ANALYTIC SMOOTHING EFFECT FOR NONLI TitleSCHRÖDINGER EQUATION IN TWO SPACE DIMENSIONS. Citation Osaka Journal of Mathematics.
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1 ANALYTIC SMOOTHING EFFECT FOR NONLI TitleSCHRÖDINGER EQUATION IN TWO SPACE DIMENSIONS Author(s) Hoshino, Gaku; Ozawa, Tohru Citation Osaka Journal of Mathematics. 51(3) Issue Date Text Version publisher URL DOI Rights Osaka University
2 Hoshino, G. and Ozawa, T. Osaka J. Math. 51 (014), ANALYTIC SMOOTHING EFFECT FOR NONLINEAR SCHRÖDINGER EQUATION IN TWO SPACE DIMENSIONS GAKU HOSHINO and TOHRU OZAWA (Received February 10, 01, revised November 16, 01) Abstract We prove the global existence of analytic solutions to the Cauchy problem for nonlinear Schrödinger equations in two dimensions, where the nonlinearity behaves as a cubic power at the origin and the Cauchy data are small and decay exponentially at infinity. 1. Introduction We consider the Cauchy problem for nonlinear Schrödinger equations of the form (1.1) i t u 1 ½u f (u), where u is a complex-valued function of (t, x) ¾ Ê Ê, ½ is the Laplacian in Ê and f is a complex-valued function on satisfying a gauge condition f (e i z) e i f (z) for any (, z) ¾ Ê and behaving as a cubic power at the origin. Typical examples of the nonlinearities f are: (1.) (1.3) f (u) u u, f (u) (exp(u ) 1)u with ¾ Ò{0} (see [, 19] for instance). There is a large literature on the Cauchy problem for the nonlinear Schrödinger equations (see for instance [3, 6, 30] and reference theorem). In this paper we study the analyticity of solutions to the Cauchy problem for (1.1). We refer the reader to [1,, 9, 10, 11, 1, 13, 14, 16, 17,, 3, 4, 5, 6, 9, 31] for the available results on the analyticity and related subjects. The purpose in this paper is to prove the global existence of analytic solutions to the Cauchy problem for (1.1) in two space dimensions for sufficiently small Cauchy data with exponential decay at infinity. In particular, we describe analytic smoothing effects for (1.1) with (1.) in the L (Ê ) setting and also 010 Mathematics Subject Classification. 35Q55.
3 610 G. HOSHINO AND T. OZAWA with (1.3) in the H 1 (Ê ) setting. The associated scaling critical space for (1.1) with (1.) and (1.3) are known respectively as L (Ê ) [4, 7, 15, 18, 33] and H 1 (Ê ) [0]. To state our result precisely we introduce the following notation. For any p with 1 p ½, L p denotes the Lebesgue space of p-th integrable function on Ê. For any s ¾ Ê, Hp s (1 ½) s L p and H È s p ( ½) s L p denote the usual Sobolev space (or the space of Bessel potentials) and the homogeneous Sobolev space (or the space of Riesz potentials), respectively. For any t ¾ Ê, U(t) exp(i(t)½) denotes the free propagator. For any t ¾ Ê, J J(t) x itö U(t)xU( t) denotes the generator of Galilei transforms. For any t ¾ ÊÒ{0}, U(t) and J(t) are represented as U(t) M(t)D(t)F M(t) and J(t) M(t)itÖ M( t), respectively, Ê where M(t) exp(ix (t)), (D(t))(x) (it) 1 (t 1 x), (F)() () 1 exp( ix )(x) dx. With the Cauchy data u(0) at t 0 the Cauchy problem for (1.1) is written as the integral equation (1.4) u(t) U(t) i t 0 U(t t ¼ ) f (u(t ¼ )) dt ¼. We introduce the following basic function spaces: with the associated norms defined by X 0 L 4 (ÊÁ L 4 ) L ½ (ÊÁ L ), X 1 L 4 (ÊÁ H 1 4 ) L½ (ÊÁ H 1 ), uá X 0 max(uá L 4 (ÊÁ L 4 ), uá L ½ (ÊÁ L )), uá X 1 max(uá L 4 (ÊÁ H 1 4 ), uá L½ (ÊÁ H 1 )). We treat (1.4) in the following function spaces with a 0: µ G a 0 (J) u ¾ X 0 Á uá G a 0 (J) a ««! J «uá X 0 ½, «0 µ G a 1 (J) u ¾ X 1 Á uá G a 1 (J) a ««! J «uá X 1 ½, «0 where J «J «1 1 J «M(it) «M 1 for any multi-index «(«1, «.) For 0, we define µ B0 a () ¾ G a 0 (x)á Á Ga 0 (x) a ««! x «Á L, «0 B a 1 () ¾ G a 1 (x)á Á Ga 0 (x) «0 a ««! x «Á H 1 µ.
4 We now state our main results. ANALYTIC SMOOTHING EFFECT FOR NLS 611 Theorem 1. There exists a constant 0 such that for any a 0 and ¾ B1 a() equation (1.4) with (1.3) has a unique solution u ¾ G a 1 (J). Theorem. There exists a constant 0 such that for any a 0 and ¾ B0 a() equation (1.4) with (1.) has a unique solution u ¾ G a 0 (J). REMARK 1. Theorems 1 and describe analytic smoothing properties of solutions since functions in the space G a 0 (J) are analytic in x for any t ¾ ÊÒ{0} (see [1, 13]). A novelty consists in the fact that minimal regularity assumption regarding scaling invariance [3, 4, 7] is imposed on the Cauchy data (compare with [13, 17, 9] for instance.) REMARK. By Proposition in [4], the norm on G a 0 (x) is described in terms of weights of exponential type. To be specific, ¼ j1 e ax j ½ Á L Á Ga 0 (x) (1 log ) ¼ j1 (1 ax j ) 1 e ax j ½ Á L. We prove Theorem 1 in Section 3 by a contraction argument of the Strichartz estimates and the Sobolev embedding in the critical case [0, 1, 7, 8]. Basic estimates for the proofs of the theorems are summarized in Section. In Section 4, we give a sketch of proof of Theorem.. Preliminaries In this section we collect some basic estimate for the Schrödinger group U(t) exp(i(t)½) and Sobolev embedding. Lemma 1 ([3, 6, 8, 30, 34]). U(t) satisfies the following estimates: (1) For any (q, r) with 0 q 1 r 1 U( )Á L q (ÊÁ L r ) CÁ L. () For any (q j, r j ) with 0 q j 1 r j 1, j 1,, the operator defined by ( f )(t) t 0 U(t t ¼ ) f (t ¼ ) dt ¼
5 61 G. HOSHINO AND T. OZAWA satisfies the estimate f Á L q 1 (ÊÁ L r 1 ) C f Á L q¼ (ÊÁ L r ¼ ), where p ¼ is dual exponent to p defined by 1p 1p ¼ 1. The following lemma is crucial in the proof of convergence of the series of exponential type. Lemma. For any r with r ½ there exists a constant C r 0 such that for any q with r q ½ the following estimate holds. uá L q C r q 1r ¼ uá È H 1 r r q uá L r 1 rq. Proof. We recall the following estimate [0, Inequality (.6)]: For any r with 1 r ½ there exists a constant Cr ¼ 0 such that for any q with r q ½ uá L q C ¼ r q1r ¼ uá È H r q r. By an interpolation inequality in the homogeneous Sobolev space [8], for any r with r ½ there exists a constant Cr ¼¼ 0 such that for any q with r q ½, uá H È r q r C ¼¼ r uá H È 1 r r q uá L r 1 rq. The lemma follows from those two inequalities. 3. Proof of Theorem 1 For 0 we define the metric space with metric X 1 () {u ¾ G a 1 (J)Á uá Ga 1 (J) } (3.1) d(u, Ú) u ÚÁ G a 0 (J). We see that (X 1 (), d) is a complete metric space. For ¾ B1 a() and u ¾ Ga 1 (J) we define (u) by ( (u))(t) U(t) i( f (u))(t). We prove that Ï u (u) is a contraction mapping on (X 1 (), d) for, 0
6 ANALYTIC SMOOTHING EFFECT FOR NLS 613 sufficiently small. By Lemma, for u ¾ X 1 we estimate f (u) in L 43 as f (u)á L 43 ½ j1 ½ j1 ½ j1 j u j uá L 43 j uá L4( j1)3 j1 j C 3 4 C ½ j1 C 4( j 1) 3 C j F 1 uá H 1 uá H 1 4 3, 1 ( j 1) uá H 1 C 4 4( j 1) ( j 1) j54 uá H 1 ( j 1) uá H uá H4 1 where F 1 () (C4 3 )È C ½ j1 ((C ) j )((43)( j 1)) j54 ( j 1) converges for any Õ with 0 3(8eC ) by d Alembert s ratio test. Similarly, for u ¾ X 1 we estimate Ö f (u) in L 43 as Ö( f (u))á L 43 ½ j1 ½ j1 ½ j1 j ( j 1)u j ÖuÁ L 43 j ( j 1)uÁ L4 j j ÖuÁ L 4 j C 4 C ½ j1 ( j 1)(C (4 j) 1 uá H 1 ) ( j 1) (C 4 (4 j) 34 uá H 1 4 ) ÖuÁ L 4 F (uá H 1 )uá H 1 4 3, (C j ) ( j 1)(4 j) j1 uá H 1 ( j 1) uá H4 1 3 È where F () (C4 C ) ½ j1 ((C ) j )( j 1)(4 j) j1 ( j 1) converges for any Õ with 0 1(4eC ) by d Alembert s ratio test. Therefore, by Lemma 1 and the Hölder inequalities in space and time, we have (u)á X 1 CÁ H 1 C(F 1 (uá L ½ (ÊÁ H 1 )) F (uá L ½ (ÊÁ H 1 )))uá X 1 3
7 614 G. HOSHINO AND T. OZAWA for any u ¾ X 1 with uá L ½ (ÊÁ H Õ1(4eC 1), where 0 ). For any multi-index «, we write J «( f (u)) as to estimate J «( f (u)) M(it) «f (M 1 u) ½ j1 ½ j1 j M(it)«(M 1 u j M 1 u) j Æ«(1) ( j) (1) ( j) J «( f (u))á L 43 ½ j1 j k1 j Æ«(1) ( j) (1) ( j) «!( 1) É j k1 (k)! (k)! Æ! «! É j k1 (k)! (k)! Æ! j k1 J (k) u J (k) u J Æ u J (k) uá L 4( j1)3 J (k) uá L 4( j1)3 J Æ uá L 4( j1)3. We use Lemma to estimate three factors in the last norms as J uá L 4( j1)3 C 4 4( j 1) 3 34 J uá H 1 4 with (1), (1), Æ. We use Lemma to estimate other factors as 4( j 1) 1 J uá L 4( j1)3 C J uá H 1 3 with (k), (k) for k. Therefore, we obtain J «( f (u))á L 43 ½ j1 j j k Æ«(1) ( j) (1) ( j) J (k) uá H 1 J (k) uá H 1 «! É C j k1 (k)!! 3 j 1) 4 j54 (k) 4C( Æ! 3 ( j 1) J (1) uá H 1 4 J (1) uá H 1 4 J Æ uá H 1 4.
8 ANALYTIC SMOOTHING EFFECT FOR NLS 615 Taking L 43 norm in time of the last inequality and using the Hölder inequality, we have J «( f (u))á L 43 (ÊÁ L 43 ) ½ j1 j j k Æ«(1) ( j) (1) ( j) «! É C j k1 (k)!! 3 j 1) (k) 4C( Æ! 4 j54 3 ( j 1) J (k) uá L ½ (ÊÁ H 1 )J (k) uá L ½ (ÊÁ H 1 ) J (1) uáx 1 J (1) uáx 1 J Æ uáx 1. Multiplying the last inequality by a ««! and taking the summation over all multiindices of the resulting inequality, we obtain (u)á G a 0 (J) «0 C «0 C a ««! J «( (u))á X 0 a ««! x «Á L ½ j1 j C3 4 C( j 1) 4 j54 3 ( j 1) uá G a j1 1 (J) CÁ G a 0 (x) C F 1(uÁ G a 1 (J))uÁ Ga 1 (J)3. We estimate Ö J «( f (u)) in L 43 by using the Hölder inequality with j(4 j) to obtain terms of the form j k1 J (k) uá L 4 j J (k) uá L 4 j J Æ uá È H 1 4 with Æ «, (1) ( j), (1) ( j). We use Lemma to estimate two factors J uá L 4 j C 4 (4 j) 34 J uá H 1 4 with (1), (1). We use Lemma to estimate two factors J uá L 4 j C (4 j) 1 J uá H 1
9 616 G. HOSHINO AND T. OZAWA with (k), (k) for k. In the same way as before, we obtain Therefore, «0 C «0 (u)á G a 1 (J) Similarly, we have a ««! Ö J «( (u))á X 0 a ««! x «Á È H 1 C F (uá G a 1 (J))uÁ Ga 1 (J)3. CÁ G a 1 (x) C(F 1(uÁ G a 1 (J)) F (u Ï G a 1 (J)))uÁ Ga 1 (J)3. (u) (Ú)Á G a 0 (J) C(F 1 (uá G a 1 (J))uÁ Ga 1 (J) F (ÚÁ G a 1 (J))ÚÁ Ga 1 (J) )u ÚÁ G a 0 (J), so that the contraction argument goes through for any ¾ G a 1 (x) with Á Ga 1 (x), provided that and satisfy C(F1 (), F () ) 1, This completes the proof of Theorem Proof of Theorem For 0 we define the metric space C C(F 1 () 3 F () 3 ). with metric X 0 () {u ¾ G a 0 (J)Á uá Ga 0 (J) } d(u, Ú) u ÚÁ G a 0 (J). We see that (X 0 (), d) is a complete metric space. We prove that Ï u (u) is a contraction mapping on (X 0 (), d) for, 0 sufficiently small. in a way similar to, and simpler than, that of the proof of Theorem 1, for any u, Ú ¾ X 0 () we have (u)á G a 0 (J) CÁ Ga 0 (J) C3, (u) (Ú)Á G a 0 (J) C u ÚÁ G a 0 (J), and the contraction argument goes through for any ¾ G a 0 (x) with Á Ga 0 (x), provided that and satisfy C 1, C C 3.
10 This completes the proof of Theorem. ANALYTIC SMOOTHING EFFECT FOR NLS 617 ACKNOWLEDGMENTS. comments. The authors would like to thank the referee for important References [1] J.L. Bona, Z. Grujić and H. Kalisch: Global solutions of the derivative Schrödinger equation in a class of functions analytic in a strip, J. Differential Equations 9 (006), [] T. Cazenave: Equations de Schrödinger non linéaires en dimension deux, Proc. Roy. Soc. Edinburgh Sect. A 84 (1979), [3] T. Cazenave: Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics 10, New York Univ., Courant Inst. Math. Sci., New York, 003. [4] T. Cazenave and F.B. Weissler: The Cauchy problem for the critical nonlinear Schrödinger equation in H s, Nonlinear Anal. 14 (1990), [5] A. DeBouard, Analytic solution to non elliptic non linear Schrödinger equations, J. Differential Equations. 104, (1993), [6] J. Ginibre, Introduction aux Équations de Schrödinger non Linéaires, Paris Onze Edition, L161. Université Paris-Sud, Paris, [7] J. Ginibre, T. Ozawa and G. Velo: On the existence of the wave operators for a class of nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor. 60 (1994), [8] J. Ginibre and G. Velo: The global Cauchy problem for the nonlinear Klein Gordon equation, Math. Z. 189 (1985), [9] N. Hayashi: Global existence of small analytic solutions to nonlinear Schrödinger equations, Duke Math. J. 60 (1990), [10] N. Hayashi and K. Kato: Analyticity in time and smoothing effect of solutions to nonlinear Schrödinger equations, Comm. Math. Phys. 184 (1997), [11] N. Hayashi and T. Ozawa: On the derivative nonlinear Schrödinger equation, Phys. D 55 (199), [1] N. Hayashi and S. Saitoh: Analyticity and smoothing effect for the Schrödinger equation, Ann. Inst. H. Poincaré Phys. Théor. 5 (1990), [13] N. Hayashi and S. Saitoh: Analyticity and global existence of small solutions to some nonlinear Schrödinger equations, Comm. Math. Phys. 19 (1990), [14] K. Kato and K. Taniguchi: Gevrey regularizing effect for nonlinear Schrödinger equations, Osaka J. Math. 33 (1996), [15] T. Kato: On nonlinear Schrödinger equations. II. H s -solutions and unconditional well-posedness, J. Anal. Math. 67 (1995), [16] T. Kato and K. Masuda: Nonlinear evolution equations and analyticity. I, Ann. Inst. H. Poincaré Anal. Non Linéaire 3 (1986), [17] K. Nakamitsu: Analytic finite energy solutions of the nonlinear Schrödinger equation, Comm. Math. Phys. 60 (005), [18] M. Nakamura and T. Ozawa: Low energy scattering for nonlinear Schrödinger equations in fractional order Sobolev spaces, Rev. Math. Phys. 9 (1997), [19] M. Nakamura and T. Ozawa: Nonlinear Schrödinger equations in the Sobolev space of critical order, J. Funct. Anal. 155 (1998), [0] T. Ozawa: On critical cases of Sobolev s inequalities, J. Funct. Anal. 17 (1995), [1] T. Ogawa and T. Ozawa: Trudinger type inequalities and uniqueness of weak solutions for the nonlinear Schrödinger mixed problem, J. Math. Anal. Appl. 155 (1991), [] T. Ozawa and K. Yamauchi: Remarks on analytic smoothing effect for the Schrödinger equation, Math. Z. 61 (009),
11 618 G. HOSHINO AND T. OZAWA [3] T. Ozawa and K. Yamauchi: Analytic smoothing effect for global solutions to nonlinear Schrödinger equations, J. Math. Anal. Appl. 364 (010), [4] T. Ozawa, K. Yamauchi and Y. Yamazaki: Analytic smoothing effect for solutions to Schrödinger equations with nonlinearity of integral type, Osaka J. Math. 4 (005), [5] L. Robbiano and C. Zuily: Microlocal analytic smoothing effect for the Schrödinger equation, Duke Math. J. 100 (1999), [6] L. Robbiano and C. Zuily: Effet régularisant microlocal analytique pour l équation de Schrödinger : le cas des données oscillantes, Comm. Partial Differential Equations 5 (000), [7] R.S. Strichartz: A note on Trudinger s extension of Sobolev s inequalities, Indiana Univ. Math. J. 1 (197), [8] R.S. Strichartz: Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (1977), [9] J.C.H. Simon and E. Taflin: Wave operators and analytic solutions for systems of nonlinear Klein Gordon equations and of nonlinear Schrödinger equations, Comm. Math. Phys. 99 (1985), [30] C. Sulem and P.-L. Sulem: The Nonlinear Schrödinger Equation, Applied Mathematical Sciences 139, Springer, New York, [31] H. Takuwa: Analytic smoothing effects for a class of dispersive equations, Tsukuba J. Math. 8 (004), [3] N.S. Trudinger: On imbeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967), [33] Y. Tsutsumi: L -solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac. 30 (1987), [34] K. Yajima: Existence of solutions for Schrödinger evolution equations, Comm. Math. Phys. 110 (1987), Gaku Hoshino Department of Applied Physics Waseda University Tokyo Japan Tohru Ozawa Department of Applied Physics Waseda University Tokyo Japan
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