The Generalized Incompressible Navier-Stokes Equations in Besov Spaces

Transcription

1 Dynamics of PDE, Vol1, No4, , 2004 The Generalize Incompressible Navier-Stokes Equations in Besov Spaces Jiahong Wu Communicate by Charles Li, receive July 21, 2004 Abstract This paper is concerne with global solutions of the generalize Navier-Stokes equations The generalize Navier-Stokes equations here refer to the equations obtaine by replacing the Laplacian in the Navier-Stokes equations by the more general operator ( ) α with α > 0 It has previously been shown that any classical solution of the -imensional generalize Navier- Stokes equations with α is always global in time Thus, attention 4 here is solely focuse on the case when α < We consier solutions 4 emanating from initial ata in several Besov spaces an establish the global existence an uniqueness of the solutions when the corresponing initial ata are comparable to the iffusion coefficient in these Besov spaces Contents 1 Introuction Besov spaces Bp, r solutions Ḃ r an Br Solutions Scaling invariance 396 References Introuction Whether or not every smooth solution of the 3D Navier-Stokes equations is global in time has been intensively investigate but yet remains open In this paper, we consier a more general form of the incompressible Navier-Stokes equations, namely (11) t u + u u + P = ν( ) α u, u = 0, 1991 Mathematics Subject Classification Primary 35Q30; Seconary 76D03 Key wors an phrases The generalize Navier-Stokes equations, global solutions, Besov spaces 381 c 2004 International Press

2 382 JIAHONG WU where ν > 0 an α > 0 are real parameters (11) becomes the Navier-Stokes equations in the case of α = 1 an will thus be referre to as the generalize Navier-Stokes (GNS) equations The goal of this paper is to establish the global existence an uniqueness of solutions of (11) when the corresponing initial ata (12) u(x, 0) = u 0 (x) are prescribe in several functional spaces etaile below We consier the general -imensional GNS equations When α , any classical solution of (11) is always global in time ([14]) In particular, smooth solutions of the 2D Navier-Stokes equations an the 3D GNS equations with α 5 4 o not evelop finite-time singularities This paper focuses its attention on the case when α < We seek global solutions emanating from initial ata in several Besov spaces Besov spaces inclue many of the frequently-use function spaces such as the Sobolev spaces an the Höler spaces an constitute a very natural setting for stuying solutions of various partial ifferential equations ([1],[4],[5],[7],[12],[13],[15]) Our stuy here covers the inhomogeneous Besov spaces Bp, r an B, r an the homogeneous Besov space Ḃ r Our major results can be roughly summarize as follows Assuming ν > 0 an α < , the GNS equations (11) have a unique an global solution when the norm of u 0 is comparable to ν in any one of the spaces: i) B r p, with 1 p 2, r > 1 an r > 1 + p 2α; ii) B r with r > 1 an r α; iii) Ḃ r with r = α > 1; iv) B r with 1 < q <, r > 1 an r > α q Precise statements an their proofs will be eferre until Section 3 an Section 4 A particular consequence of these results is the global existence of solutions starting with ata in the usual Sobolev space H r with r > α The inex r = α appears to be critical in a sense that we now explain Solutions of the GNS equations (11) are scaling invariant That is, if (u, P) is a solution of the GNS equations, then (u λ, P λ ) is also a solution of the GNS equations, where u λ (x, t) = λ 2α 1 u(λx, λ 2α t), P λ (x, t) = λ 4α 2 P(λx, λ 2α t) As we shall show in Section 5, the norm of u λ is virtually invariant in the homogeneous Besov spaces Ḃ1+ 2 2α an Ḃ1+ 2 2α, namely u λ Ḃ1+ 2 2α u Ḃ1+, 2 2α u λ Ḃ1+ 2 2α u Ḃ1+, 2 2α where means the equivalence between two norms These invariance properties allow one to argue that the inex restriction in iii) may not be relaxe to r α an that the inhomogeneous Besov space in i) with p = 2 may not be replace by the homogeneous Besov space Ḃr with r > 1 an r > α Further explanations will be provie in Section 5 We remark that Cannone, Planchon, Lemarié-Rieusset an others have previously stuie mil solutions of the Navier-Stokes equations in Besov spaces via the fixe point arguments base on the continuity of the bilinear form in these spaces

3 THE GENERALIZED NAVIER-STOKES EQUATIONS 383 One relate result is the small-ata global (in time) existence of mil solutions in the homogeneous Besov spaces Ḃ 3 p 1 p, with 1 < p < 3 ([3]) This result was later extene to Ḃ 3 p 1 p, with p > 3, but the uniqueness of such mil solutions is unknown ([8]) In comparison, this paper is mainly concerne with solutions in more regular Besov spaces in which the solutions of the GNS equations can be shown to be unique Before presenting our major results in Section 3 an Section 4, we provie the efinitions of Besov spaces an some embeing relations an inequalities in Section 2 We also nee several other inequalities involving Besov spaces, which are left to the Appenix 2 Besov spaces In this section, we provie the efinitions of the homogeneous an the inhomogeneous Besov spaces They are efine through the Littlewoo-Paley ecomposition Several relate embeing relations an inequalities will also be given here Except for Proposition 23, most of the materials in this section are classical an we refer the reaer to the books [2], [6], [9], [10] for more etails We start with the Fourier transform The Fourier transform f of a L 1 -function f is given by (21) f(ξ) = R f(x)e 2π x ξ x More generally, the Fourier transform of any f S, the space of tempere istributions, is given by ( f, g) = (f, ĝ) for any g S, the usual Schwarz class The Fourier transform is a boune linear bijection from S to S whose inverse is also boune The fractional power of the Laplacian can be efine in terms of the Fourier transform For a general exponent β R, ( ) β/2 f(ξ) = (2π ξ ) β f(ξ) For notational convenience, we will write Λ for ( ) 1/2 from now on Another important family of operators are the Riesz transforms For 1 l, R l f(ξ) = i ξ l ξ f(ξ) To efine the Besov spaces, we fix some notation { } S 0 = φ S, φ(x)x γ x = 0, γ = 0, 1, 2, R Its ual is given by S 0 = S /S0 = S /P, where P is the space of multinomials In other wors, two istributions in S 0 are ientifie as the same if their ifference is a multinomial

4 384 JIAHONG WU We now introuce a yaic partition of R We choose φ 0 S(R ) such that φ 0 is even, suppφ 0 = {ξ : 2 1 ξ 2}, an φ 0 > 0 on A 0, where A j = {ξ : 2 j 1 < ξ < 2 j+1 } for j Z For j Z, efine an efine Φ j S by φ j (ξ) = φ 0 (2 j ξ) Φ j (ξ) = φ j(ξ) j φ j(ξ) It follows that both Φ j an Φ j are even an satisfy the following properties: Furthermore, Φ j (ξ) = Φ 0 (2 j ξ), supp Φ j A j, Φ j (x) = 2 j Φ 0 (2 j x) k= Φ k (ξ) = { 1, if ξ R \ {0}, 0, if ξ = 0 Thus, for a general function ψ S, we have Φ k (ξ) ψ(ξ) = ψ(ξ) for ξ R \ {0} But, if ψ S 0, then That is, for ψ S 0, an hence (22) k= k= Φ k (ξ) ψ(ξ) = ψ(ξ) for all ξ R k= k= Φ k ψ = ψ Φ k f = f in the weak* topology of S 0 for any f S 0 Now let Ψ C0 (R ) be even an satisfy Ψ(ξ) = 1 Φ k (ξ) Then, for any ψ S, an hence Ψ ψ + (23) Ψ f + in S for any f S k=0 Φ k ψ = ψ 0 Φ k f = f k=0 To efine the homogeneous Besov spaces, we set (24) j f = Φ j f, j = 0, ±1, ±2,

5 THE GENERALIZED NAVIER-STOKES EQUATIONS 385 Suppose that s R an p, q [1, ] We say that f Ḃs p,q if f S 0 an ( 2 js ) q j f L p < j= Ḃp,q s is a Banach space when equippe with the norm ( ( ) j= 2 js q 1/q j f L p), if q <, (25) f Ḃs p,q sup <j< 2 js j f L p, if q = Ḃ s p,q with this norm will be referre to as homogeneous Besov space To introuce the inhomogeneous Besov spaces, we efine 0, if j 2, (26) j f = Ψ f, if j = 1, Φ j f, if j = 0, 1, 2, For s R an p, q [1, ], we say that f Bp,q s if f S an 1/q ( 1 f L p + 2 js ) q j f L p < j=0 Bp,q s is a Banach space with the norm ( ( ) 1 f L p + j=0 2 js q 1/q j f L p), if q <, (27) f B s p,q 1 f L p + sup 0 j< 2 js j f L p, if q = B s p,q (28) (29) (210) with this norm will be referre to as inhomogeneous Besov space We now point out several simple facts concerning the operators j : j k = 0, if j k 2; j S j k I, as j ; k= k (S j 1 f j f) = 0, if j k 4 I in (29) enotes the ientity operator an (29) is simply another way of writing (22) an (23) Finally, we caution that j with j 1 associate with the homogeneous Besov space Ḃs p,q are efine ifferently from those associate with the inhomogeneous Besov space Bp,q s Therefore, it will be unerstoo that j with j 1 in the context of the homogeneous Besov space are given by (24) an by (26) in the context of the inhomogeneous Besov space The Besov spaces efine above obey various inclusion relations In particular, we have the following theorem Theorem 21 Assume that β R an p, q [1, ] 1) If β > 0, then B β p,q Ḃβ p,q

6 386 JIAHONG WU 2) If β 1 β 2, then Bp,q β2 Bβ1 p,q This inclusion relation is false for the homogeneous Besov spaces 3) If 1 q 1 q 2, then Ḃβ p,q 1 Ḃβ p,q 2 an Bp,q β 1 Bp,q β 2 4) (Besov embeing theorem) If 1 p 1 p 2 an β 1 = β 2 +( 1 p 1 1 p 2 ), then Ḃβ1 p 1,q(R ) Ḃβ2 p (R ) an B β1 p 1,q(R ) B β2 p (R ) In aition, the usual Sobolev spaces are a special type of Besov spaces an thus follow similar embeing relations We recall that for β 0, Ḣ β = {f S : ξ β f(ξ) L 2} an H β = It is not har to check that By 3) of Theorem 21 is { f S : (1 + ξ 2 ) β/2 f(ξ) L 2} Ḣ β (R ) = Ḃβ 2,2 (R ) an H β (R ) = B β 2,2 (R ) (211) Ḃ β Ḣβ Ḃβ, Bβ Hβ B β We now turn to Bernstein s inequalities When the Fourier transform of a function is supporte on a ball or an annulus, the L p -norms of its erivatives can be boune in terms of the norms of the function itself Inequalities of this nature are referre to as Bernstein s inequalities The classical Bernstein s inequalities only allow integer erivatives They can actually be extene to involve fractional erivatives In the following, we shall first state as a proposition the classical Bernstein s inequalities an then present the fractional Bernstein inequalities Proposition 22 Let k 0 be an integer an 1 p q 1) If supp f {ξ R : ξ cλ}, then sup D γ f L q cλ k+(1/p 1/q) f L p γ =k 2) If supp f {ξ R : c 1 λ ξ c 2 λ}, then c 3 λ k+(1/p 1/q) sup D γ f L q c 4 λ k+(1/p 1/q) f L p, γ =k where c, c 1, c 2, c 3 an c 4 are constants inepenent of λ The proof of this proposition is classical an can be foun in [6] We now state the generalize Bernstein s inequalities involving fractional erivatives In the following proposition, we still use c (or c with a subinex) to enote various constants whose values may be ifferent from line to line Occasionally, we use C with a subinex to mark some crucial constants Proposition 23 Assume that β R an 1 p q 1) If β 0 an supp f {ξ R : ξ cλ}, then Λ β f L q cλ β+(1/p 1/q) f L p, 2) If supp f {ξ R : c 1 λ ξ c 2 λ}, then c 3 λ β+(1/p 1/q) Λ β f L q c 4 λ β+(1/p 1/q) f L p

7 THE GENERALIZED NAVIER-STOKES EQUATIONS 387 Proposition 23 is a simple extension of Proposition 22 The statements in Proposition 23 are communicate to the author by Davi Ullrich [11] To establish the major results of this paper, we also nee several other inequalities involving Besov spaces They inclue the logarithmic Besov inequalities, two commutator estimates, an some estimates for the usual prouct of two functions in Besov spaces Instea of presenting them here, we leave to the Appenix 3 B r p, solutions In this section, we stuy solutions of the initial-value problem (IVP) for the GNS equations, namely t u + u u + P = ν( ) α u, (31) u = 0, u(x, 0) = u 0 (x) Attention will be mainly focuse on u 0 B(R r ) Our goal is to establish the existence an uniqueness of solutions to (31) with u 0 B r satisfying suitable conitions The major results are presente in Theorem 32 For the purpose of proving this theorem, we first present an a priori estimate state in Proposition 31 We remark that Theorem 32 can be extene to cover any initial atum in Bp, s with 1 p 2 through an embeing theorem We start with an important a priori estimate Proposition 31 Let r R an s > Then any solution (u, P) of the IVP (31) obeys the following ifferential inequality (32) t u B r + cν u B r+2α c u B s u B r, where c s are constants with possible epenence on r an s only Proposition 31 contains a major ingreient in proving Theorem 32 state below As we have mentione in the introuction, the issue of global smooth solutions has been resolve for (31) with α > Therefore, we shall assume that α < here Theorem 32 Let ν > 0 an α < Assume that u 0 B r with an satisfies r > 1, r > α (33) u 0 B r C 0 ν for some suitable constant C 0 Then the IVP (31) has a unique global solution (u, P) satisfying an u L ([0, ); B) r L 1 ([0, ); B r+2α ) C([0, ); B ), P L 1 ([0, ); B r ) u(, t) B r 2C 0 ν, for all t > 0

8 388 JIAHONG WU We make two remarks Remark Using the Besov embeing theorem (Theorem 21), we can exten Theorem 32 to cover any initial atum u 0 Bp, s with 1 p 2 In fact, Theorem 21 states that for 1 p 2 an s = r + ( 1 p 1 2 ), B s p, Br Thus, u 0 B s p, with 1 p 2 an s > 1 + p 2α implies that u 0 B r with r > α Therefore, if u 0 B s p, with 1 p 2, s > 1, s > 1 + p 2α, then Theorem 32 implies that (31) has a unique global solution Remark Because of the embeing relations in (211), namely H r = B r 2,2 Br, another special consequence of Theorem 32 is the global existence an uniqueness of solutions of (31) corresponing to any initial atum in the usual Sobolev space H r with r > α We now procee to the proofs of Proposition 31 an Theorem 32 Proof of Proposition 31 For each j Z, we apply j to the GNS equations in (31), t j u + u j u + ν( ) α j u = [u, j ]u j P, where the brackets [, ] in [u, j ] represents the commutator, namely [u, j ]u = u j u j (u u) Multiplying by j u an integrating with respect to x leas to (34) where t ju 2 L2 + ν I = II + III, I = Λ α j u 2 x, II = [u, j ]u j u x, III = ( j P) j u x We now evaluate these terms By Proposition 23, I has the following lower boun (35) I c 2 2α j j u 2 L 2 To eal with II, we first apply Höler s inequality an then the commutator estimate in Proposition A2 to obtain (36) II j u L 2 [u, j ]u L 2 c u L j u 2 L 2 The estimate of III is more complex an the following lemma is evote to it

9 THE GENERALIZED NAVIER-STOKES EQUATIONS 389 Lemma 33 For j Z an any solution (u, P) of the IVP (31), we have (37) ( j P) j ux c u L ju 2 L 2 Proof of Lemma 33 secon equation, we have Applying to the first equation in (31) an using the P = R k R l (u k u l ), where R with a subinex enotes a 2D Riesz transform, an the repeate inices k an l are summe Therefore, j P = R k R l ( j (u l u k ) j (u k u l )) = R k R l ([u l, j ]u k + [u k, j ]u l ) = III 1 + III 2 R k R l (u l ( j u k ) + u k ( j u l )) Corresponingly, the integral to be boune is ivie into two parts: (38) ( j P) j u x = III 1 j u x + III 2 j u x For the first integral, we have III 1 j ux ju L 2 III 1 L 2 j u L 2 [u l, j ]u k L 2 Applying the commutator estimate in Proposition A2 yiels (39) III 1 j u x c u L ju 2 L 2 To boun the secon term in (38), we integrate by parts, III 2 j u x = R k R l (u l m ( j u k )) j u m x R k R l (u k m ( j u l )) j u m x = R k R l (( m u l ) j u k ) j u m x R k R l (( m u k ) j u l ) j u m x j,k It is then clear that (310) III 2 j u x c u L ju 2 L 2 (37) is obtaine by combining (38), (39) an (310) This completes the proof of Lemma 33 We now resume the proof of Proposition 31 Collecting the estimates in (35), (36) an (37), we obtain t ju L 2 + c ν2 2αj j u L 2 c u L j u L 2

10 390 JIAHONG WU Multiplying by 2 jr an taking sup j leas to (311) t u B r + c ν u B r+2α c u L u B r Note that we have switche t an sup j This can be justifie using the Monotone Convergence Theorem Finally, we apply Proposition A1 to boun u L in terms of u B s with s > 1 + 2, namely (312) u L c u B s 1 c u B s Inserting (312) in (311) finishes the proof Proof of Theorem 32 We apply the metho of successive approximation It consists of constructing a successive approximation sequence {(u (n), P (n) )} an showing its convergence to (u, P), the solution of the IVP (31) Consier a successive approximation sequence {(u (n), P (n) )} satisfying u (0) = 0, P (0) = 0, t u (n+1) + u (n) u (n+1) = P (n+1) ν( ) α u (n+1), (313) u (n+1) = 0, u (n+1) (x, 0) = u (n+1) 0 (x) = S n+1 u 0 (x) To show that {(u (n), P (n) )} converges, we prove that i) {(u (n), P (n) )} is boune uniformly in ( L ([0, ); B) r L 1 ([0, ); B r+2α )) L 1 ([0, ); B); r ii) {(u (n), P (n) )} is a Cauchy sequence in ( L ([0, ); B ) L1 ([0, ); B ) ) L 1 ([0, ); B ) To establish i), we procee as in the proof of Proposition 31 That is, we start with the secon equation in (313) an estimate u (n+1) in B r We eal with the term involving P (n+1) as in Lemma 37 It is boune by (314) j P (n+1) L 2 c ( u (n) L j u (n+1) L 2 + u (n+1) L j u (n) L 2) After going through the steps as in proof of Proposition 31, we arrive at t u(n+1) B r + c ν u (n+1) B r+2α c ( u (n) L u (n+1) B r + u (n+1) L u (n) B r ) Since r+2α > 1+ 2, we apply Proposition A1 to boun u(n) L an u (n+1) L Therefore, t u(n+1) B r + c ν u (n+1) B r+2α c ( u (n) B r+2α u (n+1) B r + u (n+1) B r+2α u (n) B r ) This inequality allows us to show inuctively that if (33) hols, namely u 0 B r C 0 ν,

11 THE GENERALIZED NAVIER-STOKES EQUATIONS 391 then for any t > 0, (315) sup u (n) (, τ) B r + c ν τ [0,t] t 0 u (n) (, τ) B r+2α τ 2C 0 ν Thus {u (n) } is boune uniformly in L ([0, ); B) r L 1 ([0, ); B r+2α ) To see the uniform bouneness of {P (n) } in L 1 ([0, ); B r ), we note that P (n+1) = R k R l (u (n) k u (n+1) l ) We then apply Proposition A4 an Proposition A1 to obtain P (n+1) B r k,l=1 u (n) k u (n+1) l B r c( u (n) L u (n+1) B r + u (n+1) L u (n) B r ) c( u (n) B r+2α It then follows from (315) that for a constant c, This completes the proof of i) u (n+1) B r + u (n+1) B r+2α u (n) B r ) P (n) L 1 ([0, );B r ) c ν To establish ii), we consier the ifferences v (n+1) = u (n+1) u (n), Q (n+1) = P (n+1) P (n), which satisfy t v (n+1) + u (n) v (n+1) + νλ 2α v (n+1) = Q (n+1) + v (n) u (n), (316) v (n+1) = 0, v (n+1) (x, 0) = v (n+1) 0 (x) = n+1 u 0 We shall show that for any integer n > 0 (317) sup v (n) (, τ) B + c ν τ [0,t] t 0 v (n) (, τ) B τ u 0 B r 2 (n 3) vali for any t > 0 To establish (317), we estimate {v (n) } in B After going through a similar proceure as above, we obtain t v(n+1) B + c ν v (n+1) B Q (n+1) B (318) + v (n) u (n) B + c sup 2 ()j [u (n), j ]v (n+1) L 2 j To obtain suitable bouns for the terms on the right-han sie, we apply the commutator estimate in Proposition A3 (319) sup 2 ()j [u (n), j ]v (n+1) L 2 j c ( u (n) L v (n+1) B + v (n+1) L u (n) B r ) The term involving Q (n+1) can be estimate similarly as in (314), but we apply the commutator estimate in Proposition A3 rather than the one in Proposition A2 (320) Q (n+1) B c ( u (n) L v (n+1) B + v (n+1) L u (n) B r )

12 392 JIAHONG WU Since r > 1, we apply Proposition A4 to boun the prouct v (n) u (n), (321) v (n) u (n) B c ( v (n) L u (n) B + v (n) B u (n) L ) We further apply Proposition A1 to boun the L -norms in (319), (320) an (321) an then insert the resulting estimates in (318) This leas us to the inequality t v(n+1) B + c ν v (n+1) B c( u (n) B r+2α v (n+1) B +c( v (n) B Integrating this inequality over [0, t], we obtain (322) sup v (n+1) (, τ) B + c ν τ [0,t] ( n+1 u 0 B + c t 0 u (n) B r+2ατ + sup τ [0,t] Noticing that n+1 u 0 B u (n) B r t + v (n+1) B u (n) B r ) u (n) B r + v (n) B t 0 v (n+1) (, τ) B u (n) B r+2α) τ sup v (n+1) B + sup v (n) B τ [0,t] τ [0,t] 0 ( v (n) B + v (n+1) B ) ) τ u 0 B r 2 n, (322) allows us to prove by inuction that (317) hols As a consequence, we have shown that {u (n) } is a Cauchy sequence in L ([0, ); B ) L1 ([0, ); B ) The boun for {Q (n) } can be obtaine in a similar fashion as for {P (n) } Accoring to (316), Therefore, Q (n+1) (, t) B ( +c( Q (n+1) = R k R l (u (n) k v (n+1) l ( c( v (n) B v (n) B + v (n) k u (n) l ) + v (n+1) B ) + v (n+1) B ) u (n) B u (n) B Since {u (n) } is boune uniformly in L ([0, ); B r ) L1 ([0, ); B r+2α ) an {v (n) } in L ([0, ); B ) L1 ([0, ); B ), we obtain that {Q (n) } is boune uniformly in L 1 ([0, ); B ) That is, {P (n) } is a Cauchy sequence in L 1 ([0, ); B This completes the proof of ii) We can now conclue from ii) that there exists a unique such that (u, P) ( L ([0, ); B ) L1 ([0, ); B ) ) L 1 ([0, ); B ) u (n) u in L ([0, ); B ) L1 ([0, ); B ), P (n) P in L 1 ([0, ); B ) )

13 THE GENERALIZED NAVIER-STOKES EQUATIONS 393 Because of i), (u, P) actually belongs to ( L ([0, ); B r ) L1 ([0, ); B r+2α )) L 1 ([0, ); B r ) In aition, {u (n) } an u are both absolutely continuous from [0, ) to B, or simply u (n), u C([0, ); B ) To prove this fact, we rewrite the secon equation in (313) in the integral form, with u (n+1) (x, t) = u (n+1) 0 (x) + t 0 g (n+1) (x, τ)τ g (n+1) = u (n) u (n+1) P (n+1) ν( ) α u (n+1) Since g (n+1) has the following boun g (n+1) (, t) B c( u (n) B u (n+1) B r + u (n) B + P (n+1) B + ν u (n+1) B an each term on the right is in L 1 ([0, )), we have g (n+1) (, t) B L 1 ([0, )) u (n+1) B r+2α) Therefore, {u (n+1) } is absolutely continuous from [0, ) to B an so is u Finally, letting n in (313), we obtain that (u, P) satisfies the GNS equations in (31) This completes the proof of Theorem 32 4 Ḃ r an Br Solutions We continue in this section the stuy of solutions of the IVP (31), but we now assume that u 0 is either in Ḃr or in Br with q [1, ) The major results are presente in three theorems The first theorem is on solutions in the homogeneous Besov space Ḃr while the secon one is on solutions in the inhomogeneous space B r The thir theorem concerns solutions in Br with 1 < q < It appears that the conclusion in the thir theorem is invali for Ḃr We first state the theorem for u 0 Ḃr Theorem 41 Consier the solutions of the IVP (31) with ν > 0 an α < If u 0 Ḃr with an r = α > 1 (41) u 0 Ḃr C 1 ν for some suitable constant C 1, then the IVP (31) has a unique global solution (u, P) satisfying u L ([0, ); Ḃr ) L 1 ([0, ); Ḃr+2α ) C([0, ); Ḃ ), P L 1 ([0, ); Ḃr )

14 394 JIAHONG WU an u(, t) Ḃr 2C 1 ν, for any t > 0 A similar result hols for u 0 in the inhomogeneous Besov space B r, but the conition on r can be relaxe to r α Theorem 42 Consier the solutions of (31) with ν > 0 an α < If u 0 B r with an r > 1, r α (42) u 0 B r C 2 ν for some suitable constant C 2, then the IVP (31) has a unique global solution u satisfying an u L ([0, ); B r ) L 1 ([0, ); B r+2α ) C([0, ); B ), Theorem 43 If u 0 B r with P L 1 ([0, ); B r ) u(, t) B r 2C 2 ν for any t > 0 an 1 < q <, r > 1, r > α q (43) u 0 B r C 3 ν, for some suitable constant C 3, then the IVP (31) has a unique global solution (u, P) satisfying an u L ([0, ); B r ) L q ([0, ); B r+2α q ) C([0, ); B ) P L q ([0, ); B r ) (44) u(, t) B r 2C 3 ν for any t > 0 We now prove these theorems Proof of Theorem 41 The major tool is the metho of successive approximation Since the etails resemble those in the proof of Theorem 32, it is reunant to provie a full proof of this theorem Instea, we prove a major a priori estimate, which can be easily extene into a complete proof of Theorem 41 As in the proof of Proposition 31, we have (45) t ju L 2 + cν 2 2αj j u L 2 c u L j u L 2 Multiplying (45) by 2 jr an summing over j Z yiels t u Ḃ + cν r u Ḃr+2α c u L u Ḃr

15 THE GENERALIZED NAVIER-STOKES EQUATIONS 395 Since r + 2α = 1 + 2, we have accoring to 1) of Proposition A1, This leas to the inequality u L c u Ḃr+2α 1 c u Ḃr+2α t u Ḃ + cν r u Ḃr+2α c u Ḃr+2α u Ḃr That is, for some suitable constant C 1 > 0, ) t u Ḃ c (C r 1 ν u Ḃr u Ḃr+2α If u 0 satisfies (41), this inequality then implies that u(, t) Ḃr is a non-increasing function of t for t > 0 This yiels the bouneness of u in L ([0, ); Ḃr ) L 1 ([0, ); Ḃr+2α ) To establish an a priori estimate for P, we note that P = R k R l (u k, u l ) Applying Proposition A4 an Proposition A1 yiels P(, t) B r c u Ḃ u Ḃr Therefore, P(, t) Ḃr is in L 1 ([0, )) or P L 1 ([0, ); Ḃr ) As explaine at the beginning of this proof, we omit further etails This conclues the proof Proof of Theorem 42 The proof is similar to that of Theorem 41 The major ifference is that here we use part 2) of Proposition A1 to boun u L, namely u L c u B s This inequality is vali for any s an thus allows the conition on r to be relaxe to r α We shall again omit the etails on constructing a successive approximation sequence an showing its convergence to the solution of the GNS equations Proof of Theorem 43 As we have explaine previously, it suffices to present only relevant a priori estimates For q > 1, we multiply (45) by q 2 qrj j u q 1 L an p then sum over j from 1 to to obtain (46) t u q B + c q ν u q c u r B r+2α/q L u q B r Since r + 2α q > 1 + 2, 3) of Proposition A1 implies that u L c u B r+2α q 1 c u B r+2α q Inserting this inequality an the basic embeing inequality u B r u B r+2α/q in (46) yiels t u q B c (C r 3 ν u B r ) u q, B r+2α/q where C 3 is a suitable constant epening on α, q an r only This ifferential inequality implies that u(, t) B r is a non-increasing function of t 0 Thus, if

16 396 JIAHONG WU u 0 satisfies (43), then (44) hols for all t > 0 An a priori estimate for P can be obtaine as in the proof of Theorem 41 This completes the proof of Theorem 43 5 Scaling invariance In this section, we examine some properties of the Besov spaces in which the solutions of the GNS equations have been stuie In particular, we investigate the scaling invariance property of these spaces an their implications This will help us have a better unerstaning of the results presente in the previous two sections The Besov spaces Ḃ1+ 2 2α an Ḃ1+ 2 2α are critical to solutions of the Navier- Stokes equations As mentione in the introuction, solutions of the GNS equations obey a scaling property That is, if (u, P) satisfies the GNS equations (11), then (u λ, P λ ) also satisfies (11), where u λ (x, t) = λ 2α 1 u(λx, λ 2α t), P λ (x, t) = λ 4α 2 P(λx, λ 2α t) The Besov spaces Ḃ1+ 2 2α an Ḃ1+ 2 2α are critical in the sense that the norm of u λ is essentially invariant in these spaces More precisely, we have the following lemma Lemma 51 If λ = 2 k for some k Z, then (51) u λ (, t) Ḃ1+ 2 2α (52) u λ (, t) Ḃ1+ 2 2α More generally, for any λ > 0, (53) u λ (, t) Ḃ1+ 2 2α = u(, 2 2αk t) Ḃ1+, 2 2α = u(, 2 2αk t) Ḃ1+, 2 2α u(, 2 2αk t) Ḃ1+, 2 2α u λ (, t) Ḃ1+ u(, 2 2αk 2 2α t) Ḃ1+, 2 2α where enotes the equivalence of two norms Proof For any j Z, we have j u λ (, t) 2 L 2 = R Φj (ξ) 2 û λ (ξ) 2 ξ (54) = λ 4α 2 R Φ0 (2 j λξ) 2 û(ξ, λ 2α t) 2 ξ When λ = 2 k, this equality implies Thus, (55) j u λ (, t) L 2 = 2 k(2α 1 /2) j k u(, 2 2αk t) L 2 u λ (, t) Ḃ1+ 2 2α = sup 2 (1+ 2 2α)j j u λ (, t) L 2 <j< = sup 2 (1+ 2 2α)(j k) j k u(, 2 2αk t) L 2 <j< = u(, 2 2αk t) Ḃ1+ 2 2α

17 THE GENERALIZED NAVIER-STOKES EQUATIONS 397 This proves (51) For a general λ > 0, choose k Z such that 2 k 1 < λ 2 k Since Φ 0 can be chosen to satisfy ( ) ( ) ( ξ Φ 0 2 j k Φ λξ 0 2 j Φ 0 we obtain by inserting these inequalities in (54) ξ 2 j k+1 λ 2α 1 2 j k u(, 2 2αk t) L 2 j u λ (, t) L 2 λ 2α 1 2 j k+1 u(, 2 2αk t) L 2 (53) is then establishe after following the lines as in the case λ = 2 k The proof of (52) is similar to that of (51) an the ifference is that one replaces sup j by in (55) This completes the proof j Z Now, we explore some of the implications of the scaling invariance of these spaces Theorem 32 asserts that the GNS equations have a unique global solution corresponing to any initial atum u 0 in the inhomogeneous Besov space B r an comparable to ν, where r > α If the conclusion of Theorem 32 were also true for the homogeneous Besov space Ḃr with r > α, then one woul be able to remove the smallness conition that u 0 is comparable to ν in Ḃr The reason is simple For any u 0 Ḃr, u 0λ (x, t) λ 2α 1 u 0 (λx, λ 2α t) remains in Ḃr an is comparable to ν in Ḃr for sufficiently small λ > 0 since u 0λ Ḃr λ r (1+ 2 2α) u 0 Ḃr Then u λ emanating from u 0λ leas to u, the solution corresponing to u 0 Similarly, the invariance property of Ḃ1+ 2 2α may provie another explanation as to why Theorem 42 allows for solutions in any Besov space B r with r α but the result in Theorem 41 is only for Ḃ1+ 2 2α ), Appenix As mentione before, this appenix contains several inequalities that we have use to prove our major results First, we present the logarithmic Besov type inequalities The inequalities allows us to boun the L -norm in terms of the norms in Besov spaces Proposition A1 Let 1 p 1) If f Ḃr p,1 (R ) with r = p, then (A1) f L c f Ḃr p,1 2) If f B r p,1 (R ) with r p, then (A2) f L c f B r p,1

18 398 JIAHONG WU 3) If f Bp,q r (R ) with q > 1 an r > p, then ( ) f B r (A3) f L c f B 0, 1 + log p,q 2 f B 0, Proof In particular, (A3) implies f L c f B r p,q For j Z, (28) allows us to write j f = k j f k j <2 It then follows from Proposition 22 that for any 1 p, j f L c 2 k p j f L p c 2 j p j f L p k j <2 To prove (A1), we assume f Ḃr p,1 with r = p an take the L -norm of f = j= j f to get f L j= j f L c j= 2 j p j f L p = c f p Bp,1 The proof of (A2) is similar, but the ifference is that j = 0 with j 2 is zero in the context of a inhomogeneous Besov space For f Bp,1 r with r /p, f L j f L c 2 j p j f L p c j= 1 j= 1 To prove (A3), we write for f Bp,q r (A4) f = j f = j= 1 j= 1 2 jr j f L p = c f B r p,1 N j f + j= 1 j=n+1 j f, where N is an integer to be specifie The L -norm of the first sum can be boune by N j f L (N + 2) f B 0, j= 1 while the secon sum is boune by j f L c 2 j p j f L p j=n+1 c j=n+1 j=n+1 2 jq (/p r) c 2 N q (r /p) f B r p,q, 1/q j=n+1 2 jrq j f q L p where q satisfies 1/q + 1/q = 1 Thus, for a constant c epening on p, q an r, f L (N + 2) f B 0, + c2 N q (r /p) f B r p,q 1/q

19 THE GENERALIZED NAVIER-STOKES EQUATIONS If we set N = O( r /p log f B r p,q 2 f ), (A3) is then establishe B 0, We have use extensively the commutator estimates state in the next two propositions These estimates have previously been obtaine in [16] Proposition A2 For p [1, ] an j Z, we have [u, j ]v L p c ( u L j v L p + v L j u L p), where the brackets [, ] represent the commutator, namely [u, j ]v = u j v j (u v) The estimate in Proposition A2 is suitable for situations when u an v are equally regular If v is not known to be in L, then the following commutator estimator is more useful Proposition A3 For p [1, ] an j Z, we have [u, j ]v L p c ( u L j v L p + 2 j v L j u L p) The following proposition bouns the prouct u v in a Besov space in terms of the norms of u an v in the same Besov space Proposition A4 For any s > 0 an p, q [1, ], we have uv Ḃs p,q c( u L v Ḃs p,q + v L u Ḃs p,q ), uv B s p,q c( u L v B s p,q + v L u B s p,q ) The proof of this proposition is classical (see eg[4],[6]) A special consequence of this proposition an Proposition A1 is that Ḃs p,1 with s = /p, Bs p,1 with p /p an Bp,q s with s > /p are all Banach algebras Acknowlegments I thank my colleague Davi Ullrich for iscussions on the Littlewoo-Paley theory, an the referee for pointing out the results of Cannone, Planchon an Lemarié-Rieusset References [1] H Bahouri an J-Y Chemin, Equations e transport relatives à es champs e vecteurs non-lipschitziens et mécanique es fluis, Arch Rat Mech Anal 127 (1994), [2] J Bergh an J Löfström, Interpolation Spaces, An Introuction, Springer-Verlag, Berlin- Heielberg-New York, 1976 [3] M Cannone, Onelettes, Paraprouits et Navier-Stokes, Dierot Éiteur, Paris, 1995 [4] D Chae, Local existence an blow-up criterion for the Euler equations in the Besov spaces, RIMGARC preprint no 01-7 [5] D Chae an J Lee, Global well-poseness in the super-critical issipative quasi-geostrophic equations, Commun Math Phys 233 (2003), [6] J-Y Chemin, Perfect Incompressible Fluis, Clarenon Press, Oxfor, 1998 [7] R Danchin, Local theory in critical spaces for compressible viscous an heat-conuctive gases, Commun Partial Differential Equations 26 (2001),

20 400 JIAHONG WU [8] PG Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, Chapman & Hall/CRC Research Notes in Mathematics 431, Chapman & Hall/CRC, Boca Raton, FL, 2002 [9] J Peetre, New Thoughts on Besov Spaces, Duke University Press, 1976 [10] E Stein, Singular Integrals an Differentiability Properties of Functions, Princeton University Press, 1970 [11] D Ullrich, private communications [12] M Vishik, Hyroynamics in Besov spaces, Arch Rat Mech Anal 145 (1998), [13] M Vishik, Incompressible flows of an ieal flui with vorticity in borerline spaces of Besov type, Ann Sci École Norm Sup 32 (1999), [14] J Wu, Generalize MHD equations, J Differential Equations 195 (2003), [15] J Wu, Global solutions of the 2D issipative quasi-geostrophic equation in Besov spaces, SIAM J Math Anal, in press [16] J Wu, Solutions of the 2D quasi-geostrophic equation in Höler spaces, submitte Department of Mathematics, Oklahoma State University, Stillwater, OK74078, USA aress: jiahong@mathokstateeu