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Far East J. Appl. Math. ANALYTIC AND GEVREY SOLUTIONS OF NON-LINEAR PARTIAL DIFFERENTIAL EQUATIONS G. DE DONNO, A. OLIARO and L. RODINO Dipartimento Di Matematica, Università Di Torino Via Carlo Alberto 10, 10123 Torino, Italy e-mail: dedonno@dm.unito.it oliaro@dm.unito.it rodino@dm.unito.it Abstract We prove some results of existence and regularity in analitic-Gevrey spaces for solutions of linear and non-linear partial differential equations. In particular we consider Navier-Stokes equations and perturbations of powers of the Schrödinger operator. 1. Introduction An elementary, but instructive, example of the use of the Gevrey classes for the study of the non-linear equations of the Applied Mathematics is given by the Cauchy problem: 2 ∂ t u = F (t, x , u, ∂ t u, ∂ x1 u, ..., ∂ xn u ), u(0, x ) = u0 (x ), (∂ t u ) (0, x ) = u1 (x ), (1.1) where x = (x1 , ..., x n ) are the space variables in a domain of R n . Assume for simplicity that the non-linear function F is analytic, but allow 2000 Mathematics Subject Classification: Primary 35S05. Key words and phrases: Please provide. Work supported by NATO grant PST. CLG. 979347. Communicated by Guidorzi Marcello Received February 19, 2004 2004 Pushpa Publishing House 2 G. DE DONNO, A. OLIARO and L. RODINO possibly that dx u = (∂ x1 u, ..., ∂ x n u ) is replaced in (1.1) by Adx u, where A is a matrix of classical pseudo-differential operators of order 0. Problems of the type (1.1) appear often in the applications, related to equations of fluids, or with different Physical meanings, according to the choice of F. It is well known that (1.1) is well-posed in the classical sense, i.e., for initial data in Sobolev spaces or other standard function or distribution spaces, only if restrictive assumptions are satisfied by the right-hand side F. In other relevant cases, despite the solution is expected by Physical intuition, a theorem of existence is not possible in this frame. Gevrey classes provide a different functional frame, allowing to satisfy the natural hope, that (1.1) should be always solvable, independently of F. Namely, if the initial data u0 (x ), u1 (x ) belong to the Gevrey class G s , with index 1 ≤ s ≤ 2, than (1.1) admits a solution u(t, x ), t ≤ T , in the same Gevrey class. Note that, if u0 (x ), u1 (x ) belong to G1 , i.e., are analytic functions, the existence of an analytic solution u(t, x ) is already granted by the Cauchy-Kowalevsky theorem. To be more definite, let us recall here the definition of the Gevrey classes, which play the role of intermediate spaces between the spaces of the C ∞ and analytic functions. Given Ω, open subset of R n , we say that u ∈ G s (Ω ), s ≥ 1, if u ∈ C ∞ (Ω ) and for every compact subset K of Ω we have sup ∂ α u(x ) ≤ C α + (α!)s , 1 (1.2) x ∈K for a constant C depending only on u and K. When s = 1 we recapture the analytic case, whereas for s > 1 we obtain larger spaces, containing functions with compact support. It is interesting to observe that if s u ∈ G0 (R n ), i.e., u ∈ G s (R n ) has compact support, then its Fourier transform u(ξ ) satisfies the estimates ˆ ξ1s u(ξ ) ≤ Ce − ε ˆ , ξ ∈ Rn , (1.3) for some positive constants C and ε. Actually, the estimates (1.3) ANALYTIC AND GEVREY SOLUTIONS OF NON-LINEAR … 3 characterize the G s -regularity of a Fourier transformable function, or distribution. Observe that (1.3) implies ξ1s ξ1s eτ ∫ e 2τ 2 u(ξ ) ˆ = u(ξ ) 2 dξ < ∞ ˆ (1.4) for a sufficiently small τ > 0. In the sequel we shall also consider Gevrey classes on the n-torus T n . Given a function on T n u( x ) = ∑u en α iαx , uα ∈ C, (1.5) α∈Z we have u ∈ G s (T n ), s ≥ 1, if and only if ∑e n 2τ α 1 s uα 2 < ∞ (1.6) α∈Z for some τ > 0. The problem (1.1) can be obviously reset for x ∈ T n , with identical results of existence in G s (T n ), 1 ≤ s ≤ 2. For all the related proofs we address the reader to classical works, see Kajitani [10] and the references there; for the linear case see also the monographies of Rodino [19], Mascarello and Rodino [14]. In the sequel of the paper we want to show the effectiveness of the Gevrey functional frame in the study of some other linear and non-linear partial differential equations, with emphasis on equations of the Mathematical Physics. Namely, in the next Section 2 we review a result of Foias and Temam [6] concerning Navier-Stokes equations. In that paper existence of solutions is assumed already achieved in a certain Sobolev space by other methods, let us refer to Solonnikov [20, 21], and the aim is to show the Gevrey regularity of such solutions. The other Sections of the present article are devoted to prove new results for perturbations of powers of the Schrödinger operator (without potential), in one space variable: S = i∂ t − ∂ 2 . x (1.7) In fact, the local properties of Su = F (t, x , u, ∂ x u ) (1.8) 4 G. DE DONNO, A. OLIARO and L. RODINO do not differ from those of the non-perturbed operator S, see for example Messina and Rodino [15]. On the contrary, for the equations l j S 2u = F (t, x , ∂ t ∂ x u )2l + j ≤ 3 (1.9) we have completely different phenomena, depending on the lower order terms in the (linear or non-linear) expression of F. So for example we prove that S 2u = i∂ t ∂ x u (1.10) is hypoelliptic, i.e., all the solutions are smooth, whereas the solutions of S 2u = 0 admit obviously singularities. Gevrey classes provide a frame for the unified treatment of (1.9): leaving a more detailed analysis to the future, we begin to give in the last part of this paper some simple but representative results of Gevrey solvability. The proofs rely on the techniques introduced in De Donno and Oliaro [4]. 2. Navier-Stokes Equations In the following we report the results of Foias and Temam [6] in a lightly more general form, as suggested by the final Section of that paper. We do not give proofs, since they are essentially a repetition of those in [6]. We consider the Navier-Stokes equations of viscous incompressible fluids with space periodicity boundary conditions, in space dimension n = 2 or n = 3. Setting then the problem in [0, T ] × Tn we obtain for the unknown function u = (u (1) , u (2 ) ) for n = 2, or u = (u (1) , u (2 ) , u (3 ) ) for n = 3, the abstract evolution equation du + νAu + B (u ) = f , u(0 ) = u0 , (2.1) dt where ν is the kinematic viscosity and A, B are defined as standard. Namely, we refer to the Hilbert space H defined as the closed subspace of L2 (T n )n of all functions ANALYTIC AND GEVREY SOLUTIONS OF NON-LINEAR … 5 u = ∑u e n α iαx , uα ∈ Cn , u− α = uα , u0 = 0, (2.2) α∈Z such that for every α ∈ Z n n α ⋅ uα = ∑ α u( ) = 0 j =1 j α j (2.3) (i.e., ∇u = 0). The operator A in (2.1) is the Stokes operator with space periodicity boundary conditions. It is a self-adjoint unbounded positive operator in H. The domain D ( A ) ⊂ H and more generally the domain of the positive power D ( A r 2 ), r > 0, is the set of all u ∈ H such that ∑ α 2r uα 2 < ∞. (2.4) α∈Z n We then consider (G s (T n ))n , the space of all the n-tuple of functions satisfying (1.6). More generally, given s ≥ 1, τ > 0 and r ≥ 0 we define s the space Gτ, r (Tn ) of all the (n-tuples of) functions satisfying ∑ α 2r 2 τ α 1 s e uα 2 < ∞. (2.5) α∈Z n Note that Gτ, r (T n ) ⊂ (G s (T n ))n and for every fixed r ≥ 0 s ∪G τ >0 s n τ, r (T ) = (G s (T n ))n . The spaces G1, r (T n ), τ > 0, r ≥ 0 are then subsets of the analytic class in τ T n . Finally, concerning B in (2.1), we have B (u ) = b(u, u ) where b(u, v ) is defined as usual by n ∂v (k ) (k ) (b(u, v ), w ) = ∑∫ j , k =1 T n u( j ) ∂x j w dx , which one can easily re-interpret in terms of the Fourier coefficients in (2.2). 6 G. DE DONNO, A. OLIARO and L. RODINO Let us now state the result of Gevrey regularity for the solutions of (2.1). We shall prescribe initial data u0 in D ( A1 2 ) ⊂ H , cf. (2.4). It is then well known that in dimension n = 2 the solution u = u(t ) exists and remains bounded in D ( A1 2 ) for all positive times, whereas for n = 3 this is granted only in a finite interval [0, T ]. For all t in the domain of D ( A1 2 ) existence, we have: Theorem 2.1. Let the right-hand side f in (2.1) be given in (G s (Tn ))n ∩ H , s ≥ 1. Then the above solution u(t ) is an analytic function of t, with values in (G s (T n ))n . s More precisely, assume f ∈ Gτ, 0 (Tn ) ∩ H , s ≥ 1, τ > 0. For small values of t > 0 we have u(t ) ∈ Gts, 1 (T n ) ∩ H , whereas for large values of s t we get u(t ) ∈ Gσ, 1 (T n ) ∩ H , for a suitable constant σ, σ ≤ τ. Let us observe that for f ≡ 0, or f ∈ (G1 (T n ))n ∩ H , from Theorem 2.1 we deduce analyticity of the solution with respect to space and time variables, for t > 0. 3. Gevrey Hypoellipticity of Perturbations of Powers of the Schrödinger Operator This Section deals with the properties of hypoellipticity and Gevrey hypoellipticity of perturbations of powers of the Schrödinger operator. The main properties of the operators depend heavily on the lower order terms of their symbol. In particular we consider the following class of linear differential operators: P (x , y, Dx , D y ) = (D y − Dx ) p + 2 ∑ b ( x , y )D D , ( ) l, j ∈I lj l x j y (3.1) ∂ ∂ where Dx = −i and D y = −i ; with p, l, j ∈ Z + , p ≥ 2, and blj : Ω → C ∂x ∂y are smooth functions. The subset of the indices I corresponding to lower ANALYTIC AND GEVREY SOLUTIONS OF NON-LINEAR … 7 order terms will be described in the sequel. The respective class of symbols is the following: p(x , y, ξ, η) = (η2 − ξ ) p + ∑ b (x , y)ξ η , ( ) l, j ∈I lj l j (3.2) obtained from (3.1) by replacing the couple (ξ, η) to the couple of derivatives (Dx , D y ) . We denote by z = (x , y ) the real variables in Ω, open subset in R 2 , neighborhood of a point z0 ; ζ = (ξ, η) the dual variables of z. To investigate (3.1), we follow closely the arguments in De Donno and Oliaro [4], where the class of operators D m − a(x , y ) Dx + y d ∑ b (x , y)D D , ( ) l , j ∈I lj l x j y d <m is studied in the anisotropic frame; also see De Donno and Rodino [5] about isotropic case and pseudo-differential coefficients. Hypoellipticity and solvability in [4], [5] depend only on lower order terms corresponding to a m ∗ 1 unique couple (l ∗ , j ∗ ) ∈ Z 2 with k ∗ := + l + j∗ > m − ( k ∗ > m − 1, d 2 in the case of constant coefficients), such that Im b ∗ ∗ (x , y ) ≠ 0. We also l j 1 obtained a sufficient condition for G s hypoellipticity, s ≥ in ∗ k − (m − 1) md [4], s ≥ ∗ in [5] (cf. the definition of Gevrey spaces in (1.2)). On k − (m − 1) the contrary, for P in (3.1), terms of much lower order may play a role, see the next Theorem 3.1. It is also natural to allow more couples ∗ ∗ (li , ji ), i = 1, ..., n, to give the order k ∗ = 2li∗ + ji∗ . The case of p-powers d of generic operators (D m − a(x , y )Dx ) shall be detailed in a future paper. y Let us observe that for operators of the form ∗ ∗ (D y − Dx )p + iDx D y , 2 l j (3.3) we shall prove in this Section that k ∗ := 2l ∗ + j ∗ varying in the interval 8 G. DE DONNO, A. OLIARO and L. RODINO ( p, 2 p) is allowed if p ≥ 2; while in the case p = 1 we obtained in [4], [5] and [15] that the lower order terms have no influence. We define the following sets for k ∈ Q + , 0 < k < 2 p : I k = {(l, j ) ∈ Z + × Z + : 2l + j = k} and fix k = k ∗ such that p < k ∗ < 2p. We use the notations k − for all k < k ∗ , k + for all k > k ∗ . We define I = I − ∪ I ∪ I + , with I − = ∪ I k∗ k− and I + = ∪ I . k+ m In this Section we prove Sρ, δ estimates of (3.2) in order to obtain hypoellipticity and Gevrey hypoellipticity of the class of operators (3.1). In the plane (ξ, η) we consider the conic neighborhood Λ = { η < Cξ} of the semi-axis ξ > 0, for a suitable constant C; we observe that the intersection of the half-plane ξ < 0 with the curve η2 − ξ = 0 is empty. We denote with Γ the set Ω × Λ, and we state the following: Theorem 3.1. Assume that I consists of couples (li∗ , ji∗ ), i = 1, ..., n, k∗ k ∗ = 2li∗ + ji∗ with p + 1 ≤ k ∗ < 2 p such that: n ∗ ∗ (i) ∑i =1 Im bli∗ ji∗ (x , y)ξli η ji ≠ 0, for all (z , ζ ) ∈ Γ, η ≠ 0; (ii) blj (x , y ) ≡ 0 for all (l, j ) ∈ I , (2l + j > k ∗ ); k+ n ∗ ∗ (iii) ∑i =1 Re bli∗ ji∗ (x , y)ξli η ji ≥ 0, for even p, n ∗ ∗ ∑i =1 Re bli∗ ji∗ (x , y)ξli η ji ≡ 0, for odd p. Then 1 ∗ k p(x , y, ξ, η) ≥ b ζ 2 in Ω × R 2 , for a suitable constant b > 0, and for all α = (α1 , α 2 ) ∈ Z 2 , β = (β1 , β2 ) + ANALYTIC AND GEVREY SOLUTIONS OF NON-LINEAR … 9 ∈ Z 2 and for all K ⊂⊂ Ω we have with suitable constants Lα, β and B + that: α α β ρ α −δ β Dx 1 D y 2 Dξ 1 p(x , y, ξ, η) ζ ≤ Lα, β , ξ + η > B, (3.4) p(x , y, ξ, η) where k∗ − p ρ = , δ ≥ 0, (3.5) 2p for any δ < ρ. Since we have assumed p + 1 ≤ k ∗ < 2 p, we have 1 1 0 < ≤ ρ < . 2p 2 Remark 3.2. By formula (3.4) and by Mascarello and Rodino [14, Theorem 3.3.6], we have that an operator P (z , Dz ), associated to the symbol p(z , ζ ) in (3.2), is C ∞ -microlocally hypoelliptic in Γ; i.e., Γ ∩ WFPu = Γ ∩ WFu, for all u ∈ D ′(Ω ), where W F u is the Hörmander wave front set. A microhypoelliptic operator is hypoelliptic too. Remark 3.3. If the coefficients are analytic, formula (3.4) holds for Lαβ = L α + β +1 α! β! , so by Kajitani and Wakabayashi [11, Theorem 1.9], we have that an operator P (z , Dz ), associated to the symbol p(z , ζ ) in 1 1 1 (3.2) is G s -microlocally hypoelliptic in Γ for s ≥ max , = , ρ 1 − δ ρ 2p that is, s ≥ . ∗ k −p As examples of operators satisfying Theorem 3.1, beside (3.3), we can consider the following: Example. P (x , y, Dx , D y ) = (D y − Dx )p + i( y 2 + 1) Dx D y + h − 2l , 2 l p having k∗ = p + h, 1 ≤ h ≤ p − 1. They are hypoelliptic and G s hypoelliptic 2p for s ≥ in a neighborhood of the origin. h 10 G. DE DONNO, A. OLIARO and L. RODINO Example. P (x , y, Dx , D y ) = (D y − Dx )4 + iy 2 Dx D y + (x 2 + i( y + 1)) Dx D 2 + (x + 1) Dx , 2 4 2 y 3 where k ∗ = 6, the imaginary part is given by ξη2 ( y 2 η2 + ( y + 1)ξ ) ≠ 0 in Ω × Λ, η = 0, with Ω neighborhood of the origin and ξ > 0; and we also / have ξ2 (x 2η2 + (x + 1)ξ ) ≥ 0. We obtain hypoellipticity and Gs hypoellipticity for s ≥ 4. Remark 3.4. When ρ < 1, and δ > 0, one can prove by means of interpolation theory as in Wakabayashi [22, Theorem 2.6] that (3.4) is valid for any (α, β) ∈ Z 4 , if (3.4) holds for + α + β = 1. Hence it is sufficient to verify (3.4) for α + β = 1 since we can consider δ > 0 in Theorem 3.1. Proof of Theorem 3.1. First we estimate the numerator of (3.4) and then we give some lemmas to estimate the denominator, see Lemma 3.5, 3.7 and 3.8. In view of hypothesis (ii) in Theorem 3.1 we have I + ≡ 0, so for α1 = 1, we get Dx p(x , y, ζ ) ζ −δ = (l , j ∈I ) ∑ Dx blj (x , y )ξl η j ζ −δ ≤ L1 ∑ ξ l η j ζ (l, j )∈I ∗ ∪ I − −δ ; k ∗ ∪I− k and similarly for α 2 = 1 j D y p(x , y, ζ ) ζ −δ ≤ L2 ∑ ξ l η (l, j )∈I ∗ ∪ I − ζ −δ , k for suitable constants L1 , L2 . If β1 = 1, j Dξ p(z , ξ, η) ζ ρ ≤ L3 η2 − ξ p −1 + ) ∑ξ (l , j ∈ I l −1 ∪ I− η ζ ρ; k∗ and for β2 = 1 ANALYTIC AND GEVREY SOLUTIONS OF NON-LINEAR … 11 j −1 Dη p(z , ξ, η) ζ ρ ≤ L4 η2 − ξ p −1 η + ) ∑ (l , j ∈I ∪ I− ξ l η ζ ρ , k∗ with suitable constants L3 , L4 . To prove (3.4), it will be then sufficient to show the boundedness, for ζ > B, of the functions j ∑( l, j )∈I ∗ ∪ I − ξ l η Q1 (z , ζ ) = k , p(z , ζ ) 2 j η −ξ p −1 η + ∑( l, j )∈I ∗ ∪ I − ξ l η ζ ρ Q2 (z , ζ ) = k , p(z , ζ ) 2 j η −ξ p −1 + ∑( l , j )∈I ∗ ∪ I − ξ l −1 η ζ ρ Q3 (z , ζ ) = k . p(z , ζ ) First we introduce three regions in the plane (ξ, η) : R1 : cξ ≤ η2 ≤ Cξ R2 : η2 ≥ Cξ R3 : η2 ≤ cξ, (3.6) 1 for suitable constants c, C satisfying the inequalities c << , and 2 C >> 2, cf. [4, 5]. The following estimates then hold: const. η − 2δ , ζ ∈ R1 −δ ζ ≤ const. η − δ , ζ ∈ R2 (3.7) const.ξ − δ , ζ ∈ R3 ; 12 G. DE DONNO, A. OLIARO and L. RODINO and const. η 2ρ , ζ ∈ R1 ρ ζ ≤ const. η 2ρ , ζ ∈ R2 (3.8) const.ξρ , ζ ∈ R3 . By abuse of notation, in the following we shall also denote by R1 , R2 , R3 the sets Ω × R1 , Ω × R2 , Ω × R3 ; recall that Γ = Ω × Λ. The following three lemmas give us some relevant estimates from the below of p(z , ζ ) in (3.2). Lemma 3.5. Let p(z , ζ ) be the function (3.2), such that (i), (ii) and (iii) in Theorem 3.1 hold. Then there are positive constants K1 < 1, B, such that, for (z , ζ ) ∈ R1 , ζ > B. 1 2 2 2 n ∗ li ji ∑ ∗ p(z , ζ ) ≥ K1 (η − ξ ) + 2p Im b ∗ ∗ (x , y ) ξ η . (3.9) li ji i =1 Proof. We have that 2 l j p(z , ζ ) 2 = (η2 − ξ )p + ∑( l, j )∈I ∗ ∪ I − k Re blj (x , y ) ξ η 2 ∑ ∑( n ∗ ∗ + Im b ∗ ∗ (x , y ) ξli η ji + Im blj (x , y ) ξl η j . (3.10) i =1 li ji l , j )∈ I − By developing (3.10) and removing the terms 2 2 ∑ Re blj (x , y )ξl η j (l, j )∈I ∗ ∪ I − and ∑ Im blj (x , y )ξl η j k (l, j )∈I − respectively from the real and imaginary part of p(z , ζ ), we can write 2 n ∗ 3 ∑ ∑ J (z, ζ ), ∗ p(z , ζ ) 2 2 ≥ (η − ξ ) 2p + Im b ∗ ∗ (x , y ) ξli η ji + i li ji i =1 i =1 ANALYTIC AND GEVREY SOLUTIONS OF NON-LINEAR … 13 where ∑ ∗ ∗ J1 (z , ζ ) = 2(η2 − ξ ) p Re b ∗ ∗ li , ji (x , y )ξli η ji (3.11) (li∗ , ji∗ )∈I ∗ k n ∑ ∑ Im b (x , y)ξ η , ∗ ∗ J 2 (z , ζ ) = Im b ∗ ∗ (x , y ) ξli η ji lj l j (3.12) li ji i =1 ( ) l, j ∈I − J 3 (z , ζ ) = 2(η2 − ξ ) p ∑ Re b (x , y)ξ η . ( ) l, j ∈I − lj l j (3.13) The function (3.11) is non-negative by hypothesis (iii). Let us fix attention on J 2 (z , ζ ) and J 3 (z , ζ ) defined respectively by (3.12) and (3.13). We have for all ε > 0 n 2 2 1 ∗ n ∗ ∑ 1 − ε ∑ ∗ ∗ li ji Im b ∗ ∗ (x , y ) ξ η + J 2 (z , ζ ) ≥ Im b ∗ ∗ (x , y ) ξli η ji , 2 li ji 2 i =1 li ji i =1 in R1 , ζ > B. In fact by (3.6) in R1 and hypothesis (i) in Theorem 3.1, for all ε > 0 we get for B sufficiently large k∗ + 2l + j J 2 (z , ζ ) η 2 ≤ const. ∑ ∗ < ε, ζ > B. ∑ η2k n ∗ ∗ Im b ∗ ∗ (x , y ) ξ η li ji (l , j )∈ I − i =1 li ji We remark that k ∗ = 2l ∗ + j ∗ > k − = 2l + j. Concerning (3.13) we also have that: n 2 ∗ 1 2 ∑ (η − ξ)2 p + 1 Im bl∗ j∗ (x , y )ξli η ji + J 3 (z, ζ ) ∗ 2 2 i i i =1 2 n ∗ 1 ∑ ∗ ≥ − ε (η2 − ξ )2 p + Im b ∗ ∗ (x , y ) ξli η ji 2 li ji i =1 in R1 , ζ > B, since 14 G. DE DONNO, A. OLIARO and L. RODINO J 3 (z , ζ ) < ε, ζ > B. (3.14) 2 ∑ n ∗ ∗ 2 (η − ξ) 2p + Im b ∗ ∗ (x , y ) ξ η li ji i =1 li ji In fact, considering the curves ξ = (1 − r ) η2 in the plane (ξ, η), r < 1, we obtain by (3.6) in R1 that the left part of (3.14) is estimated by 2 p + 2l + j 2 p + 2l + j −2k∗ rp η tp η ∑ (l, j )∈I − r 2p η 4p + η2k ∗ ≤ ∑ (l, j )∈I − η 4 p − 2k ∗ + t2 p < ε, t + η > B, (3.15) 1 ∗ where t = , by dividing for ηk ; since k − < k ∗ . Recall that: r For all ε > 0 there exists Bε > 0 such that for x + y > Bε , x α yβ < ε if and only if (2γ − α ) (2ν − β) > αβ. x 2 γ + y 2ν For r ≥ 1, in R1 , we have 2 p + 2l + j − 2k∗ tp η ∑ ( ) l, j ∈I − η 4 p − 2k∗ +t 2p ≤ η − 2 p + 2l + j < ε, since k − < 2p. (3.16) Then 1 2 2 2 n ∗ li ji ∑ ∗ 2p p(z , ζ ) ≥ K1 (η − ξ ) + Im b ∗ ∗ (x , y ) ξ η , ζ > B. (3.17) li ji i =1 for a suitable positive constant K1 . Remark 3.6. From the previous estimate (3.17) follows easily that 1 ∗ k p(z , ζ ) ≥ ζ 2 . Lemma 3.7. Let p(z , ζ ) be the function (3.2). Then there are positive constants K 2 < 1, B, such that: 2p p(z , ζ ) ≥ K 2 η , (z , ζ ) ∈ R2 , ζ > B. (3.18) ANALYTIC AND GEVREY SOLUTIONS OF NON-LINEAR … 15 Lemma 3.8. Let p(z , ζ ) be the function (3.2). Then there are positive constants K 3 < 1, B, such that: p p(z , ζ ) ≥ K 3 ξ , (z , ζ ) ∈ R3 , ζ > B. (3.19) For the proofs of Lemma 3.7 and Lemma 3.8 see De Donno and Rodino [5] concerning similar operators, the present case actually does not involve more complications. By the relations (3.6), (3.7) and (3.8), and the inequality η2 − ξ ≤ 2 η + ξ , we easily estimate the numerators N i , i = 1, 2, 3, of Qi , in the regions R2 and R3 in the following way: const. η 2l + j − δ , ζ ∈ R2 N1 (ζ ) ≤ (3.20) const. ξ l +1 j − δ 2 , ζ ∈ R3 ; const. η 2 p −1 + 2ρ , ζ ∈ R2 N 2 (ζ ) ≤ (3.21) const. ξ p −1 + ρ 2 , ζ ∈ R3 ; 2 p − 2(1 − ρ ) const. η , ζ ∈ R2 N 3 (ζ ) ≤ p −1 + ρ (3.22) const. ξ , ζ ∈ R3 . We have N 3 (z , ζ ) ≤ N 2 (z , ζ ) in R2 and R3 , so we can just consider the functions N1 (z , ζ ) and N 2 (z , ζ ) in those regions. Now Lemma 3.7, Lemma 3.8 and the estimates (3.20) and (3.21) show the boundedness of Q1 (z , ζ ) and Q2 (z , ζ ) in the regions R2 and R3 since in (3.5) always it 1 is ρ < ; so for Q3 (z , ζ ), too. For Q1 (z , ζ ) in R1 we have immediately 2 boundedness since we apply Remark 3.6, formula (3.8) in R1 , recalling that in Q1 , (l, j ) ∈ I ∪ I − with δ ≥ 0. k∗ Regarding Q2 and Q3 in the region R1 , we observe that η2 − ξ vanishes in it, so estimates of the previous type are not optimal. For Q2 , by (3.6), (3.8) and (3.9) we get easily: 16 G. DE DONNO, A. OLIARO and L. RODINO p −1 1 + 2ρ η2 − ξ η Q2 (z , ζ ) ≤ const. 1 2 2 2 ∑ n ∗ ∗ (η − ξ )2 p + Im b ∗ ∗ (x , y ) ξ η li ji i =1 li ji 2l + j −1 + 2ρ η + ) ∑ (l , j ∈I ∪ I− η k∗ ≤ L, (x , y ) ∈ Ω, ζ > B. k∗ 1 The second term in the right-hand side is bounded since ρ < by (3.5). 2 About the first term we can argue in the same way we have done in k∗ − p Lemma 3.5, see formulas (3.14), (3.15) and (3.16), obtaining ρ ≤ . 2p The study of the boundedness of the functions Q3 (z , ζ ) in the region R1 actually does not involve further complicated statements, so arguing like the previous step and using the estimates (3.5) on ρ, we have proved that Qi (z , ζ ), i = 1, 2, 3 is also bounded in R1 . The following Remark ends the proof: Remark 3.9. By formulas (3.9), (3.18) and (3.19), we obtain that 1 ∗ p(z , ζ ) ≥ b ζ 2k , b > 0, ζ > B. In fact we obtain that p(z , ζ ) ≥ 1 ∗ k∗ k∗ 1 k∗ 1 k∗ k∗ 2k ) const. η in R1 , then η = η + η ≥ const. ( η + ξ 2 2 1 ∗ 1 ∗ 1 ∗ 1 ∗ ≥ const. ( η 2k + ξ 2k ) ~ const. ζ 2k , so p(z , ζ ) ≥ b ζ 2k . In the same p way we get p(z , ζ ) ≥ b ζ in R2 and R3 , the result follows since we have k ∗ < 2p. 4. Gevrey Local Solvability for the Square of the Schrödinger Operator with a Nonlinear Perturbation In this Section we consider the following equation: (D y − Dx )2 u + F (x , y; ∂ lx ∂ yu ) |2l + j < 4 = µf (x , y ), 2 j (4.1) ANALYTIC AND GEVREY SOLUTIONS OF NON-LINEAR … 17 where F (x , y; z ), ∈ C6 , is a nonlinear function. Before giving the main result let us recall that, given an open set Ω ⊂ R 2 , the Gevrey anisotropic space G (λ1 , λ 2 ) (Ω ), λ1 ≥ 1, λ 2 ≥ 1, is the set of all C ∞ functions such that for every compact set K ⊂ Ω there exists a positive constant C K satisfying sup K ∂ l ∂ y f (x , y ) ≤ C K j +1l!λ1 j!λ 2 x j l+ for every ( ) l, j ∈ Z + . As standard G0λ1 , λ 2 (Ω ), λ1 , λ 2 > 1, is the set of all the functions in G (λ1 , λ 2 ) (Ω ) with compact support in Ω. We shall prove the following result. Theorem 4.1. Let us fix λ = (λ1 , λ 2 ), 1 < λ1 < 4, λ 2 > 1. We suppose λ that the datum f in (4.1) is in G0 (Ω δ ), Ω δ := {(x , y ) ∈ R 2 : x 2 + y2 < δ}. Moreover, rewriting F (x , y; z ) as G (x , y; ℜz; ℑz ) and setting w = (ℜz , ℑz ) we suppose that: G (x , y; 0 ) = 0; G (x , y; w0 ) ∈ G λ (Ω δ ) for every w0 ∈ R12 ; G (x 0 , y0 ; w ) ∈ G σ (R12 ), for every (x 0 , y0 ) ∈ Ω δ , σ < 4. Then the semilinear equation (4.1) admits a classical solution in Ω δ , for δ and µ sufficiently small. Remark 4.2. Arguing in the isotropic case we have from the previous theorem the G 4 − ε local solvability of (4.1) for every ε > 0. We want now to recall the definition of a class of Gevrey-Sobolev spaces on the strip R × (− δ, δ ), δ > 0. These spaces have been studied in the isotropic form by Gramchev and Rodino [9] and then generalized to the anisotropic case in Marcolongo and Oliaro [13], see also De Donno and Oliaro [4]. To start with, let us recall that the space H s (R × (− δ, δ )), p p < 1, s integer, is the set of the L2 functions satisfying s δ ∞ f (x , y ) 2 Hsp := ∑∫ ∫ k =0 −δ −∞ (1 + ξ 2p s−k ) k~ D y f ( y, ξ ) 2 dξ dy < ∞, (4.2) 18 G. DE DONNO, A. OLIARO and L. RODINO ~ − ixξ where f ( y, ξ ) = ∫e f (x , y )dx is the Fourier transform of f with respect to x. This definition extends to every s > 0 by interpolation. We 1 fix now s > 0, p < 1, τ > 0 and r = + ε, for a fixed (arbitrarily small) 2 1 ε; we set q = . p Definition 4.3. Let us suppose that ψ( y, ξ ) is a positive function pr belonging to the Hörmander class S1, 0 ((− δ, δ ) × R ). We define the Gevrey- ψ Sobolev space H s, q , r (R × (− δ, δ )) as the set of all functions f ∈ L2 ( R × τ, (−δ, δ)) such that f s, q is finite, where f s, q := e τψ ( y, Dx ) f (x , y ) H s ( R × ( − δ , δ )) ; (4.3) p the operator e τψ( y, Dx ) acts on the function f in the following way: 1 ixξ τψ ( y, ξ ) ~ e τψ ( y, Dx ) f (x , y ) = 2π ∫e e f ( y, ξ )dξ. Observe that the operator e τψ( y, Dx ) and its inverse e − τψ( y, Dx ) establish an isometry between Hilbert spaces: e τψ ( y, Dx ) : H s, q , r (R × (− δ, δ )) → H s (R × (− δ, δ )) τ, ψ p (4.4) e − τψ ( y, Dx ) : H p (R × (− δ, δ )) → H s, q , r (R × (− δ, δ )) . s τ, ψ (4.5) In the next theorem we summarize the most important properties of ψ H s, q , r (R τ, × (− δ, δ )). Theorem 4.4. Let us suppose that the function ψ( y, ξ ) satisfies the condition 1q 1q r ψ( y, ξ1 ) − ψ( y, ξ1 − ξ2 ) − ψ( y, ξ2 ) ≤ −b min{1 + ξ1 − ξ2 , 1 + ξ2 } (4.6) for a constant b > 0. Then there exists salg > 0 such that, for every s > salg , ψ H s, q , r is an algebra. τ, ANALYTIC AND GEVREY SOLUTIONS OF NON-LINEAR … 19 q (λ , λ ) Let us fix 1 < λ1 <, λ 2 > 1. Let f ∈ G0 1 2 (R × (− δ, δ )). Then, for r every function ψ satisfying the assumptions of Definition 4.3 for every ψ τ > 0 and s ≥ 0 we have that f ∈ H s, q , r (R × (− δ, δ )). τ, Let us consider now the linear part of the equation (4.1) P = (D 2 + y Dx )2 : we want to analyze the following operator: ~ P = e τψ( y, Dx ) Pe − τψ( y, Dx ) , (4.7) y where we choose from now on ψ( y, ξ ) = 1 + ϕ (ξ ) ξ r 2 , ϕ being a C ∞ 2δ 1 function satisfying 0 ≤ ϕ (ξ ) ≤ 1 for every ξ, ϕ (ξ ) ≡ 0 for ξ ≤ , ϕ (ξ ) ≡ 1 2 for ξ ≥ 1 (observe that ψ( y, ξ ) satisfies (4.6)). The next proposition then follows from the results of De Donno and Oliaro [4]. Proposition 4.5. The symbol of the operator e τψ ( y, Dx ) (D 2 − y Dx ) e − τψ ( y, Dx ) is given by: τ r 2 (η2 − ξ) + i ϕ (ξ) ξ η + p1 + r − ν (x , y, ξ, η), (4.8) δ where p1+r −ν (x , y, ξ, η) satisfies the following estimates in a neighborhood Γ of the set ∑ = {(ξ, η) ∈ R 2\ (0, 0 ) : η2 − ξ = 0}, Γ being of the form Γ = − {(ξ, η) ∈ R 2\ (0, 0 ) : c0 1η2 ≤ ξ ≤ c0 η2 }, c0 > 1 : l j k h Dx D y Dξ Dη p1 + r − ν (x , y, ξ, η) ≤ Cljhk (1 + ξ 12 + η )(1 + r − ν )− 2k − h , for every l, j, k, h ∈ Z + . The results in De Donno and Oliaro [4] assure us that the operator e τψ ( y, Dx ) (D y − Dx ) e − τψ ( y, Dx ) is C ∞ microlocally hypoelliptic in Γ, in 2 particular, denoting its symbol by q(x , y, ξ, η), we have: q (x , y, ξ, η) ≥ c( ξ 12 + η )1 + r , 20 G. DE DONNO, A. OLIARO and L. RODINO Dx D y Dξ Dη q (x , y, ξ, η) ( ξ 1 2 + η )r (2k + h )− (1 − r )(2l + j ) l j k h ≤ Lljkh , q(x , y, ξ, η) for every l, j, k, h ∈ Z + and ξ + η >> 0. Now we observe that, since e τψ ( y, Dx ) e − τψ ( y, Dx ) = e − τψ ( y, Dx ) e τψ ( y, Dx ) = Id ~ we can write P , cf. (4.7), in the following way: ~ P = (e τψ( y, Dx ) (D y − Dx ) e − τψ( y, Dx ) ) (e τψ( y, Dx ) (D y − Dx ) e − τψ( y, Dx ) ). (4.9) 2 2 So we can apply Proposition 4.5 to the two factors in the right hand ~ side of (4.9), obtaining that the symbol of the operator P equals, modulo lower order terms, the square of the (microlocally hypoelliptic in Γ ) ~ symbol (4.8). Then we can construct a microlocal parametrix of P in Γ. ~ Since outside Γ the operator P is microlocally quasi-elliptic, a standard technique of patching together the microlocal parametrices in Γ and outside Γ (cf. Gramchev and Rodino [9], Marcolongo and Oliaro [13]), ~ ~ s gives us a parametrix E of P in the space H1 2 (R × (− δ, δ )) : ~ E : H1 2 (R × (− δ, δ )) → H1 + 2(1 + r ) (R × (− δ, δ )), s s 2 with ~~ ~ PEu = ~(x , y )u + Ru, χ ~ where ~(x , y ) ≡ 1 in a neighborhood of (0, 0 ) and R is a regularizing χ s operator in H1 2 (R × (− δ, δ )). Then by (4.4), (4.5) and (4.7) we have the following result. Proposition 4.6. There exists a linear map E : H s, 2, r (R × (− δ, δ )) → H s,+22(r + r ), ψ (R × (− δ, δ )) τ ,ψ τ , 1 such that PEu = χ(x , y )u + Ru, ANALYTIC AND GEVREY SOLUTIONS OF NON-LINEAR … 21 ( ) 2 where χ(x , y ) ∈ G0λ1 , λ 2 (Ω δ ), 1 < λ1 < , λ 2 > 1, χ(x , y ) ≡ 1 in a neighborhood r ,ψ of the origin, and R is a regularizing operator in the space H s, 2, r . τ The nonlinearity can be treated with the technique developed by Bourdaud et al. [1]. The following proposition holds, cf. De Donno and Oliaro [4], Oliaro and Rodino [16]. Proposition 4.7. Let us consider the nonlinearity in (4.1); we write j j for shortness J (u ) = G (x , y, ℜ(∂ l ∂ yu ), ℑ(∂ l ∂ yu ))|2l + j <4 . We take u(x ) ∈ B ⊂ x x ψ ,ψ H s, 2, r (R × (− δ, δ )), where B is bounded in H s, 2, r . Then we can find a τ, τ continuous non-decreasing function Φ : (0, + ∞ ) → (0, + ∞ ), Φ(0 ) = 0, such that J (u ) s, 2 ≤ Φ( u s + 3, 2 ); (4.10) moreover, if u, v ∈ B we have: J (u ) − J (v ) s, 2 ≤ CB u − v s + 3, 2 (4.11) for every s > salg . We can now prove the solvability of (4.1). ,ψ Proof of Theorem 4.1. Let us fix the datum f ∈ H s, 2, r . Using τ Proposition 4.6 and arguments as in Gramchev and Popivanov [7], see also Gramchev and Rodino [9], Marcolongo and Oliaro [13], De Donno and Oliaro [4], we can find a positive, continuous, non-decreasing function L : [0, δ0 ] → [0, + ∞ ), L(0 ) = 0 such that, defining Ev s + 3, 2 Rv s, 2 A s (δ ) := sup , B s (δ ) := sup , v≠0 v s, 2 v≠0 v s, 2 ψ ψ v∈H s, 2, r (Ω δ ) τ, v∈H s , 2, r (Ω δ ) τ, we have A s (δ ) ≤ L(δ ), B s (δ ) ≤ L(δ ). We are looking for a solution of the form u = Ev, so the equation (4.1) becomes v(x ) = Q (v(x , y )) + µf (x , y ), j j where Q (v(x , y )) := − Rv(x , y ) − G (x , y; ℜ(∂ l ∂ y (Ev )(x , y )), ℑ(∂ l ∂ y (Ev )(x , y ))|2l + j <4 . x x 22 G. DE DONNO, A. OLIARO and L. RODINO We then have to find a fixed point of the operator Q (⋅) + f ; we choose δ and µ such that the following conditions are satisfied: B s (δ ) (1 + µ f s, 2 ) + Φ(A s (δ ) (1 + µ f s, 2 )) ≤1 (4.12) B s (δ ) + A s (δ )C B < 1, (4.13) ,ψ where Φ(⋅) and C B are the ones of Proposition 4.7, B := {w ∈ H s, 2, r (Ω δ ) τ : w − µf s, 2 ≤ 1}. Now, by (4.10) and (4.12) we have that Q (⋅) + f : B → B; moreover (4.11) and (4.13) imply that Q (⋅) + f is a contraction. We then obtain a solution as an application of the Fixed Point Theorem in the Banach space B. Taking s sufficiently large the solution is classical. By 1 Theorem 4.4 and since r = + ε, with ε arbitrarily small, we obtain the 2 λ solvability of (4.1) for f ∈ G0 (Ω δ ). References [1] G. Bourdaud, M. Reissig, and W. Sickel, Hyperbolic equations, function spaces with exponential weights and Nemytskij operators, Prépublication 302, Institut de Mathématiques de Jussieu, Unité Mixte de Recherche 7586, Universités Paris VI et Paris VII/CNRS, Septembre 2001. http://www/institut.math.jussieu.fr/~preprints/index-2001.html. [2] A. Corli, On local solvability of linear partial differential operators with multiple characteristics, J. Differential Equations 81 (1989), 275-293. [3] G. De Donno and A. Oliaro, Hypoellipticity and local solvability of anisotropic PDE’s with Gevrey non-linearity, Preprint, J. Differential Equations (2002), submitted. [4] G. De Donno and A. Oliaro, Local solvability and hypoellipticity for semilinear anisotropic partial differential equations, Trans. Amer. Math. Soc. 335(8) (2003), 3405-3432. [5] G. De Donno and L. Rodino, Gevrey hypoellipticity for partial differential equations with characteristics of higher multiplicity, Rend. Sem. Mat. Univ. Pol. Torino 58(4) (2000), 435-448. [6] C. Foias and R. Temam, Gevrey class regularity for the solutions of the Navier- Stokes equations, J. Funct. Anal. 87 (1989), 359-369. [7] T. Gramchev and P. Popivanov, Local Solvability of Semilinear Partial Differential Equations. Ann. Univ. Ferrara Sez. VII-Sc. Mat. 35 (1989), 147-154. ANALYTIC AND GEVREY SOLUTIONS OF NON-LINEAR … 23 [8] T. Gramchev and P. Popivanov, Partial differential equations: approximate solutions in scales of functional spaces, Mathematical Research, 108, Wiley-VCH Verlag, Berlin, 2000. [9] T. Gramchev and L. Rodino, Gevrey solvability for semilinear partial differential equations with multiple characteristics. Boll. Un. Mat. Ital. B 2(8) (1999), 65-120. [10] K. Kajitani, Local solutions of Cauchy problem for nonlinear hyperbolic systems in Gevrey classes, Hokkaido Math. J. 12 (1983), 434-460. [11] K. Kajitani and S. Wakabayashi, Hypoelliptic operators in Gevrey classes, Recent Developments in Hyperbolic Equations, L. Cattabriga, F. Colombini, M. K. V. Murthy (London), S. Spagnolo, ed., Longman, 1988, pp. 115-134. [12] O. Liess and L. Rodino, Linear partial differential equations with multiple involutive characteristics, Microlocal analysis and spectral theory (Dordrecht), L. Rodino, ed., Kluwer, 1997, 1-38. [13] P. Marcolongo and A. Oliaro, Local Solvability for Semilinear Anisotropic Partial Differential Equations, Ann. Mat. Pura Appl. (4) 179 (2001), 229-262. [14] M. Mascarello and L. Rodino, Partial differential equations with multiple characteristics, Wiley-VCH, Berlin, 1997. [15] F. Messina and L. Rodino, Local solvability for nonlinear partial differential equations, Nonlinear Analysis 47 (2001), 2917-2927. [16] A. Oliaro and L. Rodino, Solvability for Semilinear PDE with Multiple Characteristics, Banach Center Publ. 60 (2003), 295-303. [17] P. Popivanov, Local solvability of some classes of linear differential operators with multiple characteristics. Ann. Univ. Ferrara, VII, Sc. Mat. 45 (1999), 263-274. [18] P. Popivanov and G. S. Popov, Microlocal properties of a class of pseudo-differential operators with multiple characteristics, Serdica 6 (1980), 169-183. [19] L. Rodino, Linear partial differential operators in Gevrey spaces, World Scientific, Singapore, 1993. [20] V. Solonnikov, On estimates of the tensor Green’s function for some boundary value problems, Dokl. Akad. Nauk SSSR 130 (1960), 988-991. [21] V. Solonnikov, Estimates for solutions of nonstationary Navier-Stokes equations, J. Soviet Math. 8 (1977), 467-529. [22] S. Wakabayashi, Singularities of solution of the cauchy problem for hyperbolic system in Gevrey classes, Japan J. Math. 11 (1985), 157-201. g 24 G. DE DONNO, A. OLIARO and L. RODINO Kindly return the proof after Proof read by: …………………………………. correction to: Signature: …..……………………...…………. Date: …………..……………………………….. The Publication Manager Tel: …………..…………………………………. Pushpa Publishing House Fax: …………..………………………………… Vijaya Niwas E-mail: …………..…………………………… Number of additional reprints required 198, Mumfordganj …………………………… Allahabad-211002 (India) Cost of a set of 25 copies of additional reprints @ U. S. Dollars 15.00 per page. (25 copies of reprints are provided to the corresponding author ex-gratis)

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