ANALYTICITY AND DECAY ESTIMATES OF THE NAVIER STOKES EQUATIONS IN CRITICAL BESOV SPACES

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1 ANALYTICITY AND DECAY ESTIMATES OF THE NAVIER STOKES EQUATIONS IN CRITICAL BESOV SPACES HANTAEK BAE, ANIMIKH BISWAS, AND EITAN TADMOR Abstract In this aer, we establish analyticity of the Navier-Stokes equations with small data in critical Besov saces Ḃ The main method is so-called Gevrey estimates, which is motivated by the work of Foias and Temam 19 We show that mild solutions are Gevrey regular, ie the energy bound e tλ v(t) E < holds in E := L (, T; Ḃ ) L 1 (, T; Ḃ +1 ), globally in time for < We extend these results for the intricate limiting case = in a suitably designed E sace As a consequence of analyticity, we obtain decay estimates of weak solutions in Besov saces Finally, we rovide a regularity criterion in Besov saces Contents 1 Introduction and statement of main results 1 Notations: the Littlewood-Paley decomosition and araroducts 6 The case < : roof of Theorem 11 (existence) and Theorem 1 (analyticity) 7 4 The case = : roof of theorem 1 (existence) and Theorem 14 (analyticity) 1 5 Proof of theorem 15: decay of Besov norms 17 6 Proof of theorem 16: regularity condition in Besov saces 19 Acknowledgments 1 References 1 1 Introduction and statement of main results It is well-known that regular solutions of many dissiative equations, such as the Navier- Stokes equations, the Kuramoto-Sivashinsky equation, the surface quasi-geostrohic equation and the Smoluchowski equation are in fact analytic, in both sace and time variables 4, 14, 18, 6, 49 In fluid-dynamics, the sace analyticity radius has an imortant hysical interretation: at this length scale the viscous effects and the (nonlinear) inertial effects are roughly comarable Below this length scale the Fourier sectrum decays exonentially 1, 17, 7, 8 In other words, the sace analyticity radius yields a Kolmogorov tye length scale encountered in turbulence theory At a more ractical level, this fact can be used to show that the finite dimensional Galerkin aroximations converge exonentially fast in these cases 1 Other alications of analyticity radius occur in establishing shar temoral decay Key words and hrases Navier-Stokes equations, scaling invariant Besov saces, Gevrey regularity, decay of Besov norms, regularity condition Research was suorted in art by NSF grants, DMS1-897 and FRG

2 HANTAEK BAE, ANIMIKH BISWAS, AND EITAN TADMOR rates of solutions in higher Sobolev norms 9, establishing geometric regularity criteria for the Navier-Stokes equations, and in measuring the satial comlexity of fluid flow 4, 1, In this aer, we study analyticity roerties of the incomressible Navier-Stokes (NS) equations in R The system of equations is given by (11a) (11b) v t + v v µ v + =, v =, where v is the velocity field, is the ressure, and µ > is the viscosity coefficient which for simlicity we set µ = 1 Since v =, we can rewrite the momentum equation (11a) by rojecting it onto the divergence-free sace Let P = Id ( ) div be the orthogonal rojection of L over divergence-free vector fields By alying P to (11a), we obtain (1) v t + P (v v) v = Formally, we can exress a solution v of (1) in the integral form: (1) v(t) = e t v e (t s) P (v v)(s) ds Any solution satisfying this integral equation is called a mild solution We can find it by using a fixed oint argument for the function v F(v), F(v)(t) = e t v e (t s) P (v v)(s) ds The invariant sace for solving this integral equation corresonds to a scaling invariance roerty of the equation Assume that (v, ) solves (11) Then, the same is true for rescaled functions: (14) v λ (t, x) = λv(λ t, λx), λ (t, x) = λ (λ t, λx), λ > Under these scalings, L, Ḣ, 1 Ẇ, and Ḃ are critical saces for initial data (t = ), ie, the corresonding norms are invariants under these scaling One can find various well-osedness results for small data in these critical saces in 6, 7, 9, 1, 9,, 4 The goal of this aer is threefold: (i) analyticity of mild solutions in critical Besov saces, (ii) decay of Besov norms of weak solutions, and (iii) a new regularity condition in Besov saces More details of these three toics will be resented in section 11, section 1, and section 1 resectively 11 Analyticity of mild solution Let us begin with analyticity results of this aer Comared to revious works by 15,, 7 (derivative estimations), 4 (l sace on T ), and 5, 6 (comlexified equations), we are able to establish analyticity of the Navier-Stokes equations by obtaining Gevrey estimates in Besov saces Ḃ Namely, we will show that a solution v(t) Ḃ satisfies su t> e tλ v(t) Ḃ <, where Λ is the Fourier multilier whose

3 ANALYTICITY AND DECAY ESTIMATES OF THE NAVIER-STOKES EQUATIONS symbol is given by ξ 1 = ξ i We emhasize that here Λ Λ 1 is quantified by the l 1 norm i=1 rather than the usual l norm associated with Λ := ( ) 1 This aroach enables one to avoid cumbersome recursive estimation of higher order derivatives In order to exlain the main idea, we define V (t) = e tλ v(t) Then, V (t) satisfies the following equation: V (t) = e tλ+t v = e tλ+t v e tλ+(t s) P (e sλ V (s) e sλ V (s)) ds e ( t s)λ+(t s) P e sλ (e sλ V (s) e sλ V (s)) ds Since e t ξ 1 is dominated by e t ξ for ξ 1, the behavior of the linear term, e tλ+t v, closely resembles that of v(t) The estimates of the nonlinear term are similar to those of v(t) due to the nice boundedness roerty of the bilinear oerator B s : B s (f, g) = e sλ ( e sλ f(s)e sλ g(s) ) As noticed from the above argument, the existence result of v(t) is crucial in establishing Gevrey regularity Thus, in section (for < ) and section 4 (for = ), we will first show the existence of a mild solution and then roceed to exlain how to modify the existence roof to obtain Gevrey regularity Comared to revious works in 19, 9, where they defined Gevrey norms of the form e tλ v(t) X with a L based sace Sobolev sace X, the use of Λ instead of Λ is fundamental for the estimates of B s in L based function saces A similar aroach using Λ in L -saces was taken by 4 However, our results cover a larger Besov class of initial data in the full range of, q 1,, and the same method can be alied to other dissiative equations; see for examle 1, We now resent our existence/analyticity results for < and = searately For notational simlicity, we will suress the deendence of norms (defined below) on q 111 The case < The existence of global-in-time solutions for small data in Ḃ for < was roved by Chemin 9 The result indicates a gain of two derivatives from the maximal regularity of the heat kernel, which is realized in terms of the function sace E, where u L t = Ḃ E := { u S : u E = u L t j( )q j u q L t L j Z with the usual change for q = 1 q Ḃ, u L 1 tḃ +1 + < u L 1 +1 tḃ }, = j( +1)q j u q L 1 t L j Z 1 q

4 4 HANTAEK BAE, ANIMIKH BISWAS, AND EITAN TADMOR Theorem 11 (Existence 9) Let 1 <, 1 q and v Ḃ There exists a constant ɛ > such that for all v Ḃ with v Ḃ a global-in-time solution v E Moreover, if q <, v C t Ḃ ɛ, the NS equations (1) admit The first result of this aer is showing that solutions of Theorem 11 are, in fact, analytic in the following sense Theorem 1 (Analyticity) There exists a ositive constant ɛ > such that for all v Ḃ with v ɛ Ḃ, the NS equations (1) admit a solution v E such that e tλ v E 11 The case = The case of Ḃ,q data, corresonding to =, is much harder because the Navier-Stokes equations are ill-osed in Ḃ,q for q > (5,, 51) To circumvent the difficulty in this case, we rove the existence of solutions subject to a restricted class of initial data in Ḃ E :=,q Ḃ,, 1 q < The corresonding function sace is defined as follows } Ḃ, + su u(t) Ḃ +,q t 4 1 t> u(t) Ḃ,q { u S : u E = u L t Comared to the time-integrated gain of two derivatives we had in the case <, here we have ointwise-in-time gain of regularity of order, which is realized in E Theorem 1 (Existence) Let 1 q < and v Ḃ,q Ḃ, There exists a constant ɛ > such that for all v Ḃ,q Ḃ, with v Ḃ + v Ḃ ɛ,q, the NS equations, (1) admit a global-in-time solution v E Remark 11 We note that one can relace Ḃ, with other auxiliary saces, Ḃ,, < in Theorem 11 We choose the former because it is closely related to saces aearing in the regularity criterion 16 with initial data in L Also, we choose q < to avoid the embedding Ḃ, Ḃ, Once we show the existence of a solution of (1) in E, we can show the following analyticity result along the lines of the roof of Theorem 1 and Theorem 1 Theorem 14 (Analyticity) There exists a ositive constant ɛ > such that for all v Ḃ,q Ḃ, with v Ḃ + v Ḃ ɛ,q, the NS equations (1) admit a solution v E, such that e tλ v E 1 Decay of weak solution As an alication of analyticity of solutions addressed above, we will estimate decay rates of weak solutions Most of decay results have been based on L estimations as one can see in 8, 9, 4, 41, 4, 44, 45, 5 Here, we obtain the decay of weak solutions in Besov saces Ḃ for all Before resenting our result, we recall the usual notion of a weak solution, v(t) L t L L tḣ1, satisfying (1) in the sense of distributions and the additional energy inequality, v(t) L + v(s) L ds v L

5 ANALYTICITY AND DECAY ESTIMATES OF THE NAVIER-STOKES EQUATIONS 5 The fundamental theorem of Leray 5 states the existence of such weak solutions for initial data v L with v = We now briefly exlain the main idea of the decay estimates: (i) The energy inequality imlies that lim inf t v(t) Ḣ = If we show that Besov norms 1 v(t) Ḃ can be controlled by Ḣ1 norm, then v(t ) Ḃ becomes sufficiently small after a certain transient time t (ii) Theorem 1 and Theorem 14 tell us that if initial data are sufficiently small in critical saces Ḃ, then the solution satisfies the estimate e tλ v(t) Ḃ globally in time Combining these two observation, we will show the following decay estimates of weak solutions in section 5 Theorem 15 (Decay) Let v be a weak solution of the three dimensional Navier-Stokes equations, subject to initial data v L and, in addition ω = v L 1 in case 1 < < Then, there exists a time t > such that v(t becomes sufficiently small so that Theorem 1 or Theorem 14 hold for < and, resectively, = Moreover, the following ) Ḃ decay estimate holds for ζ >, { Λ ζ v(t) Ḃ C ζ v(t (t t ) Ḃ ) ζ, Cζ = Λ ζ q for, e Λ L 1, q for 1 < < Remark 1 By the Sobolev embedding, one can easily extend revious L decay results to obtain decay rates in L saces, > However, the decay rates for < are new Remark 1 This decay rate satisfies the Petrowsky arabolicity condition 48 Remark 14 The case = q = corresonds to the uer bound of decay rate in 9 In that aer, however they assumed the following two conditions to the solution itself: (a) there exist ositive real numbers M 1 and γ which may deend on v such that v(t) L M 1 (1 + t) γ for all t and (b) for r, lim inf v(t) Hr < By contrast, we do not need these two t conditions to obtain the decay estimate asserted in theorem 15 1 Regularity condition As a related subject, we will rovide a Serrin-tye regularity criterion in Besov saces The Serrin criterion 46 says that the Leray weak solution v is smooth for t (, T if v L r (, T; L s ), with r + = 1 We would like to show a new s regularity criterion in Besov saces with less regularity Theorem 16 There exists a smooth solution of the Navier-Stokes equation on the time interval, T for smooth initial data v if on any time interval T t, T, C, t < T, (15) (T t) 1 q v(t) Ḃσ, q + σ = 1,, < q <, t T, Remark 15 Theorem 16 generalizes the Serrin s regularity condition and other related results in two asects: (i) σ can be negative and (ii) for σ =, v(t) Ḃ, v(t) L

6 6 HANTAEK BAE, ANIMIKH BISWAS, AND EITAN TADMOR Remark 16 Comared to a new regularity criterion obtained by 1 in L q q (, T; B, ) for q q, B, < q <, where their result is better than our result in the satial variables by Ḃ for a negative regularity index q, our result imroves the criterion in the time variable because the time singularity (T t) q is in the weak L q sace, which contains the L q sace Moreover, our roof is much simler However, our method cannot cover several known results due to the missing end oint q = : for examle, L (, T; L ) in 16 and C((, T; B, ) in 1 Notations: the Littlewood-Paley decomosition and araroducts We begin with some notations L (, T; X) denotes the Banach set of Bochner measurable ( T functions f from (, T) to X endowed with either the norm f(, t) )1 X dt for 1 < or su f(, t) X for = For T =, we use L t X instead of L (, ; X) For a sequence t T {a j } j Z, {a j } l q := ( a j q) 1 q, with the usual change for q = Finally, A B means there j Z is a constant C such that A CB We next rovide notation and definitions in the Littlewood-Paley theory We take a coule of smooth functions (χ, ϕ) suorted on {ξ; ξ 1} with values in, 1 such that for all ξ R d, χ(ξ) + ψ( j ξ) = 1, ψ(ξ) = ϕ( ξ ) ϕ(ξ) j= and we denote ψ( j ξ) by ψ j (ξ) The homogeneous dyadic blocks and lower frequency cut-off functions are defined by (1) j u = jd h( j y)u(x y)dy, S j u = jd h( j y)u(x y)dy, R d R d with h = F ψ and h = F χ Then, we can define the homogeneous Littlewood-Paley decomosition by () u = j Z j u in S h, where S h is the sace of temered distributions u such that lim S ju = in S Using this j decomosition, we define stationary/ time deendent homogeneous Besov saces as follows: ( Ḃ s = {f S h ; f Ḃ := jsq s j f q )1 } q <, L () j Z ( L r (, T; Ḃ) s = {f S h ; f L r (,T;Ḃ s) := jsq j f q )1 q L }, < L r (,T) ( L r (, T; Ḃs ) = {f S h ; f L r (,T;Ḃ s) := jsq j f q L r (,T;L ) j Z j Z )1 q } <

7 ANALYTICITY AND DECAY ESTIMATES OF THE NAVIER-STOKES EQUATIONS 7 with the usual change if q = According to the Minkowski inequality, we have (4) f L r (,T;Ḃ s ) f L r (,T;Ḃ s ) if r q, f L r (,T;Ḃ s ) f L r (,T;Ḃ s ) if r q The concet of araroduct enables to deal with the interaction of two functions in terms of low or high frequency arts, 8 For two temered distributions f and g, (5) fg = T f g + T g f + R(f, g), T f g = i j i f j g = j Z S j f j g, R(f, g) = j j 1 j f j g Then, u to finitely many terms, (6) j (T f g) = S j f j g, j R(f, g) = k f k g In section, we will use the following decomosition: (7) fg = j Z S j f j g + j Z S j f j g Again, u to finitely many terms, we have (8) j (fg) = S k f k g + k fs k g We finally recall a few inequalities which will be used in the sequel Bernstein s inequality 8 For 1 q and k N, (9) su α j f L jk j f L, j f L q jd( 1 q ) j f L α =k Localization of the heat kernel 9 (1) e t j f L e tj j f L The case < : roof of Theorem 11 (existence) and Theorem 1 (analyticity) In this section, we rove Theorem 1, analyticity of the Navier-Stokes equations with small initial data in Ḃ, with < The roof is based on an adequate modification of the roof Theorem 11 Therefore, we begin with the detailed existence roof of 9

8 8 HANTAEK BAE, ANIMIKH BISWAS, AND EITAN TADMOR 1 Proof of Theorem 11 We recall the definition of the function sace E, { } (1) E = u S : u E = + u L t Ḃ < u L 1 +1 tḃ We construct a solution in the integral form: v(t) = e t v B(v, v), where the bilinear form B is () B(u, v) = e (t s) P (u v)(s) ds We only need to show that B mas E E to E We first decomose the roduct u v as araroduct (7) Then, B can be decomosed as B(u, v) = B 1 (u, v)+b (u, v), B 1 (u, v) = j Z B(S j u j v), B (u, v) = j Z B(S j v j u) We now estimate B 1 in E We aly j to B 1 and take the L norm By Bernstein s inequality (9) and localization of the heat kernel as (1), we have () j B 1 (u, v) L l k i i l k e (t s)k S k u(s) k v(s) L ds e (t s)k S k u(s) L k v(s) L ds By Bernstein s inequality (9), S k u L l u L l l u L = l l( ) l u L Therefore, we can relace the right-hand side of () by (4) j B 1 (u, v) L l k l k e j (t s)k( ) l l( ) l u(s) L k v(s) L ds By taking the L norm of (4) in time with the aid of Young s inequality in time, we obtain j B 1 (u, v) L t L j l l( ) l u L t L k v L 1 t L (5) We multily (5) by j( ) Then, (6) u L t Ḃ u E l k j( ) j B 1 (u, v) L t L u E j k k v L 1 t L j k( ) k( +1) k v L 1 t L (k j)( ) k( +1) k v L 1 t L

9 ANALYTICITY AND DECAY ESTIMATES OF THE NAVIER-STOKES EQUATIONS 9 Since >, we can use Young s inequality to estimate the right-hand side of (6) with resect to l q Namely, we let a j = j( ) and b j = j( +1) j v L 1 t L and aly Young s inequality to a k j b k to obtain (7) B 1 (u, v) L t Ḃ u E u v L1 t Ḃ +1 E v E Similarly, we can obtain the L 1 in time estimation of B 1 (u, v) By taking the L 1 norm in time to (4) and alying Young s inequality in time, (8) j B 1 (u, v) L 1 t L u E j j k( ) k( +1) k v L 1 t L We multily (8) by j( +1) Then, j( +1) j B 1 (u, v) L 1 t L u E from which we have (9) B 1 (u, u v) L1 t Ḃ +1 E v E (k j)( ) k( +1) k v L 1 t L Therefore, we conclude that B 1 (u, v) E u E v E By the symmetry of the araroduct of B(u, v), we finally have (1) which comletes the roof B(u, v) E u E v E, Remark 1 We will need to aly Young s inequality to sequences several times to estimate sequences having the convolution structure Since the structure of sequences aearing later (for examle, (45), (49), (41) and (6)) is exactly of the form used to obtain (7), we will aly Young s inequality to sequences without defining {a j } and {b j } each time Preliminaries The roof of Theorem 1 in this section and likewise, Theorem 14 in section 4, requires a coule of elementary inequalities which are summarized in the following two lemmas Lemma 1 Consider the oerator E := e t s+ s tλ for s t Then E is either the identity oerator or is an L 1 kernel whose L 1 norm is bounded indeendent of s, t Proof Clearly, a := t s + s t is non-negative for s t In case a =, E = e aλ is the identity oerator, while if a >, E = e aλ is a Fourier multilier with symbol Ê(ξ) = d i=1 e a ξi Thus, the kernel of E is given by the roduct of one dimensional Poisson kernels d a i=1 π(a +x ) The L1 norm of this kernel is bounded by a constant indeendent of a i Lemma The oerator E = e 1 a + aλ is a Fourier multilier which mas boundedly L L, 1 < <, and its oerator norm is uniformly bounded with resect to a Proof When a =, E is the identity oerator When a >, then E is Fourier multilier with symbol Ê(ξ) = e aξ + aξ 1 Since Ê(ξ) is uniformly bounded for all ξ and decays exonentially for ξ 1, the claim follows from Hormander s multilier theorem, eg, 47

10 1 HANTAEK BAE, ANIMIKH BISWAS, AND EITAN TADMOR Proof of theorem 1 We are now ready to rove Theorem 1 For the notational simlicity, we define a function sace F such that { } F = v(t) E ; e tλ v(t) E We only need to show that B in () is bounded from F F to F To this end, we first aly e tλ to B in () (11) e tλ B(u, v) = e tλ Let U(s) = e sλ u, V (s) = e sλ v, with U, V E Then, (1) We rewrite (1) as (1) e tλ B(u, v) = e e tλ B(u, v) = tλ e (t s) P (u v)(s) ds e (t s) P (e sλ U e sλ V )(s) ds e ( t s)λ e 1 (t s) e 1 (t s) e sλ P (e sλ U e sλ V )(s) ds We aly j to (1) and take the L norm By Lemma 1 and Lemma, we have (14) j e tλ B(u, v) L e (t s)j j e sλ ( j e sλ U e sλ V )(s) ) L ds To deal with the right-hand side of (14), we decomose the roduct e sλ U e sλ V as e sλ U e sλ V = ( e sλ S j U ) ( e sλ j V ) + ( e sλ j U ) ( e sλ S j V ) j Z j Z Then, (15) j e tλ B(u, v) L + e (t s)j j e sλ ( e sλ S k U e sλ k V )(s) L ds e (t s)j j e sλ ( e sλ k U e sλ S k V )(s) L ds To estimate the right-hand side of (15), we introduce the bilinear oerators B t of the form B t (f, g) = tλ e (e tλ fe tλ g) = e ix (ξ+η) e t( ξ+η 1 ξ 1 η 1 ) ˆf(ξ)ĝ(η)dξdη R R Recall that for a vector ξ = (ξ 1, ξ, ξ ), we denoted ξ 1 = ξ i As one can see below, this l 1 version of Λ instead of the usual l version of Λ is crucial to estimate B t For ξ = (ξ 1, ξ, ξ ), η = (η 1, η, η ), we now slit the domain of integration of the above integral into i=1

11 ANALYTICITY AND DECAY ESTIMATES OF THE NAVIER-STOKES EQUATIONS 11 sub-domains deending on the sign of ξ j, η j and ξ j + η j In order to do so, we introduce the oerators acting on one variable (see age 5 in ) by K 1 f = 1 π e ıxξ ˆf(ξ) dξ, K f = 1 π Let the oerators L a, and L a,1 be defined by L a,1 f = f, L a, f = 1 π R e ıxξ ˆf(ξ) dξ e ıxξ e a ξ ˆf(ξ) dξ For α = (α 1, α, α ), β = (β 1, β, β ) {, 1}, denote the oerator Z a, α, β = K β1 L t,α1 β 1 K β L t,α β and K α = K α1 K α K α The above tensor roduct means that the j th oerator in the tensor roduct acts on the j th variable of the function f(x 1, x, x ) A tedious (but elementary) calculation now yields the following identity: ( (16) B t (f, g) = K α1 K α K α Zt, α, β fz t, α, γg ), ( α, β, γ) {,1} We now note that the oerators K α, Z a, α, β defined above, being linear combinations of Fourier multiliers (including Hilbert transform) and the identity oerator, commute with Λ Moreover, they are bounded linear oerators on L, 1 < < and the corresonding oerator norm of Z t, α, β is bounded indeendent of t By taking the L norm to (16), we have B t (f, g) L Z t, α, β fz t, α, γg L We now aly this argument the right-hand side of (15) Then, (17) e tλ B(u, v) L + e (t s)j j ZS k U Z k V L ds e (t s)j j Z k U ZS k V L ds, where we denote Z without indices t, α, β for the notational simlicity By Bernstein s inequality (9), we have ZS k u L Z l u L l Zu L l k l k l k l l Zu L l k l l( ) l u L, where we use the fact that Z commutes with l and the boundedness of Z on L Therefore, we can follow the lines from (4) to (1) in roof of Theorem 11 to obtain (18) which comletes the roof B(u, v) F U E V E u F v F,

12 1 HANTAEK BAE, ANIMIKH BISWAS, AND EITAN TADMOR 4 The case = : roof of theorem 1 (existence) and Theorem 14 (analyticity) We now show the well-osedness and analyticity for the Navier-Stokes equations of the limiting case = To this end, we recall the definition of the sace E E := { u S : u E = u L t } Ḃ, + su v(t) Ḃ +,q t 4 v(t) Ḃ 1 t>,q As one can see below, we need to obtain two additional estimates t 1 4 v(t) Ḃ 1 and t 1 v(t) Ḃ,q,q (See (41), (414) and (418)) These terms can be obtained by interolating v(t) Ḃ and,q t 4 v(t) Ḃ 1,q, but we roceed the roof by establishing bounds in K whose norm is given by (41) v K = su v(t) Ḃ +,q t1 4 v(t) Ḃ 1 + t 1 v(t) Ḃ + t 4 v(t) Ḃ 1 t>,q,q,q to avoid comlicated exressions coming from the interolation The time weights aearing in (41) will be introduced below through the Gaussian bound, (4) ξ a e t ξ t a In addition, we will use the following lemma reeatedly in the roof of Theorem 1 Lemma 41 For any < a < 1 and < b < 1, (t s) a s b ds t 1 a b The result follows by decomosing the time integral into two arts, (t s) a s b ds = (t s) a s b ds + t a t s b ds + t b (t s) a ds t (t s) a s b ds 41 Proof of theorem 1 As we did in the roof Theorem 1, we need to show that B mas E E to E Since we already estimated the bilinear term B in Ḃ, in Theorem 11, we only need to show that B mas from E E to K We decomose u v as (5) Then, (4) B(u, v) = e (t s) P(T u v + T v u + R(u v)) ds := B 1 (u, v) + B (u, v) + B (u, v)

13 ANALYTICITY AND DECAY ESTIMATES OF THE NAVIER-STOKES EQUATIONS 1 Estimation of B (u, v) We estimate B first, where we need the auxiliary norm u L t Ḃ, By Bernstein s inequality (9) and localization of the heat kernel (1), (44) j B (u, v)(t) L j j B (u, v)(t) L j e (t s)j u L t Ḃ, u E k u(s) L k v(s) L j e (t s) j j e (t s)j ds j k k k v(s) L ds j k k k v(s) L ds We will reeatedly use (44) to estimate B (u, v) in K To estimate B (u, v) in L t Ḃ,q, we multily (44) by j Then, j j B (u, v)(t) L u E j e (t s) j j k k k v(s) L ds (45) u E (t s) 4s 4 j k s 4 k k v(s) L ds By taking the l q norm to (45) with the aid of Young s inequality (as mentioned in Remark 1), we have B (u, v)(t) Ḃ u E,q t 4 v(t) Ḃ 1,q (t s) 4 s 4 ds Therefore, Lemma 41 imlies that su t> (46) B (u, v)(t) Ḃ,q u E v E We will do the same calculation to estimate B for the next two terms in (41) without details (47) B (u, v)(t) Ḃ 1 u E su t 4 v(t) Ḃ 1,q t>,q (t s) s 4 ds u E v E t 1 4 (48) B (u, v)(t) Ḃ,q u E su t> t 4 v(t) Ḃ 1,q (t s) 4 s 4 ds u E v E t 1 To estimate t 4 B (u, v)(t) Ḃ1,q, we need to divide the time integration into two arts j j B (u, v)(t) L j + j t = I j + II j j e (t s)j j e (t s)j k u(s) L k v(s) L k u(s) L k v(s) L ds ds

14 14 HANTAEK BAE, ANIMIKH BISWAS, AND EITAN TADMOR We begin with I j By Bernstein s inequality (9), I j = j e (t s)j j 1 t s u L t Ḃ, u E 1 t s k u(s) L k v(s) L k u(s) L j k k k v(s) L ds 1 t s Using Young s inequality, we obtain that { (49) Ij u }l q E su t> Next, we estimate II j II j (t s) = t t from which we have { IIj } (41) t 4 v(t) Ḃ 1,q (t s) s 4 s 1 l q su t> By (49) and (41), we have (411) j k k k v(s) L ds j k k k v(s) L ds (t s) s 4 j k k k u(s) L k v(s) L ds t 1 v(t) Ḃ,q su t> u E v E Therefore, by (46), (47), (48), and (41), (41) t 4 ds ds u E v E t 4 j k 1 s k u(s) L s k 4 k v(s) L ds t 4 v(t) Ḃ 1,q B (u, v)(t) Ḃ1,q u E v E t 4 B (u, v) K u E v E t (t s) s 4 s 1 ds Estimation of B 1 (u, v) and B (u, v) Now, we estimate B 1 (u, v) By alying j to B 1 (u, v) and taking the L norm, we have j B 1 (u, v)(t) L j e (t s)j S j u(s) L j v(s) L ds We note that we used the auxiliary norm u Ḃ, to relace k u L by k u L to gain one derivative to estimate B (u, v) Here, we can gain one derivative from S j u to estimate B 1 and

15 B (u, v) as follows (41) ANALYTICITY AND DECAY ESTIMATES OF THE NAVIER-STOKES EQUATIONS 15 S j u L j l= l u L = j l= l l l u L j u Ḃ,q j u E We will use this roerty to estimate B 1 (u, v) for the first three terms in (41) We begin with the estimation of B 1 (u, v) L t Ḃ,q j j B 1 (u, v)(t) L u E j e (t s)j j v(s) L ds which imlies, by lemma 41, that (414) B 1 (u, v)(t) Ḃ,q u E (415) B 1 (u, v)(t) Ḃ 1,q u E = u E su t> u E su t> t 1 v(t) Ḃ,q (t s) j v(s) L ds (t s) s 1 1 s j v(s) L ds (t s) s 1 ds u E v E Again, we will use the same calculation to estimate B 1 (u, v) for the following two terms without details t t 4 1 v(t) Ḃ,q (t s) s 4 (416) B 1 (u, v)(t) Ḃ,q u E su t> ds u E v E t 1 4 t t 4 1 v(t) Ḃ,q (t s) 4 s 4 ds u E v E t 1 To estimate t 4 B 1 (u, v)(t) Ḃ1,q, we need to divide the time integration into two arts j j B 1 (u, v)(t) L We begin with III j By (41), Thus, we have (417) + t III j u E { IIIj }l q u E su t> j j e (t s)j S j u(s) L j v(s) L ds j j e (t s)j S j u(s) L j v(s) L ds = III j + IV j j e (t s)j j j v(s) L ds t 4 v(t) Ḃ 1,q (t s) s 4 ds u E v E t 4

16 16 HANTAEK BAE, ANIMIKH BISWAS, AND EITAN TADMOR To estimate IV j, we slightly change the estimation of S j u L as follows: (418) Therefore, S j u(s) L j l= j s 1 4 l u(s) L = IV j u E which imlies that { (419) IV u }l q E By (417) and (419), we have j l= su s 1 4 u(s) Ḃ 1 s>,q t u E su t> t l s 1 4 l s 1 4 l u(s) L j s 1 4 u E j e (t s) j s j 4 j v(s) L ds (t s) 4 s 1 j 4 j v(s) L ds t 4 v(t) Ḃ 1,q B 1 (u, v)(t) Ḃ1,q u E v E (4) t 4 Therefore, By (414), (415), (416), and (4), (41) t B 1 (u, v) K u E v E (t s) 4 s 1 4 s 4 ds u E v E t 4 Since B (u, v) is of the form of B 1 (u, v) by changing the role of u and v, we have (4) B (u, v) K u E v E Combining (41), (41), and (4), we finally have (4) which comletes the roof of Theorem 1 B(u, v) K u E v E 4 Proof of theorem 14 We can rove Theorem 14 along the lines the roof of Theorem 1 and Theorem 1 We define a function sace F such that { } F = v(t) E ; e tλ v(t) E We only need to show that B in () is bounded from F F to F Let U(s) = e sλ u, V (s) = e sλ v, with U, V E By relacing the L norm by the L norm in (14), we have (44) j e tλ B(u, v) L e (t s)j j e sλ ( j e sλ U e sλ V ) (s) L ds Then, we decomose ( e sλ U e sλ V ) as T e sλ U e sλ V + T e sλ V e sλ U + R(e sλ U e sλ V )

17 ANALYTICITY AND DECAY ESTIMATES OF THE NAVIER-STOKES EQUATIONS 17 and follows calculations in the roof of theorem 1 In general, K α and Z t, α, β do not ma L to L However, these oerators are bounded in L when localized in dyadic blocks in the Fourier saces Therefore, (45) e tλ B(u, v) K U E V E u F v F Since we already obtained e tλ B(u, v) L t Ḃ, in section, we finally have This comletes the roof of Theorem 14 B(u, v) F u F v F 5 Proof of theorem 15: decay of Besov norms In this section, we will obtain various decay estimates of weak solutions of the Navier-Stokes equations in Besov saces We need the following lemma to roceed Lemma 51 The Fourier multiliers corresonding to the symbols m(ξ) = ξ ζ e t ξ 1 are given by convolution with corresonding kernels k which is a L 1 functions with k L 1 C ζ t ζ/ Proof For a roof of the L 1 bound on k, we aly lemma 1 in under the scaling: ξ t 1 ξ Theorem 1 and Theorem 14 tell us that if initial data are sufficiently small in critical saces Ḃ, then the solution is globally in the Gevrey class Then, lemma 51 allows us to obtain the following time decay of (homogeneous) Besov norms: Λ ζ v(t) = Λ ζ Ḃ e tλ e tλ v(t) Ḃ C ζ t ζ e tλ v(t) Ḃ where we recall that Λ = ( ) 1 If we can show that a solution v(t) satisfies, ζ >, (51) lim inf t v(t) Ḃ =, then due to Theorem 1 and Theorem 14, after a certain transient time t, we have su tλ e t>t v(t) Ḃ < Consequently, we obtain (5) Λ ζ v(t) Ḃ C ζ v(t ) Ḃ (t t ) ζ, ζ >, where v(t ) Ḃ is sufficiently small to aly Theorem 1 or Theorem 14

18 18 HANTAEK BAE, ANIMIKH BISWAS, AND EITAN TADMOR 51 Proof of theorem 15 We only need to show (51) For v L, we have the following energy inequality: This imlies that (5) v(t) L + su v(t) L v t> L, v(s) L ds v L lim inf t v(t) Ḣ 1 = In order to obtain the second relation in (5), for ɛ > arbitrary, choose t large so that 1 t v L < ɛ 4 We note that the energy inequality yields 1 v(s) L t 1 t v L This immediately imlies that there exists t (, t) such that v(t ) L < ɛ Due the uniform bound on v(t) L, it also follows that lim inf u(t) t Ḣ = for < β 1 β For = q =, Ḃ 1, = Ḣ 1 Thus by (5), we have (54) Λ ζ v(t) Ḣ 1 C ζ v(t ) Ḣ 1 (t t ) ζ, ζ >, where v(t ) Ḣ 1 is sufficiently small to aly Theorem 1 For > and q, the embedding Ḣ 1 Ḃ imlies that v(t) Ḃ Therefore, (55) Λ ζ v(t) Ḃ C ζ v(t ) Ḃ (t t ) ζ, ζ >, as t where v(t is sufficiently small to aly Theorem 1 for < or Theorem 14 for ) Ḃ = and q < To deal with the case <, we will use the vorticity ω = v From the vorticity equation, ω t + v ω ω = ω v, we have (11) (56) ω(t) L 1 v L + ω L 1 We will use this L 1 vorticity bound to estimate the decay rate in Besov saces By the interolation of L 1 norm and L norm of j v(t), we have (57) j v(t) L C j v(t) α L jv 1 α, α = L 1 We multily (57) by j( +ζ) Then, (58) j( +ζ) j v(t) L C j( 1 +ζ) j v(t) α L ( j j v(t) L 1 C j( 1 +ζ) j v(t) α L ( j w(t) L 1 ) 1 α ) 1 α where we use the fact that C j( 1 +ζ) j v(t) α L, j v L 1 C j w L 1 C w L 1, C indeendent of j

19 ANALYTICITY AND DECAY ESTIMATES OF THE NAVIER-STOKES EQUATIONS 19 By taking the l q norm to (58), we have ( Λ ζ v(t) (59) C jq( 1 Ḃ +ζ) j v(t) αq We take q such as αq = Then for q q = >, we have (51) Since q (1 + ζ) = 1 () + qζ > 1 by (54) Therefore, (511) where v(t ) Ḃ Theorem 15 Λ ζ v(t) Ḃ Λ ζ v(t) Ḃ j q C v(t) Ḣ q ( 1 +ζ) ) 1 q L for <, the right-hand side of (51) goes to as t C ζ v(t ) Ḃ (t t ) ζ, ζ >, is sufficiently small to aly Theorem 1 This comletes the roof of Remark 51 By using the relation between Besov saces and Triebel-Lizorkin saces F s Ḃ s, F s,, we can obtain decay of weak solutions in Sobolev saces Ẇ s, with < (For the definition of Triebel-Lizorkin saces and embedding roerties, see ) By taking = q <, e tλ v(t v(t) Ḟ, ) Ḃ, Since l l for <, e tλ v(t v(t) Ḟ, ) Ḃ, which is equivalent to the estimate solutions in the otential function W s, such that Therefore, where v(t ) Ḃ, e tλ v(t) Ẇ s, v(t ) Ḃ, s = 1 > Λ ζ v(t) L C ζ v(t ) Ḃ (t t ) ζ s, ζ > s,, is sufficiently small to aly Theorem 1 6 Proof of theorem 16: regularity condition in Besov saces In order to rove Theorem 16, we only need to show that v(t) L aearing in the blowu criterion () can be controlled by (15) To this end,we will estimate v in Besov saces Ḃ,1 which is contained in the L As before, we exress v in the integral form: (61) v(t) = e t v e (t s) P (v v)(s) ds

20 HANTAEK BAE, ANIMIKH BISWAS, AND EITAN TADMOR By alying j to (61) and taking the L norm, we have j v(t) L e tj j v L + e tj j v L + + j e (t s)j j (v v)(s) L ds j e (t s)j( S j v(s) L j v(s) L ) k k v(s) L k v(s) L ds = e tj j v L + I + II, where we use the decomosition v v = T v v + R(v v) at the second inequality Estimation of I By Bernstein s inequality (9), j S j v(s) L l σl σl l v(s) L j( σ) v(s) Ḃσ,, l= where we use the condition σ = 1 q > for q > Then, Therefore, (6) I(t) (I) Ḃ +1,1 su <τ<t su <τ<t j(1+ σ) e (t s)j v(s) Ḃσ, j v(s) L ds (t s) (1+ σ) s q s 1 q v(s) Ḃσ ds, v(s) Ḃ +1 τ 1 q v(τ) Ḃσ, v(τ) Ḃ +1,1 τ 1 q v(τ) Ḃσ, v(τ) Ḃ +1,1,,1 (t s) (1+ σ) s q ds where we use the condition 1 + σ = q < for q <, and 1 (1 + ) σ + 1 q = 1 to aly Lemma 41 Estimation of II II(t) j e (t s)j jσ e (t s)j v(s) Ḃσ, k kσ kσ k v(s) L k( +1) k v(s) L (j k) k( 1 +1) k v(s) L ds ds

21 from which we obtain (6) ANALYTICITY AND DECAY ESTIMATES OF THE NAVIER-STOKES EQUATIONS 1 II(t) Ḃ +1,1 su <τ<t su <τ<t By (6) and (6), (64) v(t) Ḃ +1,1 (t s) (1+ σ) s 1 q s q v(s) Ḃσ ds, v(s) Ḃ +1 τ 1 q v(τ) Ḃσ, v(τ) Ḃ +1,1 τ 1 q v(τ) Ḃσ, v(τ) Ḃ +1,1 v + su Ḃ +1,1 <τ<t,1 (t s) (1+ σ) s q ds τ 1 q v(τ) Ḃσ, v(τ) Ḃ +1,1 We translate the time interval from, t to T a, T Then, (65) v(t) Ḃ +1 v(t a) Ḃ +1,1 Therefore, v(t) Ḃ +1,1 of Theorem 16,1 + su <τ<a (τ) 1 q v(t + τ) Ḃσ, v(t + τ) Ḃ +1,1 does not blow u at T as long as (15) holds This comletes the roof Acknowledgments HB gratefully acknowledges the suort by the Center for Scientific Comutation and Mathematical Modeling (CSCAMM) at University of Maryland where this research was erformed A B also thanks the Center for Scientific Comutation and Mathematical Modeling at the University of Maryland for hosting him while this work was comleted and the University of North Carolina at Charlotte for artial suort via the reassignment of duties rogram References 1 H Bae and A Biswas, Gevrey regularity for a class of dissiative equations with analytic nonlinearity, rerint (11) H Bae, A Biswas, and E Tadmor, Analyticity of the subcritical and critical quasi-geostrohic equations in Besov saces, rerint (11) A Bertozzi and A Majda, Vorticity and incomressible flow, Cambridge University Press, xii+545, 4 A Biswas and D Swanson, Gevrey regularity of solutions to the -D Navier-Stokes equations with weighted l initial data, Indiana Univ Math J 56 (7), no, J Bourgain and N Pavlovic, Ill-osedness of the Navier-Stokes equations in a critical sace in D, J Funct Anal 55 (8), no 9, 47 6 M Cannone, Ondelettes, araroduits et Navier-Stokes, Diderot Editeur, Paris (1995) 7 M Cannone and F Planchon, Self-similar solutions for Navier-stokes equations in R, Comm Partial Differential Equations 1 (1996), JY Chemin, Perfect incomressible fluids, Oxford University Press, USA, Y Chemin, Thormes d unicit our le systme de Navier-Stokes tridimensionnel, J Anal Math 77 (1999), A Cheskidov and R Shvydkoy, The regularity of weak solutions of the D Navier-Stokes equations in B,, Arch Rational Mech Anal 195 (11), P Constantin and C Fefferman, Direction of vorticity and the roblem of global regularity for the Navier- Stokes equations, Indiana Univ Math J 4, no

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