A remark on norm inflation for nonlinear Schrödinger equations

Abstract

We consider semilinear Schrödinger equations with nonlinearity that is a polynomial in the unknown function and its complex conjugate, on R d or on the torus. Norm inflation (ill-posedness) of the associated initial value problem is proved in Sobolev spaces of negative indices. To this end, we apply the argument of Iwabuchi and Ogawa (2012), who treated quadratic nonlinearities. This method can be applied whether the spatial domain is non-periodic or periodic and whether the nonlinearity is gauge/scale-invariant or not.2010 Mathematics Subject Classification. 35Q55, 35B30.1 Proposition A.1 below for the definition). Our argument, which evaluates each term in the power series expansion of the solution directly, is different from the aforementioned works. Note that, for smooth nonlinearities, Theorem 1.1 covers all the remaining cases in the range s < min{s c (d, p), 0} and extends the result to the (partially) periodic setting as well as to the case of general nonlinearities with complex coefficients. Moreover, our argument also gives another proof of the results in [6, 7] on NI s with infinite loss of regularity; see Proposition A.1 for the precise statement.The one-dimensional cubic equation with nonlinearity ±|u| 2 u has been attracting particular attention due to its various physical backgrounds and complete integrability. Note also that this is the only L 2 -subcritical case among smooth and gauge-invariant nonlinearities. In spite of the L 2 subcriticality, the equation becomes unstable below L 2 due to the Galilean invariance, both in R and in T. In fact, the initial value problem was shown to be globally well-posed in L 2 [39, 3], whereas it was shown in [23,9] for R and in [4,9] for T that the solution map fails to be uniformly continuous below L 2 . Ill-posedness below L 2 (T) was established in the periodic case by the lack of continuity of the solution map [11,32] and by the non-existence of solutions [17]. Nevertheless, one can show a priori bound in some Sobolev spaces below L 2 [27,12,28,17], which prevents norm inflation. Recent results in [29,24] finally gave a priori bound on H s for s > − 1 2 , both in R and in T. We remark that NI s at s = − 1 2 shown in Theorem 1.1 ensures the optimality of these results. 3 In [24, Theorem 4.7], Killip, Vişan and Zhang also derived a priori bound of the solutions in the norm which is logarithmically stronger than the critical H − 1 2 . Motivated by this result, in addition to Theorem 1.1 (ii) we also show norm inflation for the one-dimensional cubic equation in some "logarithmically subcritical" spaces; see Proposition B.3 below.Since the work of Kenig, Ponce, and Vega [22], non gauge-invariant nonlinearities have also been intensively studied. In [2], Bejenaru and Tao proposed an abstract framework for proving ill-posedness in the sense of discontinuity of the solution map. They considered the quadratic NLS (1.2) on R with nonlinearity u 2 and obtained a complete dichotomy of Sobolev index s into locally well-posed (s ≥ −1) and illposed (s < −1) in the sense ...

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