BLOWUP THEORY FOR THE CRITICAL NONLINEAR SCHRÖDINGER EQUATIONS REVISITED

BLOWUP THEORY FOR THE CRITICAL NONLINEAR SCHRÖDINGER EQUATIONS REVISITED TAOUFIK HMIDI AND SAHBI KERAANI Abstract. In this note we prove a refined version of compactness lemma adapted to the blowup analysis and we use it to give direct and simple proofs to some classical results of blowup theory for critical nonlinear Schrödinger equations (mass concentration, classification of the singular solutions with minimal mass...). 1. Introduction We consider the L -critical nonlinear Schrödinger equation (NLS): { i t u + u + u 4 d u = 0, x R d, t > 0, (1.1) u(0, x) = u 0 (x). Here, = d i=1 x i is the Laplace operator on R d and u 0 : R d C. It is well-known from the result by Ginibre and Velo (see [3] for a review) that Cauchy problem (1.1) is locally well-posed in H 1 : there exists T (0, + ] and a solution u C([0, T ), H 1 ), with the following blowup alternative: either T = (the solution is global) or T < + (the solution blows up in finite time) and lim u(t, ) H 1 = +. t T The unique solution has the following conservation law: (1.) N (t) = u(t, x) dx, R d E(t) = 1 u dx d 4 + d u 4 d + dx Also, if u 0 Σ = {f H 1, xf L }, then the solution satisfies the Virial identity d x u(t, x) dx = 16E(0). dt If E(0) < 0 then t x u(t, x) dx is an inverted parabola which becomes negative in finite time. Thus, the solution cannot exist globally and blows up at finite time. This was the starting point of a blowup theory of Schroödinger 000 Mathematics Subject Classification. 35Q55, 35B40, 35B05. Key words and phrases. Time dependent Schrödinger equation, blowup, mass concentration. 1

T. HMIDI AND S. KERAANI equations which has been developed in the two last decays (see [3], [13], [11] and the references therein). This theory is mainly connected to the notion of ground state: the unique positive radial solution of the elliptic problem Q Q + Q 4 d Q = 0. In [15], M. I. Weinstein exhibited the following refined Gagliardo-Nirenberg inequality (1.3) ψ 4 d + L 4 d + C d ψ 4 d L ψ L ψ H 1, with C d = d+ 4 d Q d. Combined with the conservation of energy, this L implies that Q L is the critical mass for the formation of singularities: for every u 0 H 1 such that u 0 L < Q L the solution of (1.1) with initial data u 0 is global. Also, this bound is optimal. In fact, by using the conformal invariance, one constructs u(t, x) = (T t) d/ e [(i/(t t))+( i x /T t)] x Q( T t ) a sigular solution of with critical mass. In this sens Q L is the minimal mass necessary to ignite a wave collapse. There is an abundant literature devoted to the study of the blow up mechanism (see [3] and [13] for a review). In this note we prove a compactness lemma adapted to the analysis of the blowup phenomenon of the nonlinear Schroödinger equation and use it to give elementary proofs of some classical blowup results. Our main tool is the following. Theorem 1.1 (Comactness Lemma). Let {v n } n=1 H 1 (R d ), such that be a bounded family of lim sup v n L M and lim sup v n 4 L + m. d Then, there exits {x n } n=1 Rd such that, up to a subsequence, with V L ( d d+ )d/4 m d +1 M d/ Q L. v n ( + x n ) V Remark 1.. The lower-bound on the L norm of V is optimal. In fact, if we take v n = Q then we get equality. The plan of the rest of the note is as follows. In section we restate and prove some well-known blow up results. Section 3 is devoted to the proof of Theorem 1.1.

COMPACTNESS LEMMA AND APPLICATIONS 3. Blow up theory revisited.1. Concentration. For d and spherically symmetric blow up solutions, it has been shown (see [1],[14],[17]) that there is a minimal amount pf concentration of the L norm at the origin. Below we give an easy proof for the general case. Theorem.1. Let u be a solution of (1.1) which blows up at finite time T > 0, and λ(t) > 0 any function, such that 0 as t T. Then, there exits x(t) R d, such that lim inf t T x x(t) λ(t) T t λ(t) u(t, x) dx Q. Proof. Let {t n } n=1 be a sequence such that t n T. We set ρ n = Q L u(t n, ) L The sequence {v n } n=1 satisfies: and v n = ρ d/ n u(t n, ρ n x). v n L = u 0 L, v n L = Q L. Furthermore, by conservation of the energy and blowup criteria, it ensues that E(v n ) = ρ ne(0) 0, as n which yields, in particular, v n 4 d + d + L d 4 + d Q L, as n. The family {v n } n=1 satisfies the conditions of the Theorem 1.1 above with m 4 d + = d + d Q L and M = Q L. Thus, there exists exits {x n } n=1 Rd such that, up to a subsequence, v n ( + x n ) = ρ d/ n u(t n, ρ n +x n ) V H 1 with V L Q L.[Note taht this asymptotic is proved in [16] via concentration-compactness Lemma by P.L. Lions [8].] From this, it follows that lim inf ρ d n u(t n, ρ n x + x n ) dx V dx, x A for every A > 0. Thus, (.1) lim inf Since T t λ(t) sup y R d u(t n, x) dx x y Aρ n ρ n λ(t n) x A x A V dx. 0 as t T, it ensues that 0 and then lim inf u(t n, x) dx V dx sup y R d x y λ(t n) x A

4 T. HMIDI AND S. KERAANI for every A, which means that lim inf sup y R d { x y λ(t n)} u(t n, x) dx V dx R d Since this is true for every sequence t n T, then lim inf u(t, x) dx t T as claimed. sup y R d { x y λ(t)} Q. Q... Universality of the profile with critical mass. In the context of the proof of Theorem.1 above, if we assume u 0 L = Q L then 1 Thus, V L = Q L. v n ( x n ) V in L. Also, since it s bounded in H 1, we have v n ( x n ) V in L 4. In view of Galiardo-Nirenberg inequality, this leads to V L Q L. Since V L lim sup v n L = Q L, then v n L V L. This means that the strong convergence holds in H 1. This fact implies, in particular, E(V ) = 0. Let us summarize the properties of the limit V : V H 1, V L = Q L, V L = Q L and E(V ) = 0. The variational characterization of the ground state implies that V (x) = e iθ Q(x + x 0 ), for some θ [0, π[ and x 0 R d. This prove the following result : if u is a singular solution with critical mass then there exist x(t) and θ(t) such that (.) (λ(t)) d/ e iθ(t) u(t, λ(t)x + x(t)) Q, t T. 1 Since V L u 0 L. This result is due to Weinstein [16] and Kwong [7].

COMPACTNESS LEMMA AND APPLICATIONS 5.3. Characterization of the singular solutions with critical mass. In the sequel we use the notation: A = {λ d/ e iθ R(λx + y), y R d, λ R +, θ [0, π[}. Theorem. ([9]). Let u be a blowing up solution of (1.1) at finite time T > 0 such that u 0 L = Q L. Then, there exits x 0 R d such that e i x x 0 4T u 0 A. Proof. Let t n T any sequence. It is clear that (.) implies u(t n, x) dx R L δ x=xn 0. Up to extract a subsequence and translation, one assume x n x 0 {0, }. Let φ a nonnegative radial C 0 (Rd ) function such that φ(x) = x, x < 1 and φ(x) Cφ(x). For every p N one defines φ p (x) = p φ( x p ) and g p(t) = φ p (x) u(t, x) dx. Using the Cauchy-Schwartz estimates by V. Banica [], we get (remember φ p Cφ p ) ġ p (t) = I ū(x) u(x) φ p (x)dx (8E(u 0 ) u φ p dx) 1/ C(u 0 ) g p (t), for every t [0, T [. By integration we obtain, for every t [0, T [, g p (t) g p (t n ) C t n t. We let t n goes to T ; and we get (since φ p (x 0 ) = 0 for both finite or infinite case) g p (t) C(u 0 )(T t). We fix t [0; T [ and get p goes to infinity 3 to obtain (.3) 8t E(e i x 4t u0 ) = x u(t, x) dx C(u 0 )(T t). The first identity, which is just another way the write Viriel identity, is easy. We let then t goes to T and get 4 E(e i x 4T u0 ) = 0. Since e i x 4T u 0 L = u 0 L = R L, then the variational characterization of R implies that e i x 4T u 0 A. This ends the proof of this theorem. 3 We can take first the limit at t = 0 to get u0 Σ. This implies that R x u(t, x) dx is well defined for every t [0, T [. We take then the limit in all t [0, T [. 4 We may use the uniform bound (.4) to prove that lim xn and then equal, up to a translation, to 0.

6 T. HMIDI AND S. KERAANI Appendix. Proof of Theorem 1.1 In the sequel we put = if d = 1,, and = d d if d 3. Theorem 1.1 is a consequence of a profile decomposition of the bounded sequences in H 1 following the work by P. Gérard [4] (see also [1] and [6]). More precisely, we have the following Proposition.3. Let {v n } n=1 be a bounded sequence in H1 (R d ). Then there exist a subsequence of {v n } n=1 (still denoted {v n} n=1 ), a family {x j } of sequences in Rd and a sequence {V j } of H1 functions, such that i) for every k j, x k n x j n + ; ii) for every l 1 and every x R d, we have with v n (x) = V j (x x j n) + vn(x), l (.4) lim sup vn l L p (R d ) 0 l for every < p <. Moreover, we have (.5) (.6) as n. v n L = v n L = V j L + vn l L + o(1), V j L + vn l L + o(1), Proof. Let v = {v n } n=1 be a bounded sequence in H1 (R d ). Let V(v) be the set of functions obtained as weak limits in H 1 of subsequences of the translated v n (. + x n ) with {x n } n=1 Rd. We denote η(v) = sup{ V H 1, V V(v)}. Clearly η(v) lim sup v n H 1. We shall prove the existence of a sequence {V j } {x j } of sequences of Rd, such that of V(v) and a family k j = x k n x j n,

COMPACTNESS LEMMA AND APPLICATIONS 7 and, up to extracting a subsequence, the sequence {v n } n=1 can be written as v n (x) = V j (x x j n) + vn(x), l η(v l ) 0, l such that the identities (.5)-(.6) hold. Indeed, if η(v) = 0, we can take V j 0 for all j, otherwise we choose V 1 V(v), such that V 1 H 1 1 η(v) > 0. By definition, there exists some sequence x 1 = {x 1 n} n=1 Rd, such that, up to extracting a subsequence, we have v n ( + x 1 n) V 1 in H 1. We set vn 1 = v n V 1 ( x 1 n). Since vn( 1 + x 1 n) 0 weakly in H 1, we get v n L = V 1 L + v 1 n L + o(1), v n L = V 1 L + v 1 n L + o(1), as n. Now, we replace v by v 1 and repeat the same process. If η(v 1 ) > 0 we get V, {x n} n=1 and v. Moreover, we have x 1 n x n, as n Otherwise, up to extracting of subsequence, we get for some x 0 R d. Since x 1 n x n x 0 v 1 n( + x n) = v 1 n( + (x n x 1 n) + x 1 n) and vn( 1 + x 1 n) converge weakly to 0, then V = 0. Thus η(v 1 ) = 0, a contradiction. An argument of iteration and orthogonal extraction allows us to construct the family {x j } and {V j } satisfying the claims above. Furthermore, the convergence of the series V j H implies that 1 V j H 1 However, by construction, we have 0. j η(v j ) V j 1 H 1, which proves that η(v j ) 0 as claimed. To complete the proof of Proposition.3, (.4) remains to be proved. For that purpose let us introduce χ R S(R d ) such that 0 ˆχ R 1 and ˆχ R (ξ) = 1 if ξ R, ˆχ R (ξ) = 0 if ξ R. Here ˆ denotes the Fourier transform. One has v l n = χ R v l n + (δ χ R ) v l n,

8 T. HMIDI AND S. KERAANI where stands for the convolution. Let p ], [ to be fixed. On the one hand, in view of Sobolev embedding, we get (δ χ R ) v l n L p (δ χ R ) v l n Ḣβ R β 1 v l n H 1, for β = d( 1 1 p ) < 1. On the other hand, one can estimate Now, observe that χ R vn l L p χ R vn l /p χ L R vn l 1 /p L v l n /p L χ R v l n 1 /p L. lim sup χ R vn l L (R d ) = sup lim sup χ R vn(x l n ). {x n} n + Thus, in view of the definition of V(v l ), we infer lim sup χ R vn l L (R d ) sup{ χ R ( x)v (x)dx, V V(v l )}. R d Therefore, by Hölder s inequality, it follows that lim sup χ R vn l L (R d ) C sup{ V L (R d ), V V(v l )}. Here, C (R) = χ R L (R d ). Thus, we obtains lim sup χ R vn l L (R d ) C η(v l ) for every l 1. Finally, we get v l n L p (R d ) R β 1 v l n H 1 + C(R) v l n /p L η(v l ) 1 /p. Now, we let l go to infinity, then R go to infinity and since η(v l ) l 0 and the family of sequences {v l n} are uniformly bounded in H 1 (R d ), we obtain lim sup vn l L p 0 l as claimed. This closes the proof of the proposition.3. Let us return to the proof of Theorem 1.1. According to Proposition.3, the sequence {v n } n=1 can be written, up to a subsequence, as v n (x) = V j (x x j n) + vn(x) l such that (.4) and (.5) hold. This implies, in particular, m 4 d + lim sup V j ( x j n) 4 d + L d 4 +. The elementary inequality a j 4/d+ a j 4/d+ C a j a k 4/d+1. j k

COMPACTNESS LEMMA AND APPLICATIONS 9 and the pairwise orthogonality of the family {x j } leads the mixed terms in the sum above to vanish and we get m 4 d + V j 4 d + L d 4 +. On the one hand, in view of Galiardo-Nirenberg inequality (1.3), we have V j 4 d + C L d 4 + d sup{ V j 4/d, j 1} V j L L. On the other hand, from (.4), we get V j L lim sup v n L M. Therefore, sup V j 4/d m 4 d + L j 1 (M C d ) d/4. Since the series V j L converges then the supremum above is attained. In particular, there exists j 0, such that V j 0 L m d +1 (C d M) d/4 = ( d m d +1 d + )d/4 M d/ Q L. On the other hand, a change of variables gives v n (x + x j 0 n ) = V j 0 (x) + V j (x + x j 0 n x j n) + ṽn(x), l 1 j l j j 0 where vn(x) l = ṽn(x+x l j 0 n ). The pairwise orthogonality of the family {x j } implies V j ( + x j 0 n x j n) 0 weakly for every j j 0. Hence, we get v n ( + x j 0 n ) V j 0 + ṽ l, where ṽ l denote the weak limit of {ṽn} l n=1. However, we have ṽ l L 4 d + lim sup ṽn l 4 L + = lim sup v d n l 4 L + d Thereby, by uniqueness of weak limit, we get for every l j 0. This yields ṽ l = 0 v n ( + x j 0 n ) V j 0. 0. l The sequence {x j 0 n } n=1 and the function V j 0 fulfill the conditions of Theorem 1.1.

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