LOCAL WELL-POSEDNESS OF NONLINEAR DISPERSIVE EQUATIONS ON MODULATION SPACES

Transcription

1 LOCAL WELL-POSEDNESS OF NONLINEAR DISPERSIVE EQUATIONS ON MODULATION SPACES ÁRPÁD BÉNYI AND KASSO A. OKOUDJOU Abstract. By using tools of time-frequency analysis, we obtain some improve local well-poseness results for the NLS, NLW an NLKG equations with Cauchy ata in moulation spaces M. 1. Introuction an statement of results The theory of nonlinear ispersive equations (local an global existence, regularity, scattering theory) is vast an has been stuie extensively by many authors. Almost exclusively, the techniques evelope so far restrict to Cauchy problems with initial ata in a Sobolev space, mainly because of the crucial role playe by the Fourier transform in the analysis of partial ifferential operators. For a sample of results an a nice introuction to the fiel, we refer the reaer to Tao s monograph [12] an the references therein. In this note, we focus on the Cauchy problem for the nonlinear Schröinger equation (NLS), the nonlinear wave equation (NLW), an the nonlinear Klein-Goron equation (NLKG) in the realm of moulation spaces. Generally speaking, a Cauchy ata in a moulation space is rougher than any given one in a fractional Bessel potential space an this low-regularity is esirable in many situations. Moulation spaces were introuce by Feichtinger in the 8s [6] an have asserte themselves lately as the right spaces in time-frequency analysis. Furthermore, they provie an excellent substitute in estimates that are known to fail on Lebesgue spaces. This is not entirely surprising, if we consier their analogy with Besov spaces, since moulation spaces arise essentially replacing ilation by moulation. The equations that we will investigate are: (1) (NLS) i u t + xu + f(u) =, u(x, ) = u (x), (2) (NLW ) 2 u t 2 xu + f(u) =, u(x, ) = u (x), u t (x, ) = u 1(x), (3) (NLKG) 2 u t 2 + (I x)u + f(u) =, u(x, ) = u (x), u t (x, ) = u 1(x), Date: April 2, Mathematics Subject Classification. Primary 35Q55; Seconary 35C15, 42B15, 42B35. Key wors an phrases. Fourier multiplier, weighte moulation space, short-time Fourier transform, nonlinear Schröinger equation, nonlinear wave equation, nonlinear Klein-Goron equation, conservation of energy. 1

2 2 Á. Bényi an K. A. Okoujou where u(x, t) is a complex value function on R R, f(u) (the nonlinearity) is some scalar function of u, an u, u 1 are complex value functions on R. The nonlinearities consiere in this paper will be either power-like (4) p k (u) = λ u 2k u, k N, λ R, or exponential-like (5) e ρ (u) = λ(e ρ u 2 1)u, λ, ρ R. Both nonlinearities consiere have the avantage of being smooth. The corresponing equations having power-like nonlinearities p k are sometimes referre to as algebraic nonlinear (Schröinger, wave, Klein-Goron) equations. The sign of the coefficient λ etermines the efocusing, absent, or focusing character of the nonlinearity, but, as we shall see, this character will play no role in our analysis on moulation spaces. The classical efinition of (weighte) moulation spaces that will be use throughout this work is base on the notion of short-time Fourier transform (STFT). For z = (x, ω) R 2, we let M ω an T x enote the operators of moulation an translation, an π(z) = M ω T x the general time-frequency shift. Then, the STFT of f with respect to a winow g is V g f(z) = f, π(z)g. Moulation spaces provie an effective way to measure the time-frequency concentration of a istribution through size an integrability conitions on its STFT. For s, t R an 1 p, q, we efine the weighte moulation space M p,q t,s (R ) to be the Banach space of all tempere istributions f such that, for a nonzero smooth rapily ecreasing function g S(R ), we have ( ) q/p ) 1/q f M p,q = V t,s g f(x, ω) ( R p < x > tp x < ω > qs ω <. R Here, we use the notation < x >= (1 + x 2 ) 1/2. This efinition is inepenent of the choice of the winow, in the sense that ifferent winow functions yiel equivalent moulation-space norms. When both s = t =, we will simply write M p,q = M p,q,. It is well-known that the ual of a moulation space is also a moulation space, (M p,q s,t ) = M p,q s, t, where p, q enote the ual exponents of p an q, respectively. The efinition above can be appropriately extene to exponents < p, q as in the works of Kobayashi [9], [1]. More specifically, let β > an χ S such that supp ˆχ { ξ 1} an k Z ˆχ(ξ βk) = 1, ξ R. For < p, q an s >, the moulation space M p,q is the set of all tempere istributions f such that (6) ( ( k Z R f (M βk χ)(x) p x ) q ) 1 p q < βk > sq <.

3 NONLINEAR DISPERSIVE EQUATIONS ON MODULATION SPACES 3 When, 1 p, q this is an equivalent norm on M p,q, but when < p, q < 1 this is just a quasi-norm. We refer to [9] for more etails. For another efinition of the moulation spaces for all < p, q we refer to [5, 15]. For a iscussion of the cases when p an/or q =, see [4]. These extensions of moulation spaces have recently been reiscovere an many of their known properties reprove via ifferent methos by Baoxiang et all [1], [2]. There exist several embeing results between Lebesgue, Sobolev, or Besov spaces an moulation spaces, see for example [11], [13]; also [1], [2]. We note, in particular, that the Sobolev space Hs 2 For further properties an uses of moulation spaces, the intereste reaer is referre to Gröchenig s book [8]. The goal of this note is two fol: to improve some recent results of Baoxiang, Lifeng an Boling [1] on the local well-poseness of nonlinear equations state above, coincies with M2,2. by allowing the Cauchy ata to lie in any moulation space M, p >, s, +1 an to simplify the methos of proof by employing well-establishe tools from timefrequency analysis. Ieally, one woul like to aapt these methos to eal with global well-poseness as well. We plan to aress these issues in a future work. In what follows, we assume that 1, k N, < p, λ, ρ R an s +1 are given. With p k an e ρ efine by (4) an (5) respectively, our main results are the following. Theorem 1. Assume that u M (R ) an f {p k, e ρ }. Then, there exists T = T ( u M ) such that (1) has a unique solution u C([, T ], M (R )). Moreover, if T <, then lim sup u(, t) M =. t T Theorem 2. Assume that u, u 1 M (R ) an f {p k, e ρ }. Then, there exists T = T ( u M, u 1 M ) such that (2) has a unique solution u C([, T ], M (R )). Moreover, if T <, then lim sup u(, t) M =. t T Theorem 3. Assume that u, u 1 M (R ) an f {p k, e ρ }. Then, there exists T = T ( u M, u 1 M ) such that (3) has a unique solution u C([, T ], M Moreover, if T <, then lim sup u(, t) M =. t T Remark 1. In Theorem 1 we can replace the (NLS) equation with the following more general (NLS) type equation: (7) (NLS) α i u t + α/2 x u + f(u) =, u(x, ) = u (x), (R )). for any α [, 2] an p 1. The operator α/2 x is interprete as a Fourier multiplier operator (with t fixe), α/2 x u(ξ, t) = ξ α û(ξ, t). This strengthening will become evient from the preliminary Lemma 1 of the next section. Remark 2. Theorems 1.1 an 1.2 of [1] are particular cases of Theorem 1 with p = 2 an s =.

4 4 Á. Bényi an K. A. Okoujou 2. Fourier multipliers an multilinear estimates The generic scheme in the local existence theory is to establish linear an nonlinear estimates on appropriate spaces that contain the solution u. As inicate by the main theorems above, the spaces we consier here are M, an we present the appropriate estimates in the lemmas below. In fact, we will nee estimates on Fourier multipliers on moulation spaces. As prove in [3] an [7], a function σ(ξ) is a symbol of a boune Fourier multiplier on M p,q for 1 p, q if σ W (FL 1, l ) (see the proofs of the following two lemmas for a efinition of this space). As we shall inicate below, this conition can be naturally extene to give a sufficient criterion for the bouneness of the Fourier multiplier operator on M p,q for < p, q an s. The notation A B stans for A cb for some positive constant c inepenent of A an B. Lemma 1. Let σ be a function efine on R an consier the Fourier multiplier operator H σ efine by H σ f(x) = σ(ξ) ˆf(ξ) e 2πξ x ξ. R Let χ S such that supp ˆχ { ξ 1}. Let 1, s, < q, an < p < 1. If σ W (FL p, l )(R ), i.e., σ W (FL p,l ) = sup n Z σ T βn χ FL p < for β >, then H σ extens to a boune operator on M p,q (R ). Proof. We use the efinition of the moulation spaces given by (6) (see also [9]). In particular, let χ S such that supp ˆχ { ξ 1}, an efine g S by ĝ = ˆχ 2. Denote g(x) = g( x). For f S, β >, k Z an x R we have: H σ f (M βk g)(x) = V g H σ f(x, βk) = σ ˆf, M x T βk ĝ = σ ˆf, M x T βk ˆχ 2 F 1 (σ T βk ˆχ) F 1 ( ˆf T βk ˆχ) (x) F 1 (σ T βk ˆχ) f (M βk χ) (x). Now, observe that supp ( σ T βk ˆχ ) Γ k := βk + { ξ 1} an supp ( ˆf Tβk ˆχ ) Γ k. Moreover, by assumption we know that σ W (FL p, l ) an so F 1 (σ T βk ˆχ) L p an f (M βk χ) L p. Consequently, by [9, Lemma 2.6] we have the following estimate H σ f (M βk g) L p C F 1 (σ T βk ˆχ) L p f (M βk χ) L p, where C is a positive constant that epens only on the iameter of Γ k an p. Clearly, the iameter of Γ k is inepenent of k, an this makes C a constant epening only on the imension an the exponent p. Therefore, for < q we have H σ f M p,q sup k Z F 1 (σ T βk ˆχ) L p f M p,q = σ W (FL p,l ) f M p,q.

5 NONLINEAR DISPERSIVE EQUATIONS ON MODULATION SPACES 5 The result then follows from the ensity of S in M p,q for p, q < ; see [9, Theorem 3.1]. We are now reay to state an prove the bouneness of Fourier multipliers that will be neee in establishing our main results. Lemma 2. Let 1, s, an < q be given. Define m α (ξ) = e i ξ α. If 1 p an α [, 2], then the Fourier multiplier operator H mα extens to a boune operator on M p,q (R ). Moreover, If α {1, 2} an < p, then the Fourier multiplier operator +1 H mα extens to a boune operator on M p,q (R ). Proof. First, we prove the result when 1 p, an < q. Let g S(R ) an efine χ S by ˆχ = g 2. For f S, we have V χ H mα f(x, ξ) = m α (t) ˆf(t) e 2πix t ˆχ(t ξ) t R = 1 m <ξ> s α (t) T ξ g(t) < t > s ˆf(t) <ξ>s < t ξ > N g(t ξ) e 2πix t t <t> s <t ξ> N R = 1 m <ξ> s α (t) T ξ g(t) φ N (ξ, t) < D > s f(t) T ξ g N (t) t R ( = 1 F m <ξ> s α T ξ g φ N (ξ, ) < D > s f T ξ g N )( x) = 1 F(m <ξ> s α T ξ g) F 2 (φ N (ξ, )) F( < D > s f T ξ g N )( x), where N > is an integer to be chosen later, g N (t) =< t > N g(t), φ N (ξ, t) = <ξ> s, an < D > s is the Fourier multiplier efine by < D > <t> s <t ξ> s f(ξ) =< ξ > s N ˆf(ξ). We also enote by Φ 2,N (ξ, ) := F 2 (φ N (ξ, )) the Fourier transform in the secon variable of φ N (ξ, )

6 6 Á. Bényi an K. A. Okoujou We can therefore estimate the weighte moulation norm of H mα f as follows: H mα f M p,q ( ( ) q/p ) 1/q = V χ f(x, ξ) p x < ξ > qs R R ( ( = F(m p q/p α T ξ g) Φ 2,N (ξ, ) F( < D > s f T ξ g N )( x) x) ξ R R ( ) 1/q F 1 (m α T ξ g) q L Φ 1 2,N (ξ, ) q L F( < D > s f T 1 ξ g N ) q L ξ p R ( sup ξ R F 1 (m α T ξ g) L 1 (8) sup ξ R F(m α T ξ g) L 1 sup ξ R Φ 2,N (ξ, ) L 1 sup ξ R Φ 2,N (ξ, ) L 1 f M p,q. Now, it follows from [3, Lemma 8] that, for α [, 2], ) 1/q ) 1/q F 1 ( < D > s f T ξ g N ) q L ξ p R sup F 1 (m α T ξ g) L 1 := m α W (FL 1,l ) <. ξ R Moreover (see, for example, [13, Lemma 3.1] or [14, Lemma 2.1]), we can select a sufficiently large N > such that sup Φ 2,N (ξ, ) L 1 sup Φ 2,N (ξ, x)) x <. ξ R R ξ R Hence, using (8), we get H mα f M p,q C α f M p,q. To prove the secon part of the result we shall use Lemma 1. In particular, we nee to show that for α {1, 2} an < p < 1, m +1 α W (FL p, l ). This, however, follows by straightforwar aaptations of the proofs of [3, Theorems 9 an 11], which we leave to the intereste reaer. In analogy to the proof of the previous lemma, we can prove the following weighte version of [3, Theorem 16]. Lemma 3. Let 1, s, m (1) (ξ) = sin( ξ ) ξ operators H m (1), H m (2) +1 < p an < q be given, an let an m (2) (ξ) = cos( ξ ), for ξ R. Then, the Fourier multiplier can be extene as boune operators on M p,q. A smooth version of Lemma 3 is obtaine by replacing ξ with < ξ >. Lemma 4. Let 1, s, < p an < q be given, an let +1 m(ξ) = e i<ξ>, m (1) (ξ) = sin(<ξ>) an m (2) (ξ) = cos(< ξ >), for ξ R. Then, the <ξ> Fourier multiplier operators H m, H m (1), H m (2) can be extene as boune operators on M p,q.

7 NONLINEAR DISPERSIVE EQUATIONS ON MODULATION SPACES 7 Proof. It is clear that m, m (1), m (2) are C (R ) functions an that all their erivatives are boune. Therefore, m, m (1), m (2) C +1 (R ) M,1 (R ) W (FL 1, l )(R ) [8, 11]. Thus, for 1 p, an < q the result follows from [3] an Lemma 2. For < p < 1 an < q, it can be showe that m, +1 m(1), m (2) C +1 (R ) W (FL p, l )(R ). Inee, this follows from obvious moifications to the proof of the embeing C +1 (R ) M,1 (R ) W (FL 1, l )(R ) [8, 11]. Furthermore, if we moify, for example, the multiplier m to m t (ξ) = e it<ξ>, t R, we have for < p 1 +1 (9) m t W (FL p,l ) (1 + t ) +1, an similar estimates hol for moifie multipliers m (1) t an m (2) t. Finally, we state a crucial multilinear estimate that will be use in our proofs. Although the estimate will be neee only in the particular case of a prouct of functions (see Corollary 1), we present it here in its full generality that applies to multilinear pseuoifferential operators. An m-linear pseuoifferential operator is efine à priori through its (istributional) symbol σ to be the mapping T σ from the m-fol prouct of Schwartz spaces S S into the space S of tempere istributions given by the formula T σ (u 1,..., u m )(x) (1) = σ(x, ξ 1,..., ξ m ) û 1 (ξ 1 ) û m (ξ m ) e 2πix (ξ 1+ +ξ m) ξ 1 ξ m, R m for u 1,..., u m S. The pointwise prouct u 1 u m correspons to the case σ = 1. Lemma 5. If σ M,1 (R(m+1) ), then the m-linear pseuoifferential operator T σ efine by (1) extens to a boune operator from M p 1,q 1 M pm,qm into M p,q when 1 p p m = 1 1 p, q q m = m q, an < p i, 1 q i for i m. This result is a slight moification of [4, Theorem 3.1]. Its proof procees along the same lines, an therefore it is omitte here. Note that if σ M,1, an we pick u 1 = = u m = u (some of them coul be equal to ū since the moulation norm is preserve), p 1 = = p m = mp, < p, an q 1 = = q m = 1 we have (11) T σ (u,..., u) M u m M m u m, M where we use the obvious embeing M M m. The notation A B stans for A cb for some positive constant c inepenent of A an B. In particular, if we select σ = 1 (the constant function 1), then σ M,1 M,1, an we obtain Corollary 1. Let < p. If u M, then um M. Furthermore, u m M u m. M

8 8 Á. Bényi an K. A. Okoujou This is of course just a particular case of the more general multilinear estimate m m (12) u i M u i p M i,q i, p,q i=1 i=1 where the exponents satisfy the same relations as in Lemma 1. When we consier the power nonlinearity f(u) = p k (u) = λ u 2k u = λu k+1 ū k, Corollary 1 becomes Corollary 2. Let < p. If u M, then p k(u) M. Furthermore, p k (u) M u 2k+1. M For a ifferent proof of the estimate in Corollary 2, see [1, Corollary 4.2]. It is important to note that the previous estimate allows us to control the exponential nonlinearity e ρ as well. Inee, since e ρ (u) = λ(e ρ u 2 1)u = k=1 ρ k k! p k(u), if we now apply the moulation norm on both sies an use the triangle inequality, we arrive at Corollary 3. Let < p. If u M, then e ρ (u) M. Furthermore, e ρ (u) M u M (e ρ u 2 M 1). 3. Proofs of the main results We are now reay to procee with the proofs of our main theorems. We will only prove our results for the power nonlinearities f = p k, by making use of Corollary 2. The case of exponential nonlinearity f = e ρ is treate similarly, by now employing Corollary 3. In all that follows we assume that u : [, T ) R C where < T an that f(u) = p k (u) = λ u 2k u The nonlinear Schröinger equation: Proof of Theorem 1. We start by noting that (1) can be written in the equivalent form (13) u(, t) = S(t)u iaf(u) where (14) S(t) = e it, A = Consier now the mapping J u = S(t)u i S(t τ) τ. S(t τ)(p k (u))(τ) τ. It follows from Lemma 2 (see also [3, Corollary 18]) that S(t)u M C (t 2 + 4π 2 ) /4 u M,

9 NONLINEAR DISPERSIVE EQUATIONS ON MODULATION SPACES 9 where C is a universal constant epening only on. Therefore, (15) S(t)u M C T u M, where C T = sup C (t 2 + 4π 2 ) /4. Moreover, we have t [,T ) (16) S(t τ)(p k (u))(τ) τ M T C T S(t τ)(p k (u))(τ) M τ sup t [,T ] p k (u)(t) M. By using now Corollary 2, we can further estimate in (16) to get (17) S(t τ)(p k (u))(τ) τ C T T u(t) 2k+1. Consequently, using (15) an (17) we have M (18) J u C([,T ],M ) C T ( u M M + ct u 2k+1 ), M for some universal positive constant c. We are now in the position of using a stanar contraction argument to arrive to our result. For completeness, we sketch it here. Let B M enote the close ball of raius M centere at the origin in the space C([, T ], M ). We claim that J : B M B M, for a carefully chosen M. Inee, if we let M = 2C T u M we obtain J u C([,T ],M ) M 2 + cc T T M 2k+1. an u B M, from (18) Now let T be such that cc T T M 2k 1/2, that is, T T ( u M ). We obtain J u C([,T ],M ) M 2 + M 2 = M, that is J u B M. Furthermore, a similar argument gives J u J v C([,T ],M ) 1 2 u v C([,T ],M ). This last estimate follows in particular from the following fact: p k (u)(τ) p k (v)(τ) = λ(u v) u 2k (τ) + λv( u 2k v 2k )(τ). Therefore, using Banach s contraction mapping principle, we conclue that J has a fixe point in B M which is a solution of (13); this solution can be now extene up to a maximal time T ( u M ). The proof is complete.

10 1 Á. Bényi an K. A. Okoujou 3.2. The nonlinear wave equation: Proof of Theorem 2. Equation (2) can be written in the equivalent form (19) u(, t) = K(t)u + K(t)u 1 Bf(u) where (2) K(t) = sin(t ), K(t) = cos(t ), B = Consier the mapping J u = K(t)u + K(t)u 1 Bf(u). K(t τ) τ Recall that f = p k. If we now use Lemma 3 (see also [3, Corollary 21]) for the first two inequalities below an Corollary 2 for the last estimate, we can write (21) K(t)u M K(t)u 1 M Bf(u) M C T u M, C T u 1 M, ct C T u 2k+1, M where c is some universal positive constant. The constants T an C T have the same meaning as before. The stanar contraction mapping argument applie to J completes the proof The nonlinear Klein-Goron equation: Proof of Theorem 3. The equivalent form of equation (3) is (22) u(, t) = K(t)u + K(t)u 1 + Cf(u) where now (23) K(t) = Consier the mapping sin t(i )1/2, K(t) (I ) = cos t(i ) 1/2, C = K(t τ) τ. 1/2 J u = K(t)u + K(t)u 1 + Cf(u). Using Lemma 4 an the notations above, we can write (24) K(t)u M K(t)u 1 M Cf(u) M C T u M, C T u 1 M, ct C T u 2k+1, M The stanar contraction mapping argument applie to J completes the proof.

11 NONLINEAR DISPERSIVE EQUATIONS ON MODULATION SPACES 11 References [1] W. Baoxiang, Z. Lifeng, an G. Boling, Isometric ecomposition operators, function spaces Ep,q λ an applications to nonlinear evolution equations, J. Funct. Anal. 233 (26), no. 1, [2] W. Baoxiang an H. Huzik, The global Cauchy problem for the NLS an NLKG with small rough ata, J. Diff. Equations 232 (27), [3] Á. Bényi, K. Gröchenig, K. A. Okoujou, an L. G. Rogers, Unimoular Fourier multipliers for moulation spaces, J. Funct. Anal. (27), to appear. [4] Á. Bényi, K. Gröchenig, C. Heil, an K. Okoujou, Moulation spaces an a class of boune multilinear pseuoifferential operators, J. Operator Theory 54 (25), no. 2, [5] Y. V. Galperin, an S. Samarah, Time-frequency analysis on moulation spaces M p,q m, < p, q, Appl. Comput. Harmon. Anal., 16 (24), [6] H. G. Feichtinger, Moulation spaces on locally Abelian groups, in: Proc. Internat. Conf. on Wavelets an Applications (Raha, R.;Krishna, M.;Thangavelu, S. es.), New Delhi Allie Publishers (23), [7] H. G. Feichtinger an G. Narimani, Fourier multipliers of classical moulation spaces, Appl. Comput. Harmon. Anal. 21 (26), no. 3, [8] K. Gröchenig, Founations of Time-Frequency Analysis, Birkhäuser, Boston MA, 21. [9] M. Kobayashi, Moulation spaces M p,q for < p, q, J. Func. Spaces Appl. 4 (26), no. 2, [1] M. Kobayashi, Dual of moulation spaces, J. Func. Spaces Appl., to appear. [11] K. A. Okoujou, Embeings of some classical Banach spaces into the moulation spaces, Proc. Amer. Math. Soc., 132 (24), no. 6, [12] T. Tao, Nonlinear ispersive equations: Local an global analysis, CBMS Regional Conference Series in Mathematics, no. 16, American Mathematical Society, 26 [13] J. Toft, Convolutions an embeings for weighte moulation spaces, Avances in pseuoifferential operators, , Oper. Theory Av. Appl. 155, Birkhauser, Basel, 24. [14] J. Toft, Continuity properties for moulation spaces, with applications to pseuo-ifferential calculus, II, Ann. Global Anal. Geom. 26 (24), no. 1, [15] H. Triebel, Moulation spaces on the eucliean n space, Z. Anal. Anwenungen, 2 (1983), no. 5, Árpá Bényi, Department of Mathematics, 516 High Street, Western Washington University, Bellingham, WA 98225, USA aress: arpa.benyi@wwu.eu Kasso A. Okoujou, Department of Mathematics, University of Marylan, College Park, MD 2742, USA aress: kasso@math.um.eu