k L (0 ,T ) → , or lim n →∞ (cid:16) k f n − µ ( t ) k L (0 ,T ) −k f ( t ) k L (0 ,T ) (cid:17) < lim n →∞ k f n − µ ( t ) − f ( t ) k L (0 ,T ) → , ′ ) Remark 3.6.
Let us note that for any n = 1,2,...the functions f n − µ ( t ) ∈ L +1 − µ (0 , T ) . From Remark 3.4 follows that the functions F n µ ( t ) = µ − µ · R t f n − µ ( τ )( t − τ ) µ dτ are7increasing on [0,T]. And from (3,34) in Lemma 3.3 follow the following estimates: k w ( t ) k L (0 ,T ) < · k f n − µ ( t ) k L (0 ,T ) . ◭◮ We prove Lemma 3.4 by the contradiction method. Let there exists a function f ( t ) : f ( t ) ≥ , f ( t ) = 0 , f ( t ) ∈ L (0 , T ) , such that for all functions f ( t ) : f ( t ) ≥ , f ( t ) ∈ L (0 , T ) the following equality is valid Z T f ( t ) · Z t f ( τ ) dτ ( t − τ ) − µ dt = Z T f ( τ ) · (cid:16) Z Tτ f ( t ) dt ( t − τ ) − µ (cid:17) dτ = 0 Since f ( τ ) ≥ is an arbitrary function, by this equality it follows that for all τ ∈ [0 , T ] : R Tτ f ( t ) dt ( t − τ ) − µ ≡ . Then f ( t ) ≡ . Lemma is proved. ◭ Let ǫ n : 0 < ǫ n ≪ , lim n →∞ ǫ n = 0 are an arbitraries real numbers. Then for anyfunction f ( t ) ∈ L (0 , T ) , f ( t ) > by (3,35’) and for n ≫ it follows that: f ( t ) = f ( t ) − f n − µ ( t ) + f n − µ ( t ) < | f ( t ) − f n − µ ( t ) | + f n − µ ( t ) < ǫ n + f n − µ ( t ) Then: F µ ( t ) = 2Γ µ − µ · Z t f ( τ )( t − τ ) µ dτ < µ − µ · Z t ǫ n + f n − µ ( τ )( t − τ ) µ dτ == 2Γ µ − µ · (cid:16) t − µ − µ · ǫ n + Z t f n − µ ( τ ( t − τ ) µ dτ (cid:17) Let us denote: F nµ ( t ) = 2Γ µ − µ · (cid:16) t − µ − µ · ǫ n + Z t f n − µ ( τ )( t − τ ) µ dτ (cid:17) (3 , Since the functions f n − µ ∈ L +1 − µ (0 , T ) and the function t − µ is increasing, the functions F nµ ( t ) are increasing on [0,T] and F µ ( t ) < F nµ ( t ) . Therefore, from the basis inequality(3,25) we have: z (0) · t < (cid:16) Z t z ( τ ) dτ (cid:17) · e k R t F nµ ( τ ) dτ (3 , ′ ) and Z t F nµ ( τ ) dτ = 2Γ µ − µ · (cid:16) t − µ (1 − µ )(2 − µ ) ǫ n + 11 − µ · Z t ( t − τ ) − µ f n − µ ( τ ) dτ (cid:17) Further similarly to the proof of an estimate (3,34), we obtain: k Z t w ( τ )( t − τ ) − µ dτ < (1 − µ ) · ln Z T F nµ ( τ ) dτ (3 , ′ ) Using the following limits: lim µ → Γ(1 − µ ) · (1 − µ ) = 1 , lim n →∞ k f n − µ ( t ) k L (0 ,T ) = k f ( t ) k L (0 ,T ) , lim n →∞ ǫ n = 0 and Remark 3.6 , passing to the limits µ → and n →∞ , from the inequality (3,34’) we get: Z T w ( τ ) dτ < · Z T f ( τ ) dτ (3 , Theorem 3,1 is proved. ◭◭ Proof of Theorem 2,1.Definition 3.
If the sequence of vector-functions { w n ( x, t ) } weakly converges tothe vector-function w ( x, t ) in the space L ( Q t ) , then we denote : w n ( x, t ) ⇀ w ( x, t ) . I.e. for an arbitrary vector-function u ( x, t ) ∈ L ( Q t ) the following convergence isvalid: ( w n ( x, t ) , u ( x, t )) L ( Q t ) → ( w ( x, t ) , u ( x, t )) L ( Q t ) as n → ∞ If the sequence of vector-functions { w n ( x, t ) } strongly converges to the vector-function w ( x, t ) in the space L ( Q t ) , then we denote: w n ( x, t ) ⇒ w ( x, t ) . I.e. lim n →∞ k w n ( x, t ) − w ( x, t ) k L ( Q t ) → . Lemma 3.4
Let the vector-function w ( x, t ) = ( w i ( x, t )) i =1 , , ∈ L ( Q t ) . I.e. k w ( x, τ ) k L ( Q t ) = P i =3 i =1 ( R Q t ( w i ) ( x, τ ) dxdτ ) / < ∞ . We define the following nonlinearoperator K on the vector-space L ( Q t ) : K ∗ (cid:16) w ( x, t ) (cid:17) = Gw j ( x, t ) G x j w ( x, t ) = (cid:16) X j =1 Gw j ( x, t ) G x j w i ( x, t ) (cid:17) i =1 , , Let us prove that K ∗ w n ( x, t ) ⇒ K ∗ w ( x, t ) (3 . as w n ( x, t ) ⇀ w ( x, t ) .It follows from this proposition that the operator K is compact on the vector-space L ( Q t ) . It is follows from book [3 p.42] ◮ Then k K ∗ w n ( x, t ) − K ∗ w ( x, t ) k L ( Q t ) ≤ c X i,j =1 k Gw jn ( x, t ) G x j w in ( x, t ) − Gw j ( x, t ) G x j w i ( x, t ) k L ( Q t ) Let us estimate a each member: (cid:13)(cid:13)(cid:13) Gw n ( x, t ) G x w n ( x, t ) − Gw ( x, t ) G x w ( x, t ) (cid:13)(cid:13)(cid:13) L ( Q t ) ≤ (3 . ≤ (cid:13)(cid:13)(cid:13) G x w n ( x, t ) (cid:16) Gw n ( x, t ) − Gw ( x, t ) (cid:17)(cid:13)(cid:13)(cid:13) L ( Q t ) + (cid:13)(cid:13)(cid:13) Gw ( x, t ) (cid:16) G x w n ( x, t ) − G x w ( x, t ) (cid:17)(cid:13)(cid:13)(cid:13) L ( Q t ) We obtain the following estimates, using the formula (2.12.2) in Proposition 5, the inequality(2,4) in Proposition 1 and Proposition 3 for µ : 5 / < µ < , : (cid:13)(cid:13)(cid:13) G x w n ( x, t ) (cid:16) Gw n ( x, t ) − Gw ( x, t ) (cid:17)(cid:13)(cid:13)(cid:13) L ( Q t ) ≤≤ c (cid:13)(cid:13)(cid:13) G x w n ( x, t ) (cid:13)(cid:13)(cid:13) L − µ , µ − ( Q t ) · (cid:13)(cid:13)(cid:13) G (cid:16) w n ( x, t ) − w ( x, t ) (cid:17)(cid:13)(cid:13)(cid:13) L µ − , − µ ( Q t ) ≤≤ c (cid:13)(cid:13)(cid:13) w n ( x, t ) (cid:13)(cid:13)(cid:13) L ( Q t ) · (cid:13)(cid:13)(cid:13) G (cid:16) w n ( x, t ) − w ( x, t ) (cid:17)(cid:13)(cid:13)(cid:13) L µ − , − µ ( Q t ) → , We obtain the following estimates , using the formula (2.12.1) in Proposition 5 and theinequality (2,4) in Proposition 1 for µ : 5 / < µ < , : (cid:13)(cid:13)(cid:13) Gw ( x, t ) (cid:16) G x w n ( x, t ) − G x w ( x, t ) (cid:17)(cid:13)(cid:13)(cid:13) L ( Q t ) ≤ ≤ c (cid:13)(cid:13)(cid:13) Gw ( x, t ) (cid:13)(cid:13)(cid:13) L − µ , µ − ( Q t ) · (cid:13)(cid:13)(cid:13) G x (cid:16) w n ( x, t ) − w ( x, t ) (cid:17)(cid:13)(cid:13)(cid:13) L µ , − µ ( Q t ) ≤≤ c (cid:13)(cid:13)(cid:13) w ( x, t ) (cid:13)(cid:13)(cid:13) L ( Q t ) · (cid:13)(cid:13)(cid:13) G x (cid:16) w n ( x, t ) − w ( x, t ) (cid:17)(cid:13)(cid:13)(cid:13) L µ , − µ ( Q t ) → , Note that by the weakly convergence w n ( x, t ) − w ( x, t ) ⇀ it follows that for anynumber n there is a constant c such that: k w n ( x, t ) k L ( Q t ) < c k w ( x, t ) k L ( Q t ) . Since G (cid:16) w n ( x, t ) − w ( x, t ) (cid:17) ∈ W , ( Q t ) , the space W , ( Q t ) is compactly enclosed into the space L µ − , − µ ( Q t ) [5 p.78], and on the space L µ − , − µ ( Q t ) the compact operator G translates theweekly convergence w n ( x, t ) − w ( x, t ) ⇀ to the strongly convergence, then the stronglyconvergence (3.40) is valid.Since G x (cid:16) w n ( x, t ) − w ( x, t ) (cid:17) ∈ W ( Q t ) , the space W ( Q t ) is compactly enclosed into thespace L µ , − µ ( Q t ) [5 p.78], and on the space L µ , − µ ( Q t ) the compact operator G x translatesthe weakly convergence w n ( x, t ) − w ( x, t ) ⇀ to the strongly convergence, then the stronglyconvergence (3.41) is valid . The strongly convergence (3,38) follows by (3,39), (3,40), (3,41).Lemma is proved. ◭ The proof of Theorem 2.1.
Let z ( x, t ) = ( z i ( x, t )) i =1 , , ∈ L ( Q t ) be an arbitraryvector-function. And the sequence vector-functions w n ( x, t ) weakly converges to the vector-function w ( x, t ) , i.e. w n ( x, t ) ⇀ w ( x, t ) . On the vector-space L ( Q t ) we define the followingnonlinear operator K p : K p ∗ (cid:16) w ( x, t ) (cid:17) = (cid:16) X j =1 G (cid:16) w j ( x, t ) + ∂p ( x, t ) ∂x j (cid:17) · G x j (cid:16) w i ( x, t ) + ∂p ( x, t ) ∂x i (cid:17)(cid:17) i =1 , , (3 , where the functions dp ( x,t ) dx i are defined by the functions w i ( x, t ) from the formula (2,16) inProposition 8. On the vector space w ( x, t ) = (cid:16) w i ( x, t ) (cid:17) i =1 , , ∈ L ( Q t ) by the formula(2,18) in Proposition 8 we have defined the following linear and bounded operator: P ∗ (cid:16) w i ( x, t ) (cid:17) i =1 , , = (cid:16) dp ( x, t ) dx i (cid:17) i =1 , , ∈ L ( x, t ) (3 , where the functions dp ( x,t ) dx i is defined by the functions w i ( x, t ) from the formula (2,16) inProposition 8. Since P is linear and bounded operator on L ( Q t ) , there exists the linear andbounded connected operator P ∗ . Let z ( x, t ) ∈ L ( Q t ) is an arbitrary vector-function and w n ( x, t ) ⇀ w ( x, t ) . Then: (cid:16)(cid:16) w ni ( x, t )+ dp ( x, t ) dx i (cid:17) i =1 , , , z ( x, t ) (cid:17) L ( Q t ) = (cid:16) ( w ni ( x, t )+ P ∗ ( w ni ( x, t )) i =1 , , , z ( x, t ) (cid:17) L ( Q t ) == (cid:16) w n ( x, t ) , z ( x, t ) + P ∗ ∗ z ( x, t ) (cid:17) L ( Q t ) → (cid:16) w ( x, t ) , z ( x, t ) + P ∗ ∗ z ( x, t ) (cid:17) L ( Q t ) == (cid:16) w ( x, t ) + P ∗ w ( x, t ) , z ( x, t ) (cid:17) L ( Q t ) as n → ∞ . I.e, it is proved that: w n ( x, t ) + P ∗ w n ( x, t ) ⇀ w ( x, t ) + P ∗ w ( x, t ) . Then, similarly to the proof in Lemma 3.4, proves that: K p ∗ w n ( x, t ) ⇒ K p ∗ w ( x, t ) . L ( Q t ) the nonlinear operator K p is compact. [3p.42]. Therefore, it follows by the Leray -Schauder’s theorem in Proposition 6, that thebasis equation (2,20) has at least one solution w ( x, t ) ∈ L ( Q t ) and it follows from Theorem3.1 that: k w ( x, t ) k L ( Q T ) ≤ √ · k f ( x, t ) k L ( Q T ) . Then, it follows by Proposition 8 that thereexists the smooth solution u ( x, t ) ∈ W , ( Q T ) T H ( Q t ) . But the Navier-Stokes problem hasthe unique smooth solution [3p.139]. Therefore, w ( x, t ) ∈ L ( Q T ) is the unique solution to(2,20). The existence and smoothness of the solution to Navier-stokes equation is proved. ◭
4. The Navier-Stokes problem for the inhomogeneous boundary condition.
Let Ω ⊂ R be a finite domain bounded by a Lipschitz surface ð Ω and Q T = Ω × [0 , T ] , S = ð Ω × [0 , T ] , x = ( x , x , x ) and u ( x, t ) = ( u i ( x, t ) i =1 , , , f ( x, t ) = ( f i ( x, t ) i =1 , , are vector-functions. Here T > 0 is an arbitrary real number. The Navier-Stokes equationsare given by: ∂u i ( x, t ) ∂t − ρ △ u i ( x, t ) − X j =1 u j ( x, t ) ∂u i ( x, t ) ∂x j + ∂p ( x, t ) ∂x i = f i ( x, t ) (4 . ,div u ( x, t ) = X i =1 ∂u i ( x, t ) ∂x i = 0 , i = 1 , , The Navier-Stokes problem 1.
Find a vector-function u ( x, t ) = ( u i ( x, t )) i =1 , , : Ω × [0 , T ] → R , the scalar function p ( x, t ) : Ω × [0 , T ] → R satisfying the equation (4.1) andthe following initial condition u ( x,
0) = a ( x ) , u ( x, t ) | ∂ Ω × [0 ,T ] = 0 (4 . where div a ( x ) = da ( x ) dx + da ( x ) dx + da ( x ) dx = 0 and a ( x ) ∈ W (Ω) . Theorem 4.1.
For any right-hand side f ( x, t ) ∈ L ( Q t ) in equation (4.1) and forany real numbers ρ > , t > , the Navier-Stokes problem-1 has a unique smooth solution u ( x, t ) : u ( x, t ) ∈ W , ( Q t ) ∩ H ( Q t ) , the scalar function p ( x, t ) : p x i ( x, t ) ∈ L ( Q t ) satisfyingto (4.1) almost everywhere on Q t , and to the initial conditions (4,2). The following estimateholds: k u ( x, t ) k W , ( Q t ) + (cid:13)(cid:13)(cid:13) ∂p ( x, t ) ∂x i (cid:13)(cid:13)(cid:13) L ( Q t ) ≤ c (cid:16) k f k L ( Q t ) + k a ( x ) k W (Ω) (cid:17) (4 . ◮ In 1941 Hopf proved that this problem has a weak solution u ( x, t ) : k u ( x, t ) k L (Ω) + 2 ρ Z t k u x ( x, τ ) k L (Ω) dτ < k a ( x ) k W (Ω) + c Z t k f ( x, τ ) k L (Ω) dτ (4 . and lim t → k u ( x, t ) − a ( x ) k L (Ω) = 0 . [3 p. ◭ The problem 2.
Find a vector-function u ( x, t ) = ( u i ( x, t )) i =1 , , : Ω × [0 , T ] → R satisfying the following equation and the initial condition: d u ( x, t ) dt − ρ △ u ( x, t ) = 0; u ( x,
0) = a ( x ) , u ( x, t ) | ∂ Ω × [0 ,T ] = 0 (4 , ◮ It follows by div a ( x ) = 0 and u ( x, t ) | ∂ Ω × [0 ,T ] = 0 that: div u ( x, t ) = 0 for any t > 0.And it follows by a ( x ) ∈ W (Ω) that: | u ( x, t ) | ≤ c k a ( x ) k w (Ω) , | u ( x,t ) dx i | ≤ c k a ( x ) k w (Ω) . ◭ v ( x, t ) = u ( x, t ) − u ( x, t ) satisfies the following system ofequations: ∂v i ( x, t ) ∂t − ρ △ v i ( x, t ) − X j =1 v j ( x, t ) ∂v i ( x, t ) ∂x j − X j =1 u j ( x, t ) ∂v i ( x, t ) ∂x j −− X j =1 v j ( x, t ) ∂u i ( x, t ) ∂x j − X j =1 u j ( x, t ) ∂u i ( x, t ) ∂x j + ∂p ( x, t ) ∂x i = f i ( x, t ) (4 . ,div v ( x, t ) = X i =1 ∂v i ( x, t ) ∂x i = 0 , i = 1 , , and the following initial conditions: v ( x,
0) = 0 , v ( x, t ) | ∂ Ω × [0 ,T ] = 0 (4 , Similarly, we introduce the unknown vector-function (cid:16) w i ( x, t ) (cid:17) i =1 , , ∈ L ( Q t ) : ∂v i ( x, t ) ∂t − ρ △ v i ( x, t ) − ∂p ( x, t ) ∂x i = w i ( x, t ) (4 . div v ( x, t ) = X i =1 ∂v i ( x, t ) ∂x i = 0 , i = 1 , , and the following initial conditions: v ( x,
0) = 0 , v ( x, t ) | ∂ Ω × [0 ,T ] = 0 (4 , ′ ) In the Proposition 8 we have proved that for any right-side w ( x, t ) ∈ L ( Q t ) this problemhas a unique solution (cid:16) v i ( x, t ) (cid:17) i =1 , , ∈ W , ( Q t ) and k v ( x, t ) k W , ( Q t ) < c · k w ( x, t ) k L ( Q t ) . It follows from (4,8) that. v i ( x, t ) = Z t Z Ω G ( x, t ; ξ, τ ) (cid:16) w i ( ξ, τ ) + ∂p ( ξ, τ ) ∂ξ i (cid:17) dξ dτ ; i = 1 , , . Using (4,8) and this formula, we rewrite the Navier-Stokes equation (4,6) as: w i ( x, t ) − X j =1 G (cid:16) w i ( ξ, τ ) + ∂p ( ξ, τ ) ∂ξ i (cid:17) · G x j (cid:16) w i ( ξ, τ ) + ∂p ( ξ, τ ) ∂ξ i (cid:17) −− X j =1 G x j (cid:16) w i ( ξ, τ ) + ∂p ( ξ, τ ) ∂ξ i (cid:17) · u j ( x, t ) − X j =1 G (cid:16) w j ( ξ, τ ) + ∂p ( ξ, τ ) ∂ξ j (cid:17) · du i ( x, t ) dx j − (4 , − X j =1 u j ( x, t ) · du i ( x, t ) dx j = f i ( x, t ) w ( t ) < f ( t ) + b (cid:16) Z t w ( τ ) + p ( τ )( t − τ ) µ dτ (cid:17) + c ( T ) · Z t w ( τ ) + p ( τ )( t − τ ) µ dτ (4 , where w ( τ ) = k w ( x, τ ) k L (Ω) ≥ p ( τ ) = X (cid:13)(cid:13)(cid:13) ∂p ( x, τ∂x i (cid:13)(cid:13)(cid:13) L (Ω) ≥ f ( t ) = k f ( x, t ) k L (Ω) + X j =1 (cid:13)(cid:13)(cid:13) u j ( x, t ) · du i ( x, t ) dx j (cid:13)(cid:13)(cid:13) L (Ω) ≥ . ′ ) c ( T ) = k a ( x ) k L (Ω) + c Z T k f ( x, τ ) k L (Ω) dτ From the inequality a · b < a + b the following inequality holds: Z t w ( τ ) + p ( τ )( t − τ ) µ dτ = Z t w ( τ ) + p ( τ )( t − τ ) µ/ · t − τ ) µ/ dτ < Z t ( w ( τ ) + p ( τ )) ( t − τ ) µ dτ + t − µ − µ By this inequality and the inequality (3,4) in Lemma 3.2 we rewrite the inequality (4,11) as: w ( t ) < (cid:16) f ( t ) + t − µ − µ (cid:17) + (cid:16) b + c ( T ) (cid:17) · Z t ( w ( τ ) + p ( τ )) ( t − τ ) µ dτ Similarly, based on the proof of Lemma 3.2, we derive the following inequality: w ( t ) < (cid:16) f ( t ) + t − µ − µ (cid:17) + (cid:16) b + c ( T ) (cid:17) · (1 + c ) Z t w ( τ )( t − τ ) µ dτ (4 . From the Theorem 3.1,using this inequality, similarly we obtain: k w ( x, t ) k L ( Q T ) ≤ √ · c ( T ) · k f u ( x, t ) k L ( Q T ) (4 , where the vector-function f u ( x, t ) = (cid:16) f i ( x, t ) (cid:17) i =1 , , + (cid:16) P u j ( x, t ) du i ( x,t ) dx j (cid:17) i =1 , , and c(T)are defined by (4,11’). And we present the basic equation (4,10) as: w ( x, t ) − (cid:16) K p + K (cid:17) ∗ (cid:16) E + P (cid:17) w ( x, t ) = f ( x, t ) + X u j ( x, t ) · d u ( x, t ) dx j (4 , where the operator (cid:16) K p (cid:17) is defined by (3,42) , the operator P is defined by (3,43) and E isan identify operator , i.e. E (cid:16) w ( x, t ) (cid:17) = w ( x, t ) . The operator (cid:16) K (cid:17) ∗ (cid:16) E + P (cid:17) is definedas: K ∗ (cid:16) E + P (cid:17)(cid:16) w ( x, t ) (cid:17) = X j =1 (cid:16) du i ( x, t ) dx j · G + u j ( x, t ) · G x j (cid:17) ∗ (cid:16) E + P (cid:17)(cid:16) w ( x, t ) (cid:17) (4 , As the proof of Theorem 2.1(p.19) and definitions of the operators K p , P, K by the formulas(3,42), (3,43) and (4,15) it is proves that the following operators: (cid:16) K p (cid:17) ∗ (cid:16) E + P (cid:17) ; (cid:16) K (cid:17) ∗ (cid:16) E + P (cid:17) are compact on the vector-space L ( Q t ) . Similarly, using Lerau-Schauder’s theorem and the estimate (4.13), the existence and smoothness of solution to theNavier-Stokes problem with u ( x,
0) = a ( x ) = 0 is proved. ◭ Thank you for your attention.3
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