GEVREY REGULARITY FOR A CLASS OF DISSIPATIVE EQUATIONS WITH ANALYTIC NONLINEARITY

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1 GEVREY REGULARITY FOR A CLASS OF DISSIPATIVE EQUATIONS WITH ANALYTIC NONLINEARITY HANTAEK BAE AND ANIMIKH BISWAS Abstract. In this aer, we establish Gevrey class regularity of solutions to a class of dissiative equations on the whole sace R d, for initial data in certain otential saces. The equation we consider has an analytic nonlinearity and the dissiation oerator is a ower (ossibly fractional) of the Lalacian. This generalizes the results in [15] to the L setting, where the sace eriodic case was considered. Additionally, we allow for rougher initial data and extend the results to the case of the dissiation oerator being a fractional Lalacian. The main tool is a generalization of the Kato-Ponce inequality ([28]) to Gevrey saces. As an alication, we obtain temoral decay of solutions in higher Sobolev norms for a large class of equations including the Navier-Stokes equations, the subcritical quasi-geostrohic equations, a variant of the Burger s equation with a olynomial nonlinearity, and the generalized Cahn-Hilliard equation. 1. Introduction It is well-known that regular solutions of many dissiative equations, such as the Navier-Stokes equations(nse), the Kuramoto-Sivashinsky equation, the surface quasi-geostrohic equation and the Smoluchowski equation are in fact analytic, in both sace and time variables ([33], [17], [3], [14], [45]). 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 ([16], [25], [26], [13]). In other words, the sace analyticity radius yields a Kolmogorov tye length scale encountered in turbulence theory. This fact concerning exonential decay can be used to show that the finite dimensional Galerkin aroximations converge exonentially fast in these cases. For instance, in the case of the comlex Ginzburg- Landau equation, Doelman and Titi ([12]) used radius of analyticity estimates to rigorously exlain numerical observation that the solutions to this equation can be accurately reresented by a very low-dimensional Galerkin aroximation. Other alications of analyticity radius occur in establishing shar temoral decay rates of solutions in higher Sobolev norms ([36]), establishing geometric regularity criteria for the Navier-Stokes equations, and in measuring the satial comlexity of fluid flow (see [3], [29], [23]). Date: July 23, Mathematics Subject Classification. Primary 76D3, 35Q35, 76D5; Secondary 35J6, 76F5. Key words and hrases. Navier-Stokes equations, Quasigeostrohic equations, Dissiative equations, nonlinear analytic equations, Gevrey regularity. Corresonding author ( abiswas@uncc.edu). The first author was artially suorted by NSF grants, DMS and FRG

2 2 HANTAEK BAE AND ANIMIKH BISWAS In this aer, we consider a nonlinear evolution equation on a Banach sace L of the form (1.1) u t + Λ κ u = G(u), (κ > 1, t ), u L. The Banach sace L is a otential sace of adequate (ositive or negative) regularity. The oerator Λ = ( ) 1/2 in our case, and is assumed to be defined on dense subsace D(Λ) L. The term G(u) is a ossibly nonlocal, analytic function of u; see equation (2.3) for a recise descrition. We will also assume that L is a Banach sace contained in the Schwartz sace of distributions S (R d ), where S(R d ) denotes the usual Schwartz sace of test functions. The equation (1.1) will model dissiative evolutionary artial differential equations on the whole sace R d, either with no boundary or with the sace eriodic boundary condition. In this setting, we establish analyticity of solutions to (1.1) and rovide a (shar) lower estimate for the time evolution of the sace analyticity radius. It should be noted that the class of equations we consider encomasses many nonlocal equations (for instance, the Navier-Stokes and the quasi-geostrohic equations). Therefore even the fact that the solutions are analytic for ositive times, do not follow from Cauchy-Kovaleskaya tye theorems. In case the dissiation oerator is Λ κ = Λ 2 = ( ) (i.e., κ = 2), equation (1.1) with an analytic nonlinearity was first studied by Ferrari and Titi ([15]) and Cao, Rammaha, and Titi ([6]) in the sace eriodic case and the shere case resectively. Their aroach was based on energy technique and their sace of initial data comrised of sufficiently smooth functions belonging to adequate (L 2 -based) Sobolev saces. By contrast, we work on certain L -based Banach saces of initial data which allows us to consider much rougher (even distributional) initial data in certain situations. We take the mild solution aroach initiated by Fujita and Kato ([21]), Giga and Miyakawa ([22]) and Weissler ([46]) for the Navier-Stokes equations. Following Foias and Temam ([18]), we study the evolution of Gevrey norms e tγλ u L for a suitable Banach saces L and γ >. The use of Gevrey norms was ioneered by Foias and Temam ([18]) for estimating sace analyticity radius for the Navier-Stokes equations and was subsequently used by many authors (see [5], [15], [2], [4], and the references there in); a closely related aroach can be found in [24]. This aroach enables one to avoid cumbersome recursive estimation of higher order derivatives. We show that solutions to (1.1) are Gevrey regular (i.e., they satisfy the estimate su <t<t e tγλ u L < ), locally in time for initial data of arbitrary size, and globally in time if the initial data is small in critical regularity saces. In many cases, this critical regularity sace recisely corresonds to a scale invariance roerty of the equation. Our key tool in this endeavor is a generalization of the Kato-Ponce inequality ([28]) to Gevrey saces. One of our main emhasis in this aer is to obtain global (in time) Gevrey regular solutions to (1.1) for small initial data in critical regularity saces. This has several alications in the study of long term dynamics. It turns out (as we show here) that many of the equations encountered in fluid dynamics has the roerty that for large times, the solutions have small norm in these critical regularity saces. Thus, as a consequence of our result, we obtain exonential decay of Fourier coefficients in the eriodic case and algebraic decay of higher order L based Sobolev norms for a wide class of equations including the Navier-Stokes equations, sub-critical quasi-geostrohic equation, a variant of Burger s equation with a

3 GEVREY REGULARITY FOR DISSIPATIVE EQUATIONS 3 higher order olynomial nonlinearity and the generalized Cahn-Hilliard equation. In some cases (examle, Navier-Stokes and the subcritical quasigeostrohic equations) this generalizes known results while in some others (examle, Burger s and Cahn-Hilliard), the results are new to the best of our knowledge. For instance, following Gevrey class technique, Oliver and Titi ([36]) established shar uer and lower bounds for the (time) decay of L 2 -based higher order Sobolev norms for the Navier-Stokes equations in the whole sace R d, for a certain class of initial data. Our result yields decay for L -based ( > 1) higher order (homogeneous) Sobolev norms for the Navier-Stokes equations, and for a larger class of initial data. As another corollary of our general result, we also recover a similar decay result of Dong and Li ([14]) for the quasi-geostrohic equations with a roof that avoids the iterative estimation of higher order derivatives, and consequently the intricate combinatorics resent there. In summary, our main results rovide an unified aroach to a variety of decay results ([36], [37], [4], [14]) thus generalizing them to: a wider class of equations; to L decay; and allowing for a larger class of initial data. A more detailed comarison of our results vis a vis some known results is made in the remarks subsequent to the relevant theorems. The organization of this aer is as follows. In Section 2, we state our main results while in Section 3, we state our main alications concerning decay. Sections 4 and 5 are devoted to the roofs of these results while in the Aendix, we have included some requisite background on Littlewood-Paley decomosition of functions. 2. Main results Before describing our main results, we start by establishing some notation and concets. A function u( ) C([, T ]; L) is said to be a strong solution of (1.1) if t u exists and u( ) D(Λ κ ) a.e. and the equation (1.1) is satisfied a.e. A mild solution of (1.1) is a solution of the corresonding integral equation (2.1) u(t) = (Su)(t), (Su)(t) := e tλκ u + t e (t s)λκ G(u(s))ds, where u( ) is assumed to belong to C([, T ]; L). The integral on the right hand side of (2.1) is interreted as a Bochner integral. Henceforth, by a solution to (1.1) we will mean a mild solution, which is a fixed oint of the ma S. For a discussion on the connection between weak, strong, mild and classical solutions see [41]. With Λ = ( ) 1/2, the homogeneous otential saces are defined by Ḣ α = {f S (R d ) : Λ α f L (R d ), f Ḣα := Λ α f L < }, α R. For each i =,, N, let T i be a bounded oerator maing Ḣα+α T i, α Ti to Ḣα such that T i commutes with Λ and its oerator norm is bounded uniformly with resect to α and. More recisely, we will assume that there exists constants C > and α Ti such that for all α R, 1 < < and v Ḣα, we have (2.2) T i v Ḣα C v Ḣα+αTi, T i Λv = ΛT i v. Examles of such oerators are Fourier multiliers with symbols m i (ξ) = Ω i (ξ)ξ α T i or m i (ξ) = Ω i (ξ) ξ α T i,

4 4 HANTAEK BAE AND ANIMIKH BISWAS where Ω i ( ) are bounded homogeneous functions of degree zero (i.e., Ω i (cξ) = Ω(ξ), c R) and α Ti = (α (1) T i,, α (d1) T i ), α (j) T i. Other examles also include shift oerators on the sace of distributions (in which case α Ti = ). The nonlinearity G that we consider has the form (2.3) G(u) = T F (T 1 u, T 2 u,, T N u), F (z 1,, z N ) = a α z α, α Z N + where F above is an analytic function defined on a neighborhood of the origin in R n. More secific assumtions on F will be made in subsequent sections A degree n nonlinearity. In this section, we will assume that F is a monomial of degree n, i.e., (2.4) G(u) = T F (T 1 u, T 2 u,, T n u), F (z 1,, z n ) = z 1 z n, for some n 2, n N. However, see Remark 2.2 for an extension. For ξ R d, denote ξ 1 = d i=1 ξ i while ξ = ( i ξ2 i )1/2 denotes the usual Euclidean norm on R d. Recall that the norms 1 and on R d are equivalent. Let Λ 1 be a Fourier multilier whose symbol is given by m Λ1 (ξ) = ξ 1. Choose (and fix) a constant c > such that c ξ 1 < 1 4 ξ for all ξ Rd. This is ossible since all norms on R d are equivalent. With this notation, we define the Gevrey norm (in Ḣβ ) to be (2.5) v Gv(s,β,) = e cs 1/κ Λ 1 Λ β v L = e cs 1/κ Λ 1 v Ḣβ, where s, β R, and 1. Before stating our result here, we need to determine an aroriate function sace for the initial data. The idea is to choose a function sace where the linear term u t + Λ κ u and the nonlinear term G(u) have the same regularity. This can be interreted in terms of the scaling invariance. The equation (1.1) with the nonlinearity (2.4) satisfies the following scaling. Assume that u is a solution of (1.1). Then, the same is true for the rescaled functions Therefore, Ḣ βc (2.6) u λ (t, x) = λ s u(λ κ t, λx), s = κ n i= α T i n 1 is the scaling invariant sace for initial data, with β c = d κ n i= α T i n 1 and we can exect the local existence for β > β c for large data and the global existence for β = β c for small data. Theorem 2.1. Let G be a nonlinearity as in (2.4). Let u Ḣβ, with d κ n i= α T i = β c β < d n 1 + min α T i. 1 i n Assume that the following condition holds: { n min α i=1 T i > max α T (2.7) i d 1 i n n n,, n i= α T i κ n 1. }.

5 GEVREY REGULARITY FOR DISSIPATIVE EQUATIONS 5 Then there exists an adequate T = T (u ) and β > with β +β >, and a solution of (2.1) belonging to the sace C([, T ]; Ḣβ ) which additionally satisfies (2.8) max { su o<t<t u(t) Gv(t,β,), su t> t β/κ u(t) Gv(t,β+β,)} 2 u Ḣβ. Moreover, if β > β c, then the time of existence T is given by T C u (β βc)/κ Ḣ β for some constant C indeendent of u. On the other hand, in case u Ḣβc, there exists ɛ > such that whenever u Ḣβc < ɛ, we can take T =. Remark 2.2. The above theorem can be readily generalized to the case of a nonlinearity where F in (2.4) is a homogeneous olynomial of degree n with the following roerty: there exists an α R such that F (z 1,, z N ) = α S a α z α,, where S = {α = (α 1,, α N ) Z N + : N α Ti χ {αi } = α and i=1 N α i = n}. This for instance is satisfied if F is a homogeneous olynomial of degree n and α Ti = α Tj for all i, j. Remark 2.3. A version of Theorem 2.1, for the secial case of a quadratic nonlinearity, was established in [1]. In contrast to the set u of the real sace here, the norms on the initial data sace there were defined in the Fourier sace. This enables one to comletely avoid the detailed harmonic analysis machinery used in the roof here. However, due to the Hausdorff-Young inequality, even restricted to the quadratic nonlinearity case, our consideration here yields a larger sace of initial data in several alications. As we will see later, due to this, we can obtain decay of L ( > 1) based higher Sobolev norms (for instance for the Navier-Stokes equations) not available in [1] Analytic Nonlinearity. In this section, we will consider the more general case of an analytic nonlinearity. Let F (z) = a k z k be a real analytic function in k Z n + a neighborhood of the origin. Here z = (z 1,, z n ) R n and we emloy the multiindex convention z k = z k1 1 zkn n for k = (k 1,, k n ). The majorizing function for F is defined to be (2.9) F M (r) = a k r k, r <, k Z d + where for any multi-index k Z d +, k = k k d. The functions F and F M are clearly analytic in the oen balls (in R d and R resectively) with center zero and radius (2.1) R M = su{r : F M (r) < }. i=1

6 6 HANTAEK BAE AND ANIMIKH BISWAS We will assume that the set in the right hand side of (2.1) is nonemty. The derivative of the function F M, denoted by F M, is also analytic in the ball of radius R M. The nonlinearity G is of the tye (2.11) G(u) = T F ( T 1 u,, T n u ), where T i are as defined in (2.2). We consider the inhomogeneous Sobolev sace H α = {f : R d R d1 : f H α := (I + Λ) α f L < }, with Λ = ( ) 1/2. We recall that from the standard Sobolev inequalities and (4.4), we have (2.12) f L f H β, fg H β C f H β g H β for β > d, 1 < <. The Gevrey norm here is defined as (2.13) v Gv(s,β,) = e 1 2 s1/κ Λ 1 (1 + Λ) β v L. We will show that the Gevrey sace Gv(s, β, ) with β > d is a Banach algebra. We are now ready to state our main result concerning analytic nonlinearity. Theorem 2.4. Let β > d and assume that α T +α κ < 1, where α := max {α T i }. 1 i n Let u H β+α < R. Assume that 2RC < R M where C as in Lemma 4.9 and R M as in (2.1). There exists a time T > and a solution u of (2.1) such that su u(s) Gv(s,β,) <. s (,T ) Note that unlike Theorem 2.1, we do not get a global existence result here even in case of small initial data. However, in case of the eriodic boundary condition, we do obtain a global existence result for small data. We need to assume however that the nonlinearity G has the roerty that it leaves the sace of mean zero eriodic functions invariant (this haens for instance if T = ). This is due to the fact that in the eriodic case, Λ has a minimum eigenvalue, denoted by λ >, and the Fourier sectrum of all eriodic functions with sace average zero is contained in the comlement of a ball with radius λ. More recisely, we have the following result. Theorem 2.5. Consider the equation (1.1) with sace eriodic boundary condition. Assume that the sace of mean zero functions are invariant to the analytic nonlinearity G which moreover satisfies a =, a k < δ, k Z d +, k =1 where δ is suitably small. Then, there exists ɛ > such that if u H β < ɛ, with β > d, and 1 < <, we can obtain an unique solution to (2.1) satisfying su <t< u Gv(t,β,) <.

7 GEVREY REGULARITY FOR DISSIPATIVE EQUATIONS Equation in Fourier sace. In Subsection 2.2, we worked under the restriction 1 < <. In this section, we consider the case =. The harmonic analysis tools used in the revious sections do not work here since the singular integrals are not bounded in L. We therefore resort to working in the frequency sace using th Fourier transform. Recall that if the Fourier transform of a distribution in S (R d ) is in L 1 in the Fourier saces, then it is an L function. The develoment here is in the sirit of [3] and [4]. The other borderline case of = (in sace variables) is similar and discussed in Remark 2.7. We will denote by F the Fourier transform (in the sace variables) and by F 1 its inverse. By a notational abuse, letting u = F(u), we can reformulate (1.1) as (2.14) u t (ξ, t) + ξ κ u(ξ, t) = G ( u(, t) ) (ξ), u(ξ, ) = u (ξ), where G is as in (2.11). Recalling that the Fourier transform converts roducts in real sace to convolutions in the frequency sace, the analytic function F in (2.9) takes the form F (v) = a n v n1 1 v n d d, v = (v 1,, v d ) n where denotes convolution. Denoting by D the multilication oerator (Dv)(ξ) = ξ κ v(ξ), we can write the mild formulation of (1.1) as (2.15) u(t) = e td u + t e (t s)d G ( u(, s) ) ds. In this section, we will denote the L 1 norm in the Fourier sace, i.e., v = v(ξ) dξ. R d 1 In case v(ξ) is a vector, v(ξ) will denote its usual Euclidean norm. We also recall that L 1 is a Banach algebra under convolution and u v u v. For s and β R, we will now introduce the Gevrey norms as (2.16) v Gv(s,β) = e 1 2 s1/κ ξ ξ β v(ξ) dξ. R d 1 For notational simlicity, we will suress the deendence of the Gevrey norm on κ (since it is fixed), and when s =, we denote v Gv(,β) = v β and when β =, we will write v Gv(s,) = v Gv(s). Also, denote V β = {v : v β < }. When β =, we will simly write V = V. For u = (u 1,, u d1 ), v = (v 1,, v d1 ) vector valued functions, we denote u v = ( ij b ijk u i v j ) d k=1, b ijk R. It is easy to see, alying the Cauchy-Schwartz inequality, that (u v)(ξ) C( u v )(ξ), where C = max i,j,k b ijk. We now state the existence of a solution to (2.14) with resect in the Gevrey saces (2.16).

8 8 HANTAEK BAE AND ANIMIKH BISWAS Theorem 2.6. Let α = max 1 i n {α T i } and assume that α T + α κ < 1 and max{ u, u α } < R with 2RC < R M, where C is as in Lemma 4.11 and R M as in (2.1). There exists a time T > and a solution u of (2.14) such that (2.17) su u(s) Gv(s) <. s (,T ) Additionally, suose that we are in the sace eriodic setting and the nonlinearity G leaves the sace of functions with zero sace average invariant. Then, if we consider (2.14) on that subsace, there exists ɛ > such that we can take T = in the above inequality rovided max{ u, u α } < ɛ. Remark 2.7. In this subsection, we worked with L 1 norm of the Fourier transform of the function. Due to the Hausdorff-Young inequality, this corresonds to the L norm in the sace variable. However, the other borderline case of Theorem 2.4 occurs when we have L 1 norm in the sace variable. In the frequency sace, this corresonds to the L norm (in the sense that F(u) L u L 1). We can obtain an analogue of Theorem 2.6 if instead of the L 1 norm, we let u = su ξ R d (1 + ξ ) α u(ξ), α > d. The conditions with max{ u, u α } in Theorem 2.5 will simly be relaced with analogous conditions with u α+α. The roof of this fact is very similar to the roof of Theorem 2.6 once one notes that like L 1, the Gevrey sace based on this norm is also a Banach algebra under convolution (see [4]). Remark 2.8. Exonential decay of Fourier coefficients: In the sace eriodic setting (with eriod L), the oerator Λ restricted to the subsace of functions with sace average zero, has a discrete sectrum with its lowest eigenvalue being 2π L. Then, rovided max{ u, u α } in small enough (or in view of Remark 2.7, if u α+α is small enough), we have the exonential decay of the Fourier coefficients F(u)(ξ, t) Ce t1/κ ξ. This is a generalization of the results in [16] and [13] for the secial case of the 3d Navier-Stokes equations. In fact, for the 3d Navier-Stokes equations, following similar techniques as is resented here, one can sharen this decay result (see [3]) by demanding only that su k 2 F(u )(k) < ɛ. k Z 3 + This fact was roven in [3] using similar methods, and indeendently in [42]. 3. Alications: Decay of Sobolev Norms Theorem 2.1 tells us that if the initial data is small in the corresonding critical sace (i.e. in Ḣβc ), then the solution is globally in the Gevrey class. Due to Lemma 4.1 (or Lemma 4.2), this allows us to obtain the following time decay of

9 GEVREY REGULARITY FOR DISSIPATIVE EQUATIONS 9 (homogeneous) Sobolev norms as follows: (3.1) (3.2) Λ ζ u(t) L = Λ ζ βc e ct1/κ Λ 1 e ct1/κ Λ 1 Λ βc u(t) L C ζ t 1 κ (ζ βc) u(t) Gv(t,βc,), (C ζ C ζ ζ ζ, ζ > β c ). If we can show that a solution u( ) of (1.1) satisfies lim inf u(t) t Ḣ =, βc then due to Theorem 2.1, after a certain transient time t, we have Consequently, we obtain (3.3) su u(t) Gv(t t,β c,) <. t>t Λ ζ u(t) L C ζ u(t ) Ḣβ (t t ) 1 κ (ζ βc), ζ > β c where u(t ) Ḣβc is sufficiently small to aly Theorem 2.1. In the next subsections, we will rovide several examles where this can be achieved Navier-Stokes equations. The first alication of Theorem 2.1 is the three dimensional Navier-Stokes equations, with β c = 3 1, max {, 1 2} 3 < β < 1. Comared with revious works by [38, 39, 37, 36, 4, 35] and others, where decay of L 2 -based Sobolev norms have been achieved, we will rovide decay of L -based Sobolev norms. Theorem 3.1. Let 1 < < and u be a weak solution of the three dimensional Navier-Stokes equations where u L 2. In case 1 < < 2, we additionally assume that ω = u L 1. Let β c = 3 1. We then have the following decay estimate: (3.4) where ɛ = u(t ) Ḣβc for = 2. Λ ζ u(t) L C ζ ɛ(t t ) 1 2 (ζ βc), ζ > max{, β c }, is sufficiently small to aly Theorem 2.1, and C ζ C ζ ζ ζ Remark 3.2. Existence of solutions to the NSE in Gevrey classes was first roven for the eriodic boundary condition by Foias and Temam ([18]) (for initial data in H 1 ) and subsequently by Oliver and Titi on the whole sace, with initial data in H s, s > n/2, n = 2, 3 (see also [31] for initial data in Ḣ1/2 for 3D NSE). By following a slightly different aroach, namely interolating the L norms of the solution and its analytic extension, Grujic and Kukavica ([24]) roved analyticity of solutions to the 3D NSE for initial data in L, > 3 (see also [32] for a different roof). Analyticity of solutions for initial data in homogeneous otential saces Ḣ α, 1 < <, α 3 1, which includes the above mentioned L saces, follows from Theorem 2.1. The decay in L 2 -based (homogeneous) Sobolev norms u Ḣζ for the NSE were, to the best of our knowledge, first given in [37] and [4]. However, the constants C ζ were not exlicitly identified there. The shar (and otimal, in the sense of roviding lower bounds as well) decay results were rovided by Oliver and Titi ([36]) following the Gevrey class aroach. The constants C ζ in (3.4) is of the same order as in [36]. Thus, (3.4) is a L version of the shar decay result in [36]. In [37] and [36], there is also an assumtion of the decay of the L 2 norm of the solution. This is circumvented here due to our working in the critical sace Ḣ1/2.

10 1 HANTAEK BAE AND ANIMIKH BISWAS 3.2. Subcritical dissiative surface quasi-geostrohic equations. These equations are given by { ηt + u η + Λ κ η =, (x, t) R 2 (, ), (3.5) η(x, ) = η (x), x R 2, where, u = ( R 2 η, R 1 η) := ( x2 Λη, x1 Λη). We will consider the sub-critical case when the arameter κ satisfies κ (1, 2]. For initial data in L, 2 κ 1, the global existence of a unique, regular solution to (3.5) in the sub-critical case is known (see [7] and the references there in). The following theorem concerning the long time behavior of higher Sobolev norms was first given in [14]. The roof in [14] involves iterative estimation of higher order derivatives involving elaborate combinatorial arguments. We obtain this result as an alication of Theorem 2.1. Theorem 3.3. Let κ (1, 2] in (3.5) and denote = 2 κ 1. Let η be the unique regular solution to for initial data η L (R 2 ). The following estimate then holds with a constant (3.6) η Ḣα Cα α α t α κ for all t >, α >, where the constant C may deend on η, but is indeendent of t, α. Remark 3.4. The higher order decay result in Theorem 3.3 was first roven in [14]. The aroach there was an iterative estimation of higher order derivatives. The use of Gevrey class technique resented here eliminates the necessity of involved combinatorial arguments which is now encoded in the Gevrey norm Burger tye equation with higher order nonlinearity. We now give an alication to decay for a viscous Burger s equation on R of the form (3.7) t u u = x (u n ), x R, n 3. Here we take = 2 and, as is customary, we denote Ḣβ 2 = Ḣβ, β R, β and Ḣ = L 2. For = 2, κ = 2, α T = 1 and α Ti =, 1 i n, the critical sace for (3.7) is Ḣ n 1, i.e., βc = n 1. Theorem 3.5. Let u Ḣ 1 L 2 for n = 3 and u L 2 for n 4. Then, for any weak solution u of (3.7), there exists t > such that Λ ζ u(t) L 2 C ζ ɛ(t t ) 1 2 (ζ βc), ζ > β c := n 1. Here, C ζ ζ ζ C ζ and ɛ = u(t ) Ḣβc is sufficiently small to aly Theorem 2.1. Remark 3.6. In case n = 2, the critical sace is Ḣ 1 2. If (3.8) lim inf t u Ḣ 1 2 =, then, decay of higher Sobolev norms hold.

11 GEVREY REGULARITY FOR DISSIPATIVE EQUATIONS Cahn-Hilliard equation (secial case). The Cahn-Hilliard equation is given by (3.9) u t = 2 u α u + β (u 3 ), x R d, β >, α. Here we will consider the secial case when α = and β >. Here, in the notation of Theorem 2.1, we have n = 3, κ = 4, α T = 2, α Ti =, i = 1, 2, 3, from (2.6), β c := d 2 1. We have the following decay result. Theorem 3.7. Let u be a (weak) solution of (3.9)with initial data u L 2 in sace dimensions d 1. Then, there exists t > such that Λ ζ u(t) L 2 C ζ ɛ(t t ) 1 2 (ζ βc), ζ > β c := d 2 1. Here, C ζ ζ ζ C ζ when = 2 and ɛ = u(t ) Ḣβc is sufficiently small to aly Theorem 2.1. The generalized Cahn-Hilliard equation has been studied in [44]. Due to Theorem 2.5, in the eriodic case, we can imrove the revious decay result to include the generalized Cahn-Hilliard equation. Theorem 3.8. Let u : R d R, d = 1, 2, 3 be a sace eriodic solution of the general Cahn-Hilliard equation u t 2 u = f(u), f(u) = Assume further that (3.1) 2N 2 j=1 2N 1 j=1 j a j < δ, a j u j and a 2N 1 = A >. for δ suitably small and let u Ḣβ with d 2 < β < 2. Then, any solution u satisfies the decay estimate Λ ζ u(t) L 2 C ζ u(t ) Ḣβ (t t ) ζ β κ, ζ > β, where u(t ) Ḣβ is sufficiently small to aly Theorem 2.5. Remark 3.9. The decay results resented in Theorems 3.5, 3.7 and 3.8 are new to the best of our knowledge. 4. Proofs of Main Results 4.1. Degree n nonlinearity. We will start with some rearatory results. Here we will follow the notation in Section 2.1. The following lemma is well-known. We rovide a roof for comleteness and to give recise estimates of the constants involved. Lemma 4.1. Let β, t and κ >. The Fourier multiliers corresonding to the symbols m 1 (ξ) = ξ β e t ξ κ 1 and m2 (ξ) = ξ β e t ξ κ are given by convolution with corresonding kernels k 1 and k 2 both of which are L 1 functions with k i L 1 2Cβ/κ β β/κ, i = 1, 2, t β/κ where C is a constant indeendent of β and κ.

12 12 HANTAEK BAE AND ANIMIKH BISWAS Proof. We only need to obtain the desired estimate for m 1 and m 2 with t = 1. Then, time deendent bounds can be obtain by the scaling: ξ t 1 κ ξ. To estimate m i at t = 1, we will use the Littlewood-Paley decomosition (see Aendix for the definition of this decomosition). Since the estimation is the same for m 1 and m 2, we only estimate m 2. We aly j to m 2 and take the L 1 norm. j m 2 L 1 C2 jβ j e ξ κ L 1 C2 jβ e 2jκ, where the second inequality can be found in [27]. Since m 2 L 1 C j Z j m 2 L 1, we have (4.1) m 2 L 1 C j Z 2 jβ e 2jκ. The series j Z 2jβ e 2jκ is readily seen to be bounded by 2C β/κ β β/κ. From this lemma and the definitions of the oerators T i and the Gevrey norms, we immediately have the following estimates concerning the Gevrey norms. Lemma 4.2. Denote C β = C β β β. For any s, t, β, we have T i v Gv(s,β,) C v Gv(s,β+αTi,), e 1 2 tλk v Gv(s,β+β,) C β where α Ti is as defined in (2.2). (4.2) We will need the following elementary inequality: (x + y) γ x γ + y γ, x, y, γ [, 1]. t β /κ v Gv(s,β,), In the inequality above, by convention, we take = 1 when γ = and min(x, y) =. We also have the following result concerning the semigrou. Lemma 4.3. Let s t <. Let E = e c((t s)1/κ +s 1/κ t 1/κ )Λ 1 The oerator E is either the identity oerator or is a Fourier multilier with an L 1 kernel and its L 1 norm is bounded indeendent of s, t. Proof. Note that since κ 1, by (4.2) we have t 1/κ s 1/κ + (t s) 1/κ. Thus, E = e aλ1 where a = c{(t s) 1/κ +s 1/κ t 1/κ }. In case a =, E is the identity oerator while if a >, E is a Fourier multilier with symbol m E (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 2 +x 2). The L1 norm of this kernel is bounded by a constant indeendent i of a. We will also need the following lemma to roceed. Lemma 4.4. Let κ 1. The oerator Ẽ = e 1 2 aλκ +a 1/κ cλ 1 is a Fourier multilier which acts as a bounded oerator on all L saces (1 < < ) and its oerator norm is uniformly bounded with resect to a.

13 GEVREY REGULARITY FOR DISSIPATIVE EQUATIONS 13 Proof. When a =, Ẽ is the identity oerator. On the other hand, if a >, then Ẽ is Fourier multilier with symbol mẽ(ξ) = e 1 2 a1/κ ξ κ +c a 1/κ ξ 1. Since mẽ(ξ) is uniformly bounded for all ξ the claim now follows from Hormander s multilier theorem (see [43]). We also have the following lemma concerning the linear term. Lemma 4.5. Let u Ḣβ (4.3) Then M C β u Ḣβ M = for some β R and < T, β >. Denote su t β/κ e tλκ u Gv(t,β+β,). <t<t and M as T. Proof. Due to Lemmas 4.2 and 4.4, for any < t T, we have e tλκ u Gv(t,β+β,) = e ct1/κ Λ tλκ Λ β e 1 2 tλκ Λ β u L which roves the first assertion. u Ḣβ+β such that u u Ḣβ using the fact that u Ḣβ+β t β/κ e tλκ u Gv(t,β+β,) C Λ β e 1 2 tλκ Λ β u L C β t β/κ Λ β u L = C β t β/κ u Ḣβ, Concerning the second, note that there exists, we obtain < ɛ for any ɛ >. Proceeding as above, and t β/κ e tλκ (u u ) Gv(t,β+β,) + t β/κ e tλκ u Gv(t,β+β,) Cɛ + Ct β/κ u Ḣβ+β. Letting T and noting that ɛ > is arbitrary, the claim follows. We will need the following versions of the Kato-Ponce inequality ([28]). Γ = Λ or Γ = (I + Λ), we have (4.4) Γ β (ϕψ) L C [ Γ β ϕ L 1 ψ L q 1 + ϕ L 2 Γ β ψ L q 2 ], where β and 1 <, i, q i < are such that 1 = q 1 = q 2. For comleteness, a roof is rovided in the Aendix. The following lemma, which shows that the Kato-Ponce inequality holds for Gevrey norms, is crucial to estimate the nonlinear term in (1.1). Lemma 4.6. Let t, β and 1 <, i, q i < such that 1 = q 1 = q 2. Then, we have the following estimate: [ ] fg Gv(t,β,) C f Gv(t,β,1) g Gv(t,,q1) + f Gv(t,,2) g Gv(t,β,q2). Proof. For notational convenience, denote (4.5) a = ct 1/κ and ϕ = e aλ1 f, ψ = e aλ1 g. By a density argument, it will be enough to rove the result for ϕ, ψ S(R d ). Note that, using the Fourier inversion formula, we have B a (f, g) := e aλ1 (f g) = e aλ1 ((e aλ1 ϕ) (e aλ1 ψ)) = 1 (2π) d e ıx (ξ+η) e a( ξ+η 1 ξ 1 η 1) ˆf(ξ)ĝ(η) dξ dη. For

14 14 HANTAEK BAE AND ANIMIKH BISWAS Recall that for a vector ξ = (ξ 1,, ξ d ), we denoted ξ 1 = d i=1 ξ i. For ξ = (ξ 1,, ξ d ), η = (η 1,, η d ), we now slit the domain of integration of the above integral into 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 253 in [31]) by K 1 f = 1 2π e ıxξ ˆf(ξ) dξ, K 1 f = 1 2π e ıxξ ˆf(ξ) dξ. Let the oerators L a, 1 and L a,1 be defined by L a,1 f = f, L a, 1 f = 1 e ıxξ e 2a ξ ˆf(ξ) dξ. 2π For α = (α 1,, α d ), β = (β 1,, β d ) { 1, 1} d, denote the oerator Z a, α, β = K β1 L t,α1β 1 K βd L t,αd β d and K α = k α1 K αd. 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 d ). A tedious (but elementary) calculation now yields the following identity: B a (f, g) = K α ((Z a, α, β f) (Z a, α, γ g)). α, β, γ { 1,1} d 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 1 and Λ. Moreover, they are bounded linear oerators on L, 1 < < and the corresonding oerator norm of Z a, α, β is bounded indeendent of a. Thus, we can write fg Gv(t,β,) = Λ β e aλ1 (fg) L = Λ β e aλ1 ((e aλ1 ϕ)(e aλ1 ψ)) L = Λ β[ ( (Za, α, β ϕ)( Z a, α, γ ψ ))] L α, β, γ { 1,1} d K α [ = ( K α Λ (Za, α, β β ϕ)( Z a, α, γ ψ ))] L α, β, γ { 1,1} d C Λ β( ( Za, α, β ϕ )( Z a, α, γ ψ )) L [ Λ C β Z a, α, β ϕ L 1 Za, α, γ ψ) L q 1 + Za, α, β ϕ L Λ β 2 Z a, α, γ ψ) ] L q 2 [ C Za, α, β Λβ ϕ L 1 Za, α, γ ψ L q 1 + Za, α, β ϕ Za, α, γ L 2 Λ β ψ ] L q 2 [ ] C f Gv(t,β,1) g Gv(t,,q1) + f Gv(t,,2) g Gv(t,β,q2), where to obtain the above relations, we used the commutativity of Λ β with the oerators K α, Z a, α, β, the L -boundedness of these oerators (with uniformly bounded oerator norms with resect to a ) and (4.4). We will also need the following lemma. Lemma 4.7. All functions are defined on R d. Let s, 1 < < and < α, β < d and α + β > d. We have R (4.6) fg Gv(s,γ,) C f Gv(s,α,) g Gv(s,β,), γ = α + β d.

15 GEVREY REGULARITY FOR DISSIPATIVE EQUATIONS 15 Proof. Alying Lemma 4.6, we have (4.7) fg Gv(s,γ,) C( f Gv(s,γ,r1) g Gv(s,,r2) + f Gv(s,,r3) g Gv(s,γ,r4)), where 1 r r 2 = 1 r r 4 = 1. We now need the Sobolev inequalities ([31]) (4.8) f L q C Λ δ f L, δ < d, δ = d d q. By taking r i, i = 1,, 4 in (4.7) with 1 = 1 r 1 α γ, d 1 = 1 r 2 β d, 1 = 1 r 3 α d, 1 = 1 r 4 β γ, d and alying the Sobolev inequalities given above, we obtain (4.6). An iterative alication of this lemma yields the following: n Gv(s,γ,) n (4.9) f i C f i Gv(s,αi,) rovided γ = i=1 n α i i=1 (n 1)d i=1 >, max 1 i n {α i} < d. Proof of Theorem 2.1. Let γ = β + β for adequate β > to be secified later and define the Banach sace { } E = u C([, T ); Ḣβ ) : u( ) E := max{ u( ) E1, u( ) E2 } <, where Additionally, u( ) E1 := su u(s) Gv(s,β,), u( ) E2 := s β/κ u(s) Gv(s,γ,). s (,T ) E = {u E : u e tλκ u E M}, where M is as in (4.3). It is easy to see that E is a comete metric sace (it is a closed, bounded subset of the Banach sace E). For S as in (2.1), note that (Su)(t) = e tλκ u + (Bu)(t) where (4.1) (Bu)(t) = t We will rove an inequality of the form (4.11) e (t s)λκ G(u(s)) ds. (Bu) E CT µ u n E 2 for adequate µ satisfying µ = if and only if β = β c. Note that this imlies Su e tλκ u E CT µ M n, Su Sv E nct µ M n 1 u v E. The first inequality immediately follows from (4.11) while the second follows by noting n (4.12) G(u) G(v) = T F (T 1 u,, T i 1 u, T i (u v), T i+1 v,, T n (v)). i=1 If β > β c, then µ > and S is a contractive self ma of E rovided T is sufficiently small. On the other hand, in case β = β c and µ =, in view of Lemma 4.5, we

16 16 HANTAEK BAE AND ANIMIKH BISWAS can either choose T small in case the initial data is arbitrary or the initial data sufficiently small if T =, to reach the same conclusion. We now roceed to estimate Bu E2, the estimate for Bu E1 being similar. Note that e (t s)λκ G(u(s)) Gv(t,γ,) = e ct1/κ Λ 1 Λ γ e (t s)λκ G(u(s)) L (4.13) C e c(t s)1/κ Λ 1 e cs1/κ Λ 1 e (t s)λκ Λ γ G(u(s)) L = e c(t s)1/κ Λ (t s)λκ e cs1/κ Λ 1 Λ γ e 1 2 (t s)λκ G(u(s)) L C Λ γ e cs1/κ Λ 1 e 1 2 (t s)λκ G(u(s)) L. By (4.9) with α i = γ α Ti and Lemma 4.2 with β = n i= α T i +(n 1) d (n 1)γ, we have (4.14) where t β/κ (Bu)(t) Gv(t,γ,) C u n E 2 t β/κ t C u n E 2 t µ, 1 1 s (nβ)/κ (t s) 1 κ [ n ds i= α T i +(n 1) d (n 1)γ] µ = (n 1)(β β c ). κ In order to aly (4.9) and Lemma 4.2, and to ensure the finiteness of the integral in (4.14), we need n ( d ) β = α Ti +(n 1) γ, γ min α T i < d n 1 i n, nγ α Ti (n 1) d >. i= For the convergence of the integral in (4.14), we also need nβ < κ, β < κ. A choice of β > satisfying all these conditions can be made rovided the conditions on the arameters stated in the theorem hold Analytic Nonlinearity. Here we follow the notation in subsection.. We will once again start with some auxiliary results. Lemma 4.8. Let β > d. For any t, we have fg Gv(t,β,) C f Gv(t,β,) g Gv(t,β,). Proof. For notational convenience, denote a = ct 1/κ and ϕ = e aλ1 f, ψ = e aλ1 g. Proceeding as in the roof of Lemma 4.6 (with the notation there in) and using (2.12), we obtain fg Gv(t,β,) [ C Za, α, β (I + Λ)β ϕ L Za, α, γ ψ L + Za, α, β ϕ L Za, α, γ (I + Λ) β ψ ] L [ (I C + Λ) β ϕ L (I + Λ) β Z a, α, γ ψ L + (I + Λ) β Z a, α, β ϕ L (I + Λ) β ψ ] L C ϕ H β ψ H β C f Gv(t,β,) g Gv(t,β,), i=1

17 GEVREY REGULARITY FOR DISSIPATIVE EQUATIONS 17 where we have used the fact that (I + Λ) commutes with Z a, α, β. Lemma 4.9. Let β > d, s and u, v be such that max { T iu Gv(s,β,), T i v Gv(s,β,) } R. 1 i n For notational simlicity, denote F (T 1 u,, T n u) = F (u). There exists a constant C indeendent of s, u, v such that (4.15) F (u) Gv(s,β,) F M (RC), F (u) F (v) Gv(s,β,) CF M (RC) max 1 i n T i(u v) Gv(s,β,). Proof. The first inequality in (4.15) is an immediate consequence of Lemma 4.8. Concerning the second, roceeding as in (4.12), we have F (u) F (v) Gv(s,β,) a j j C j R j 1 max T i(u v) Gv(s,β,) 1 i n This comletes the roof. j Z d + CF M (RC) max 1 i n T i(u v) Gv(s,β,). Proof of Theorem 2.4. We recall α = max {α T i }, and set 1 i n { E = u C ( (, T ); H β+α) : u( ) E = su u(s) Gv(s,α,) 2R s (,T ) Proceeding as in the roof of Theorem 2.1 and using Lemma 4.9, we obtain C (4.16) e (t s)λκ G(u(s)) Gv(t,α,) (t s) F (α T +α)/κ M (2RC). Thus, t e (t s)λκ G(u(s)) ds CF M (2RC)T 1 α T E +α κ. Similarly, using the second inequality in (4.15), we can also obtain Su Sv E CF M (2RC)T 1 α T +α κ u v E. The roof is now comleted along the lines of the roof of Theorem The eriodic case. In this case, Λ has a minimum eigenvalue, denoted by λ >, and the Fourier sectrum of all eriodic functions with sace average zero is contained in the comlement of a ball with radius λ. Thus, following [11, 27], we can show that e aλκ u L Ce aλκ u, a >. This fact can be easily roven for = 2 using Plancheral theorem. Thus, instead of (4.16), we can obtain (4.17) e (t s)λκ G(u(s)) Gv(t,β+α,) = e 1 4 (t s)λκ e 3 4 (t s)λκ G(u(s)) Gv(t,β+α,) e 1 2 λκ (t s) e 3 4 (t s)λκ G(u(s)) Gv(t,β+α,) Ce b(t s) (t s) (α T +α)/κ F M (2RC), }

18 18 HANTAEK BAE AND ANIMIKH BISWAS where b = 1 2 λκ. Using now the elementary fact (see for instance Proosition 7.5 in [3]) it follows that su t> t e b(t s) s a ds C <, b >, < a < 1, Su E CF M (2RC) and Su Sv E CF M (2RC) u v E. The roof of Theorem 2.5 follows immediately from the above discussion and by noting that lim F M (2RC) =, R lim su F M (2RC) δ. R We can thus ensure that S is a contractive self ma for T = rovided u Ḣβ+α and δ are small Equation in Fourier sace. As before, we start with some estimates on Gevrey norms. Lemma 4.1. For any s and β =, we have u v Gv(s) C u Gv(s) v Gv(s). Proof. Let γ = 1 2 s1/κ. By triangle inequality ξ ξ η + η, u v Gv(s) C e γ ξ u (ξ η) v (η)dξdη R d 1 R d 1 C e γ ξ η u (ξ η)e γ η v (η)dξdη C u Gv(s) v Gv(s), R d 1 R d 1 which comletes the roof. Lemma 4.1 tells that for any s, Gv(s) is a Banach algebra. Therefore, this sace can be used to estimate analytic functions G(u) as follows. Lemma Let G be as in (2.11) and u, v be such that max { T iu Gv(s,β,), T i v Gv(s,β,) } R. 1 i n For notational simlicity, denote F (T 1 u,, T n u) = F (u). There exists a constant C indeendent of s, u, v such that (4.18) F (u) Gv(s) C 1 F M (RC), F (u) F (v) Gv(s) CF M (RC) max 1 i n T i(u v) Gv(s). Proof. The first inequality in (4.18) is an immediate consequence of Lemma 4.1. Concerning the second, we have F (u) F (v) = a j (u j v j ) j Z d + = j Z d + j 1 [ a j u ( j k ) v k u ( j k 1) v ( k +1)]. k=

19 GEVREY REGULARITY FOR DISSIPATIVE EQUATIONS 19 Alying now Lemma 4.1, we readily obtain F (u) F (v) Gv(s) j 1 a j jc j R j 1 max 1 i n T i(u v) Gv(s) This comletes the roof. CF M (RC) max 1 i n T i(u v) Gv(s). Lemma From the assumtions on T i, for s, t, β, (4.19) T i v Gv(s) v Gv(s,αTi ), T i e 1 2 td v Gv(s) C α Ti t α T i /κ v Gv(s). Proof. By definition of the Gevery norm, the first term in (4.19) can be obtained by T i v Gv(s) ξ α 1 T i e 2 s κ 1 ξ v(ξ) dξ = v Gv(s,αTi ). Since for all α R d 1 ξ α e t ξ κ C α t α κ, where C is indeendent of ξ, t, α, we can rove the second term in (4.2) by T i e 1 2 td v Gv(s) ξ α T i e 1 2 ξ κt e 1 2 s κ 1 ξ v(ξ) dξ R d 1 C α Ti e 1 t α 2 s κ 1 ξ v(ξ) dξ = C T i /κ t v α T i /κ Gv(s). which comletes the roof. R d 1 Proof of Theorem 2.6 We set { E = u C ( (, T ); V ) } } : u( ) E = su max { u(s) Gv(s), u(s) Gv(s,α) R s (,T ) For u E, we define (4.2) (Su)(t) = e td u + t e (t s)d G(u(s))ds. Note that since κ 1, we have t 1/κ s 1/κ + (t s) 1/κ. Consequently, (4.21) e (t s)d G(u(s)) Gv(t) e 1 2 t1/κ ξ e (t s) ξ κ (T F (u(s)))(ξ) dξ R d 1 C e 1 2 (t s)1/κ ξ e 1 2 (t s) ξ κ e 1 2 s1/κ e (t s) 2 ξ κ (T F (u(s)))(ξ) dξ R d 1 C e 1 2 s1/κ ξ e (t s) 2 ξ κ (T F (u(s)))(ξ) dξ, R d 1 where we use the fact that (4.22) e 1 2 τ 1/κ ξ e τ 2 ξ κ = e 1 2 τ 1/κ ξ e 1 2 τ 1/κ ξ κ C, (τ, ξ R d ),

20 2 HANTAEK BAE AND ANIMIKH BISWAS with the constant C indeendent of τ, ξ. Therefore, (4.23) C αt e (t s)d G(u(s)) Gv(t) (t s) F (u) α T /κ Gv(t) C αt (t s) α T /κ C 1F M (RC), where we used (4.19) to obtain the first inequality, and we used the first inequality in (4.18) to obtain the second inequality in (4.23). Similarly, one can obtain also the estimate e (t s)d C αt +α (4.24) G(u(s)) Gv(t,α) (t s) C (α T +α)/κ 1F M (RC). Thus, (4.25) t e (t s)d G(u(s)) ds C αt +αf M (RC) max E {T 1 α T +α κ, T 1 α T } κ. Proceeding along similar lines but using the second inequality in (4.18) in the last ste, we can also obtain Su 1 Su 2 E F M (RC) max {T 1 α T +α κ, T 1 α T } (4.26) κ u 1 u 2 E. Concerning the linear term, using (4.22), it is easy to see that } max { e (4.27) td u Gv(t), e td u Gv(t,α) C max { } R u, u α 2. From (4.25), (4.26) and (4.27), it follows that S is a (strictly) contractive self ma of E rovided T is suitably small. Thus we can find a fixed oint. The adjustment in the above argument necessary to obtain the global result in the eriodic setting is similar to Subsection 4.3 above. 5. Proof of Alications We will now rovide the roof of the decay results stated in Section 3. We will follow the notation in Section Proof of Theorem 3.1. It is enough to rove (3.2). For u L 2, we have the following energy estimate This imlies that (5.1) t u(t) 2 L + u(s) 2 ds u 2 Ḣ 1 2 L 2. su u(t) 2 L u 2 2 L2, lim inf u(t) t> t Ḣ =. 1 In order to obtain the second relation in (5.1), for ɛ > arbitrary, choose t large so that 1 t u 2 L < ɛ/4. We note that the energy inequality yields 1 t 2 t u(s) 2 ds Ḣ 1 1 t u 2 L. This immediately imlies that there exists t 2 (, t) such that u(t ) 2 < Ḣ ɛ. Recall now the interolation inequality 1 u Ḣβ u θ Ḣ β 1 u 1 θ Ḣ β 2, β = θβ 1 + (1 θ)β 2, θ (, 1), β i R, i = 1, 2.

21 GEVREY REGULARITY FOR DISSIPATIVE EQUATIONS 21 Due the uniform bound on u(t) 2 L, it also follows that lim inf u(t) 2 t Ḣ = for β < β 1. When = 2, we have β c = 1 2 and consequently (3.2) holds, which imlies (3.4). For > 2, (3.4) follows from a direct alication of the Sobolev inequalities (4.8). We will now focus on the case 1 < < 2. We will use the the vorticity ω = u. Note first that lim inf t ω 2 L2 = lim inf t u 2 L2 = lim inf t u 2 =. Ḣ 1 From the vorticity equation, ω t + u ω ω = ω u, we have the uniform L 1 bound (see [9]) (5.2) ω(t) L 1 C( u 2 L 2 + ω L 1). The uniform L 1 bound and the interolation inequality imlies (5.3) ω L q ω θ L 1 ω (1 θ) L 2, 1/q = θ + (1 θ)/2 lim inf ω Lq =, 1 < q 2. t Recall that u = ( ( ) 1 ω). The oerator ( ( ) 1 ) is a Fourier multilier of homogeneous degree zero, and consequently, by the Calderon-Zygmund theorem, we have (5.4) ω Ḣζ q = Λ ζ ω L q Λ ζ u L q u Ḣ1+ζ, 1 < q <, ζ. q Consequently, (5.5) lim inf u t Ḣ =, 1 < q < 2. 1 q For = 3/2, β c = 1 and (3.2) follows directly from (5.5). For 3/2 < < 2 and β c as above, alying (4.8), we have u Ḣβc u Ḣ1 and once again (3.2) follows 3/2 from (5.5). We now consider the case 1 < < 3/2. From (5.4) and (3.4) with = 2 (which we already established), it follows that (5.6) lim ω t Ḣ lim ζ u Ḣ1+ζ =, ζ. t Recall now the generalization of the Gagliardo-Nirenberg inequalities for the fractional (homogeneous) Sobolev saces (see [34]), namely, (5.7) ω Ḣα C ω θ Ḣ α 1 q 1 ω 1 θ L q 2, θ (, 1), α = θα 1, Combining (5.7) with (5.3) and (5.6) we have Consequently, by (5.4), we also have lim inf ω t Ḣ =, β >, 1 < < 2. β lim inf u t Ḣ =, 1 < < 3/2, β >. 1+β 1 = θ + 1 θ. q 1 q 2 Noting that for 1 < < 3/2, we have β c > 1. By taking β such that 1 + β = β c, (3.2) follows.

22 22 HANTAEK BAE AND ANIMIKH BISWAS 5.2. Proof of Theorem 3.3. For notational simlicity, we will write Gv(t,,) = Gv(t). Note that in the notation of Theorem 2.1, we have T =, T 1 = R, T 2 = I. It is known that solution to (3.5) satisfies (see [1]) lim η L =, 1. t We will aly the local existence art of Theorem 2.1 with d = 2, n = 2, α T = 1, α T1 = α T2 =, = 2 κ 1, β c =. Let t 1 be such that η(t 1 ) L < ɛ where ɛ is as in Theorem 2.1. Alying the global existence art of this theorem, we have From (3.1), we obtain η(t) Ḣα Cα α α (t t 1 ) α κ su η(t) Gv(t t1) <. t>t 1 Cα 1 α α t α κ for all t t 1 + 1, α >. Also from the local existence art of the theorem, there exists t 2 >, β > such that (5.8) max{ η Gv(t), t β/κ η Ḣβ } 2 η L for all < t t 2. Thus (3.6) holds for all t (, t 2 ] [t 1 + 1, ). To comlete the roof, we will need to show (3.6) for t [t 2, t 1 + 1]. In fact, since t lies in the comact interval [t 2, t 1 + 1], it will be enough to show an estimate of the form (5.9) η(t) Ḣα C α α α for a constant C that does not deend on t or α. Note that due to (5.8), we have η(t 2 /2) Ḣβ <. Due to the global well-osedness for the sub-critical quasigeostrohic equations, we have (see [47, 7]) M := su η(t) Ḣβ <. t 2 t t 1+1 Thus, alying the non-critical case of Theorem 2.1, there exists a time < t 3 < t 2 /2 deending on M, such that η(t) Gv(t3) 2 η(t t 3 ) Ḣβ 2M for all t 2 t t Once again due to (3.1), this yields (5.9) Proof of Theorem 3.5. As noted reviously, it will suffice to rove (3.2). We begin with the the L 2 energy estimate. We multily (3.7) by u and integrate over R. Using integration by arts, we get t u(t) 2 L + u(s) 2 ds u 2 Ḣ 1 2 L 2. As in the revious subsection, this imlies lim inf t u(t) Ḣ1 =. Consequently, due to the uniform bound on u 2 L, it also follows that lim inf u(t) 2 t Ḣ = for all β < β 1. If n 4, β c (, 1) and so the decay now follows.

23 GEVREY REGULARITY FOR DISSIPATIVE EQUATIONS 23 For n = 3, the critical sace is L 2. Therefore, we need to show lim inf u(t) L t 2 =. To do this, it will be enough to obtain a time indeendent bound for u(t) Ḣ 1. To this end, we multily (3.7) by Λ 2 u and integrate by arts over R to get (5.1) d dt Λ 1 u(t) 2 L 2 + u(t) 2 L 2 = R Λ 2 u(t) x (u(t) 3 )dx Λ 1 u(t) L 2 u(t) L 2 u(t) 2 L ] C [ Λ 1 u(t) 2 L 2 u(t) 4 L u(t) 2 L 2, where we have used the fact that the Hilbert transform x Λ 1 is a bounded oerator on L 2. In one dimension, we have the inequality [2]: (5.11) u(t) L C u(t) 1 2 L 2 u(t) 1 2 L 2. Using (5.1), (5.11) and Gronwall s inequality, we obtain This finishes the roof. t Λ 1 u(t) 2 L 2 C Λ 1 u 2 L ex u(s) 4 2 L ds t C Λ 1 u 2 L ex u(s) 2 2 L 2 u(s) 2 L 2ds [ t ] C Λ 1 u 2 L ex u 2 2 L u(s) 2 2 L 2ds C Λ 1 u 2 L 2 ex [ u 4 L 2 ] Proof of Theorem 3.7. We only need to show (3.2). Multilying (3.9) by u and integrating by arts, we arrive at 1 d 2 dt u 2 L + 2 u 2 L 3β u 2 u 2 dx. 2 R d Noting β >, as before, this immediately yields t u(t) 2 L + u(s) 2 2 L 2ds u 2 L 2. This yields an time indeendent uniform bound for u(t) L 2 and also that lim inf u L t 2 =. Interolation immediately yields (3.2) in case d 3. For cases d = 1, 2, multilying (3.9) by Λ 2 u (recall Λ 2 = ) and integrating by arts, we arrive at 1 d 2 dt Λ 1 u 2 L + 2 Λu 2 L 3β u 4 dx. 2 R d This yields u Ḣ 1 u Ḣ 1, lim inf t u Ḣ 1 =. Noting that for d = 1 and d = 2, we have β c = 1/2 and β c = resectively, we have (3.2).

24 24 HANTAEK BAE AND ANIMIKH BISWAS 5.5. Proof of Theorem 3.8. Multilying the equation by u and integrating by arts, we have the energy inequality 2N 2 1 d 2 dt u 2 L + 2 u 2 L j a 2 j u 2 u j 1 dx R d Let I 1 = 2N 2 j=1 j=1 (2N 1)a 2N 1 R d u 2 u 2(N 1) dx. 2N 2 j a j u 2 u j 1 dx, I 2 = u: u 1 j=1 j a j u 2 u j 1 dx u: u >1 and note that alying (3.1) and Poincare inequality, we get I 1 δ u 2 dx δλ u 2 L 2, I 2 δ u 2 u 2(N 1) dx. R d R d Provided δ is sufficiently small, from the energy inequality, we get 1 d 2 dt u 2 L + 2 γ u 2 L, 2 where γ > is an adequate constant. As before, from here we obtain u 2 L u 2 2 L2, lim su u L 2 =. t By interolation, this imlies that lim inf t u Ḣβ = for all β < 2. By Poincare inequality, this in turn imlies lim inf t u L 2 = as well. Consequently, lim inf t u H β = for all β < 2. This finishes the roof. 6. Aendix We now resent the Littlewood-Paley theory and its alication to the Kato- Ponce inequality. For more details of the Littlewood-Paley theory, see [8, 11] 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(2 j ξ) = 1, j= where ψ(ξ) = ϕ( ξ 2 ) ϕ(ξ). We denote ψ(2 j ξ) by ψ j (ξ). The homogeneous dyadic blocks and lower frequency cut-off functions are defined by j u = 2 jd h(2 j y)u(x y)dy, S j u = 2 jd h(2 j y)u(x y)dy, R d R d where h = F 1 ψ and h = F 1 χ. We note that u = j Z ju in S h, where S h is the sace of temered distributions u such that lim S ju = in S. This is called j the Littlewood-Paley decomosition. This decomosition allows us to characterize a large range of functions saces in a unified way. In this section, we only consider the homogeneous Triebel-Lizorkin saces ([19]): f F =,q α ( 2 2jα j f 2) 1 2 L j Z