Blowup for fractional NLS

Boulenger, Thomas; Himmelsbach, Dominik; Lenzmann, Enno

doi:10.48550/arxiv.1509.08845

Cited by 2 publications

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“…Inflation of H 1/2 -norm for P(f ) < 0 and E hw (f ) < I r . In this section we follow the approach of [3]. In the sequel the radially symmetric function ϕ : R n → R is defined as…”

Section: 2mentioning

confidence: 99%

“…We point out that at the best of our knowledge in the literature there exist very few results about the global existence of solutions to both HW and sNLS. In particular we mention the result in [13] where it is considered HW in 1 − d with nonlinearity u|u| 3 and initial data in H 1 (R), without any further symmetry assumption. Indeed the aforementioned result can be extended to 1 − d sNLS with quartic nonlinearity.…”

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confidence: 99%

“…where ϕ R is a rescaled cut-off function such that ∇ϕ R (x) ≡ x for |x| ≤ R and ∇ϕ R (x) ≡ 0 for |x| >> R. This approach has been further extended by Boulenger-Himmelsbach-Lenzmann [3] in the non-local context with dispersion (−∆) s for 1 2 < s < 1. In this paper we shall take advantage of similar computations in the case of HW.…”

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confidence: 99%

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Long time dynamics for semi-relativistic NLS and half wave in arbitrary dimension

2018

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We consider the Cauchy problems associated with semirelativistc NLS (sNLS) and half wave (HW). In particular we focus on the following two main questions: local/global Cauchy theory; existence and stability/instability of ground states. In between other results, we prove the existence and stability of ground states for sNLS in the L 2 supercritical regime. This is in sharp contrast with the instability of ground states for the corresponding HW, which is also established along the paper, by showing an inflation of norms phenomenon. Concerning the Cauchy theory we show, under radial symmetry assumption the following results: a local existence result in H 1 for energy subcritical nonlinearity and a global existence result in the L 2 subcritical regime.

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“…Inflation of H 1/2 -norm for P(f ) < 0 and E hw (f ) < I r . In this section we follow the approach of [3]. In the sequel the radially symmetric function ϕ : R n → R is defined as…”

Section: 2mentioning

confidence: 99%

mentioning

confidence: 99%

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Long time dynamics for semi-relativistic NLS and half wave in arbitrary dimension

2018

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“…It is by no means obvious how one can establish such formulas for the present nonlocal operators. Interesting generalizations have recently been obtained for translation-invariant operators by Ros-Oton and Serra, partly with Valdinoci, in [RS14b,RSV15], and applied to nonlinear equations P u = f (u) there as well as in [RS14c,RS15a]; they have also been applied to nonlinear time-dependent Schrödinger equations by Boulenger, Himmelsbach and Lenzmann in [BHL16].…”

Section: Introductionmentioning

confidence: 99%

Integration by parts and Pohozaev identities for space-dependent fractional-order operators

Grubb

2016

Journal of Differential Equations

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Abstract. Consider a classical elliptic pseudodifferential operator P on R n of order 2a (0 < a < 1) with even symbol. For example, P = A(x, D) a where A(x, D) is a secondorder strongly elliptic differential operator; the fractional Laplacian (−∆) a is a particular case. For solutions u of the Dirichlet problem on a bounded smooth subset Ω ⊂ R n , we show an integration-by-parts formula with a boundary integral involving (d −a u)| ∂Ω , where d(x) = dist(x, ∂Ω). This extends recent results of Ros-Oton, Serra and Valdinoci, to operators that are x-dependent, nonsymmetric, and have lower-order parts. We also generalize their formula of Pohozaev-type, that can be used to prove unique continuation properties, and nonexistence of nontrivial solutions of semilinear problems. An illustration is given with P = (−∆ + m 2 ) a . The basic step in our analysis is a factorization of P , P ∼ P − P + , where we set up a calculus for the generalized pseudodifferential operators P ± that come out of the construction.

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Blowup for fractional NLS

Cited by 2 publications

References 0 publications

Long time dynamics for semi-relativistic NLS and half wave in arbitrary dimension

Long time dynamics for semi-relativistic NLS and half wave in arbitrary dimension

Integration by parts and Pohozaev identities for space-dependent fractional-order operators

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