# Global well-posedness for semi-linear Wave and Schrödinger equations. Slim Ibrahim

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1 Global well-posedness for semi-linear Wave and Schrödinger equations Slim Ibrahim McMaster University, Hamilton ON University of Calgary, April 27th,

2 1 Introduction Nonlinear Wave equation: ( 2 t x )u = u p 1 u, u : ( T, T ) R d R u(x, 0) = u 0 (x), t u(x, 0) = u 1 (u 0, u 1 ) Ḣ s Ḣ s 1 (R d x) (1) Such equations arise in quantum mechanics. Nonlinear Schrödinger equation: i t v + v = v p 1 v, v : ( T, T ) R d C v(0, x) = v 0 (x) v 0 Ḣ s (R d x) (2) 2

3 The Schrödinger equation describes the propagation of an electromagnetic signal through a standard isotropic optical fibre. We will refer to the initial value problems (1) and (2) with the notation NLW p (R d ) and NLS p (R d ), respectively. Interested in the following questions: local (in time) well-posedness of the Cauchy problems (1) and (2). are the local solutions global? persistence of regularity i.e. does singularity develop? Long-time behavior or scattering i.e. does the (global) non-linear solution approach a linear solution when time t ±? 3

4 Facts about equations (1) and (2). These equations are Hamiltonian E(u(t, )) := t u(t, ) 2 L + u(t, ) L + 2 R p + 1 u p+1 (t, x)dx. d H(v(t, )) := v(t, ) 2 L 2 + R d 1 p + 1 v p+1 (t, x)dx. Equations (1) and (2) have a scaling property i.e. If u (respectively v) solves (1) (respectively (2)) then, for λ > 0, u λ : ( T λ 2, T λ 2 ) R d defined by u λ (t, x) := λ 2/(1 p) u(λ 2 t, λ 1 x) (3) also solves (1) (respectively (2)). Let s c := d 2 2 p 1. The Banach spaces Ḣs c (R d x) and L p c (R d x) are relevant in the theory of the initial value problems (1) and (2), since they are invariant under the mapping (3) 4

5 Definition 1 The Cauchy problems (1) and (2) are said subcritical if s c < s, critical if s c = s, and supercitical if s c > s. 2 Local well-posedness Definition 1 is inspired by the following complete trichotomy for the local well-posedness. Theorem 1 The Cauchy problems (1) and (2) are: locally well-posed if s c < s with T lwp = T ( u 0 H s) (Cazenave-Weissler 90) locally well-posed if s c = s with T lwp = T (u 0 ), (Ginibre-Velo 85, Cazenave 03) and is ill-posed if not. (Christ-Colliander-Tao 04, Lebeau 01) 5

6 3 Global well-posedness To simplify the results, let us restrict our selves to the case when the initial data are in Ḣ1 i.e. energy critical case. 3.1 The case of NLW P Theorem 2 The Cauchy problem (1) with initial data (u 0, u 1 ) H 1 L 2 is: globally well-posed if s c < s = 1 (or equivalently p < p c := d+2 d 2 ). (Ginibre-Velo 85). globally well-posed if s c = s = 1 (or equivalently p = p c ). (Grillakis 90, Shatah-Struwe 94) Moreover, if p > p c and the initial data is in H s H s 1 with s < s c, then the Cauchy problem (1) is ill-posed. (Christ-Colliander-Tao 04, Lebeau 01-05) 6

7 Remark 1 Shatah- Struwe result s was extended to the variable coefficients case by (Ibrahim-Majdoub 03) with a conservative Laplacian : A u := div(a( ) u), where A 1 is a Riemannian metric on R d which is flat outside a fixed compact set. 7

8 3.2 The case of NLS P Theorem 3 The Cauchy problem (2) with initial data v 0 H 1 is: globally well-posed if s c < s = 1 (or equivalently p < p c := d+2 d 2 ). (Ginibre-Velo 85, Cazenave 04) globally well-posed if s c = s = 1 (or equivalently p = p c ). (Bourgain 99, Colliander-Keel-Staffilani-Takaoka-Tao 05) Moreover, if p > p c and the initial data is in H s with s c > s, then the Cauchy problem N LSp is ill-posed. (Christ-Colliander-Tao 04, Burq-Gérard-Tzevtkov 02) Remark 2 A refinement and a generalization to the variable coefficients case is now proved by (Burq-Gérard-Ibrahim 06) for both NLW p and NLSp in any space dimension. 8

9 3.3 Energy criticality in two space dimensions In dimension two, p c = + and therefore, the initial value problems NLW p and NLS p are energy subcritical for all p > 1. To identify an energy critical nonlinear Wave/Schrödinger initial value problem on R 2, it is thus natural to consider problems with exponential nonlinearities. Consider, ( t 2 x )u + u = f(u), u : ( T, T ) R 2 R (4) u(x, 0) = u 0 (x), t u(x, 0) = u 1 (u 0, u 1 ) H 1 L 2 (R 2 ) where i t v + v = f(v), v(x, 0) = v 0 (x) H 1 (R 2 ) u : ( T, T ) R 2 C (5) f(z) = z ( e 4π z 2 1 ). (6) 9

10 Conserved quantities: Solutions to the nonlinear wave equation (4) formally satisfy the energy conservation E(u(t, )) := t u(t, ) 2 L + u(t, ) 2 2 L L1 (R 4π 2 ) = E(u(0, )). and we have conservation of mass and Hamiltonian for Schrödinger equation M(u(t, )) := u(t, ) 2 L 2 = M(u(0, )), H(u(t, )) := u(t, ) 2 L π u(t, ) 2 L 4π 1 (R 2 ) = H(u(0, )). 10

11 Definition 2 The Cauchy problem associated to (4) and with initial data (u 0, u 1 ) H 1 L 2 (R 2 ) is said to be subcritical if E(u 0, u 1 ) < 1, critical if E(u 0, u 1 ) = 1 and, supercritical if E(u 0, u 1 ) > 1. Definition 3 The Cauchy problem associated to (5) and with initial data v 0 H 1 (R 2 ) is said to be subcritical if H(v 0 ) < 1, critical if H(v 0 ) = 1 and, supercritical if H(v 0 ) > 1. 11

12 4 Results in two space dimensions Theorem 4 (Ibrahim-Majdoub-Masmoudi 05) Assume that E 0 1, then problem NLW exp has an unique global solution u in the class. C(R, H 1 (R 2 )) C 1 (R, L 2 (R 2 )). Moreover, u L 4 loc (R, C1/4 (R 2 )) and satisfies the energy identity. Remark 3 It is important here to note that contrary to problems NLW p and NLSp, we have an unconditional uniqueness results for this type of equations. 12

13 Theorem 5 (Colliander-Ibrahim-Majdoub-Masmoudi 06) Assume that H(u 0 ) 1; then problem NLSexp has an unique global solution v in the class C(R, H 1 (R 2 )). Moreover, u L 4 loc (R, C1/2 (R 2 )) and satisfies the conservation of mass and hamiltonian. Theorem 6 (Colliander-Ibrahim-Majdoub-Masmoudi 06) Assume that E 0 > 1 and H(v 0 ) > 1; then problems (4) and (5) are ill-posed 13

14 5 Ideas of proofs ( the case of NLSexp) The local-well-posedness idea is: N LSexp LSexp How does the proof of the local well-posedness go? Let v 0 be the solution of the free Schrödinger equation i t v 0 + v 0 = 0 v 0 (0, x) = u 0. Fix T > 0 and define a map i t ṽ + ṽ = (v + v 0 ) ( e 4π v+v ), ṽ(0, x) = 0, (7) on a closed neighborhood X(T ) around 0 included in the energy space C([0, T ], H 1 ). 14

15 f(u), f(u) in L 1 T (L2 ) f(u), f(u) in L 4 tx Inhomogeneous Strichartz estimates Duhamel s formula u, u in C T (L 2 ) e it u 0 u 0, u 0 in L 2 u, u in L 4 tx Homegeneous Strichartz estimates Figure 1: Local well-posedness scheme. 15

16 The local well-posedness is obtained by combining the following three fundamental ingredients: Lemma 1 (Moser-Trudinger Inequality) Let α [0, 4π). A constant c α exists such that exp(α u 2 ) 1 L1 (R 2 ) c α u 2 L 2 (R 2 ) (8) for all u in H 1 (R 2 ) such that u L 2 (R 2 ) 1. Moreover, if α 4π, then (8) is false. Lemma 2 (Strichartz estimates) Let v 0 be a function in H 1 (R 2 ) and F L 1 (R, H 1 (R 2 )). Denote by v the solution of the inhomogeneous linear Schrödinger equation i t v + v = F with initial data v(0, x) = v 0 (x). Then, a constant C exists such that for any T > 0 and any admissible couple of Strichartz exponents (q, r) i.e 16

17 0 2 q = 1 2 r < 1, we have v Lq ([0,T ],B 1 r,2 (R2 )) C [ v 0 H 1 (R 2 ) + F L 1 ([0,T ],H 1 (R 2 ))]. Lemma 3 (Log Estimate) (Ibrahim-Majdoub-Masmoudi 05). Let β ]0, 1[. For any λ > 1 2πβ and any 0 < µ 1, a constant C λ > 0 exists such that, for any function u H 1 (R 2 ) C β (R 2 ), we have u 2 L λ u 2 µ log(c λ + 8β µ β u C β u µ ), where we set u 2 µ := u 2 L 2 + µ 2 u 2 L 2. In the subcritical case, using only the conserved quantities, we can iterate the local-well-posedness result infinitly many times, thus the solution is global. 17

18 In the critical case, it is no longer sufficient to use only the conserved quantities. We prove a result about the distribution of the local mass at different times. Lemma 4 Let u be a solution of (5) on [0, T ) with 0 < T + and suppose that E := H(u 0 ) + M(u 0 ) <. For any two positive real numbers R and R and for any 0 < t < T, a constant C(E) exists such that the following holds: u(t, x) 2 dx u 0 (x) 2 dx C(E) t R. (9) B(R+R ) B(R) 18

19 The instability in the super-critical case is based on the fundamental idea: N LSexp ODEexp Theorem 7 There exist a sequence of positive real numbers (t k ), t k 0 and tow sequences (U k ) and (V k ) solutions of NLW exp and satisfying the following: for any k N (U k V k )(t = 0, ) 2 H 1 + t (U k V k )(t = 0, ) 2 L 2 = (1), k +. For any ν > 0, and 0 < E(U k, 0) 1 e 3 ν 2 and 0 < E(V k, 0) 1 ν 2, lim inf k t(u k V k )(t k, ) 2 L 2 π 4 (e2 + e 3 8π )ν 2. 19

20 How to prove Theorem 7? 1st step: An ODE analysis Let Φ k and Ψ k be the two solutions of ODEexp: with initial data d 2 dt 2 y + ye4π y2 = 0. and Φ k (0) = (1 + 1 k ) k 4π, d dt Φ k(0) = 0, Ψ k (0) = k 4π, d dt Ψ k(0) = 0. 20

21 Note that Φ k is periodic with period T k k e (1+ 1 k )2k/2. We choose time t k ]0, T k /4[ such that ( ) 1 k k Φ k (t k ) = (1 + 1/k) 4π (1 + 1/k). 4π Then for any ν > 0 and for k large enough, we have and t k c ν 2 e k/2, t Φ k (t k ) t ψ k (t k ) 2 ce k, R 2 t (Φ k (t k ) Ψ k (t k )) 2 cν 2 21

22 2nd step: PDE-ODE approximation We construct the following initial data ((1 + 1 k )f k( x ν ), 0 ) and (f k ( x ν ), 0 ), where f k is the sequence that violate the sharp moser-trudinger ineqaulity when the exponent is 4π. The paramete ν is arbitrary. Using the special form of the sequence f k, an enormous gift is provided by the finite speed of propagation: in the backward light cone N LW exp = ODEexp {(x, t) : x t νe k/2 }. 22

23 Remark 4 Note that the data are slightly supercritical For NLSexp, the analogous to Theorem 7 is harder to prove. Theorem 7 result says no better than the flow map is not uniformly continuous. 23

24 Conclusions The novel approach based on the discussion with respect to the size of the initial data in the energy space allows us to obtain a trichotomy almost similar to the power nonlinearity case. We argue that NLW exp and NLSexp are the H 1 -critical problems in R 2. The long-time behavior of solutions remains unknown. Also in the focusing case, there is no a qualitative study of blow-up. The very interesting question of global existence for supercritical problems remains open. Solving such problem may give a good insight to solve the Navier-Stokes system (which is supercritical). 24

25 24-1 References [1] J. Bourgain : Global well-posedness of defocusing critical nonlinear Schrödinger equation in the radial case, J. Amer. Math. Soc. 12, No. 1, , [2] N. Burq, P. Gérard and N. Tvzetkov: An instability property of the nonlinear Schrödinger equation on S d, Math. Res. Lett. 9, no. 2-3, pp , [3] N. Burq, P. Gérard and S. Ibrahim: Ill posedness for supercritical NLW and NLS equations, In preparation. [4] T. Cazenave and F.B. Weissler: The Cauchy problem for the critical nonlinear Schrödinger equation in H s, Nonlinear Anal., 14 (1990), MR (91j:35252) [5] M. Christ, J. Colliander and T. Tao: Ill-posedness for nonlinear Schödinger and wave equations., To appear in Annales de L Institut Henri Poincaré, 2005.

26 24-2 [6] J. Colliander, S. Ibrahim, M. Majdoub and N. Masmoudi: From well to ill-posedness for a class of 2D NLS and NLKG equations, Preprint. [7] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao: Global Well-posedness and scattering for the energy-critical nonlinear Scrödinger equation in R 3, To appear in Annals of Mathematics. [8] J. Ginibre and G. Velo: The global Cauchy problem for nonlinear Klein-Gordon equation, Math.Z, 189, pp , [9] M. Grillakis: Regularity and asymptotic behaviour of the wave equation with a critical nonlinearity, Annal. of math., 132, pp , [10] S. Ibrahim and M. Majdoub: Existence, en grand temps, de solutions pour l équation des ondes semi-linéaire critique à coefficients variables, Bull. Soc. Math. France 131 No 1, 1-22, 2003.

27 24-3 [11] S. Ibrahim, M. Majdoub and N. Masmoudi: Double logarithmic inequality with sharp constant, To appear in Proceedings de la Soc. Math. Amer. [12] S. Ibrahim, M. Majdoub and N. Masmoudi: Global Well- Posedness for a 2D Semi-Linear Klein-Gordon Equation, To appear in Comm. in Pure and App. Math. [13] G. Lebeau: Nonlinear optics and supercritical wave equation, Bull. Soc. R. Sci. Liège, 70, 4-6, , [14] J. Shatah and M. Struwe: Regularity results for nonlinear wave equations, Ann. of Math., 2, n 138 pp , 1993.

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