Well-posedness and scattering for the mass-energy NLS on ℝn × ℳk

Mirko Tarulli

doi:10.1515/anly-2016-0013

Article

Well-posedness and scattering for the mass-energy NLS on ℝⁿ × ℳ^k

Mirko Tarulli

Published/Copyright: May 18, 2017

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From the journal Analysis Volume 37 Issue 3

Abstract

We study the nonlinear Schrödinger equation posed on product spaces ℝn×ℳk, for n≥1 and k≥1, with ℳk any k-dimensional compact Riemannian manifold. The main results concern global well-posedness and scattering for small data solutions in non-isotropic Sobolev fractional spaces. In the particular case of k=2, H1-scattering is also obtained.

Keywords: Nonlinear Schrödinger equation on product manifolds; well-posedness; scattering theory; energy spaces; non-isotropic Sobolev spaces; Strichartz estimates

MSC 2010: 35J10; 35Q55; 35P25; 35R01

Funding statement: The author is supported by the project FIRB 2012 Dispersive Dynamics: Fourier Analysis and Calculus of Variations.

A A fractional inequality on compact manifolds

The target of this Appendix, having its own interest, is to present fundamental tools giving the way, in the end, to the proof of inequality (3.4). Given any compact manifold ℳk, we invoke the following basics:

the curvature tensor (with its derivatives) is bounded,
the Ricci curvature tensor is bounded from below,
the injectivity radius is positive.

These facts enable to represent the fractional derivative |∇y|σ=(-Δy)σ2 when 0<σ<1 as

(A.1)|∇y|σ⁢f⁢(x)=(∫0∞(1tσ⁢vg⁢(B⁢(x,t))⁢∫B⁢(x,t)|f⁢(x)-f⁢(y)|⁢𝑑vg⁢(y))2⁢d⁢tt)12,

where B⁢(x,t) denotes the open ball with center x∈ℳk and radius t>0 (for additional details we refer to [1] or [11]). Therefore we can recall:

Lemma A.1.

Assume that Mk is a compact manifold of dimension k≥1 and let ϕ be a Hölder continuous function of order 0<μ<1. Thus, for any 0<s<μ, 1<q<∞ and sμ<σ<1, we get

∥|∇y|s⁢ϕ⁢(f)∥Lq≤C⁢∥|f|μ-sσ∥Lq1⁢∥|∇y|σ⁢f∥Lq2⁢sσsσ

with C>0, 1q=1q1+1q2 and (1-sμ⁢σ)⁢q1>1.

Proof.

The proof is the same as [37, proof of Proposition A.1] and works in our framework without any changes. It comes out from the pointwise inequality

|∇y|s⁢ϕ⁢(f)⁢(x)≤C⁢(M⁢(|f|μ)⁢(x))1-sμ⁢σ⁢(|∇y|s⁢f⁢(x))sσ,

where

M⁢(f)⁢(x)=supt>0⁡1vg⁢(B⁢(x,t))⁢∫B⁢(x,t)|f⁢(y)|⁢𝑑vg⁢(y),

is the Hardy–Littlewood maximal operator defined on ℳk (for additional details we refer to [2] and [10]). ∎

At this point we can shape the main result of this section (we refer to [11], see also [17] and [27] for an analogue property on the flat manifold ℝn).

Proposition A.1.

Assume that Mk is a compact manifold of dimension k≥1. For any f∈Hyσ∩Ly∞ let G⁢(f)=f⁢|f|μ be a real function with μ>0. Then one has

(A.2)∥f⁢|f|μ∥Hyσ≤C⁢∥f∥Hyσ⁢∥f∥Ly∞μ

with C>0, provided that 0<σ<1+μ.

Proof.

We consider three cases.

Case 0<σ<1. Because in this regime one has

∥f∥Hyσ∼∥f∥Ly2+∥|∇y|σ⁢f∥Ly2

with |∇y|σ as in (A.1), the result is given by an application of the elementary inequality

|f⁢(x)⁢|f⁢(x)|μ-f⁢(y)⁢|f⁢(y)|μ|≤C⁢∥|f|μ∥Ly∞⁢|f⁢(x)-f⁢(y)|

≤C⁢∥f∥Ly∞μ⁢|f⁢(x)-f⁢(y)|

(for the proof of (A.2) in the specific case of ℳk=𝕋1 we refer to [35, Lemma 4.1]).

Case σ=1. This is given by the fact that L∞∩Hyσ is an algebra.

Case σ>1. We will only give the details for μ<1. The argument works also in the case μ>1 observing that if G⁢(f)=f⁢|f|μ, then G⁢(0)=…=G(d)⁢(0)=0 for d=[μ], being [μ] the integer part of μ. Because of σ>1 we can write σ=1+s with 0<s<1; by the definition of the Hyσ-norm we arrive at

∥f⁢|f|μ∥Hyσ≤C⁢∥f⁢|f|μ∥Hys+C⁢∥∇y⁡(f⁢|f|μ)∥Hys

(A.3)≤C⁢∥f⁢|f|μ∥Hys+C⁢∥|f|μ⁢∇y⁡f∥Hys.

Then we can see that it is enough to carry on with the last term in the previous estimate (A.3), that is,

(A.4)∥|f|μ⁢∇y⁡f∥Hys≤C⁢∥∇y⁡f∥Hys⁢∥f∥Ly∞μ+C⁢∥∇y⁡f∥Ly2⁢σ⁢∥|f|μ∥Wys,2⁢σs,

where we used the first estimate of [11, Theorem 27]. Since we have the interpolation bound

(A.5)∥∇y⁡f∥Ly2⁢σ≤C⁢∥f∥Hyσ1σ⁢∥f∥Ly∞1-1σ

(see again [11, Theorem 27]), we need only prove that

(A.6)∥|∇y|s⁢|f|μ∥Ly2⁢σs≤C⁢∥|∇y|σ¯⁢f∥Ly2⁢σσ¯sσ¯⁢∥f∥Ly∞μ-sσ¯

for some sμ<σ¯<1. Assume that (A.6) is true; then by the interpolation estimate (we refer to [11, Proposition 31, 32])

∥|∇y|σ¯⁢f∥Ly2⁢σσ¯≤C⁢∥|∇y|σ⁢f∥Ly2σ¯σ⁢∥f∥Ly∞1-σ¯σ,

one achieves

∥|∇y|s⁢|f|μ∥Ly2⁢σs≤C⁢(∥|∇y|σ⁢f∥Ly2σ¯σ⁢∥f∥Ly∞1-σ¯σ)sσ¯⁢∥f∥Ly∞μ-sσ¯

≤C⁢∥|∇y|σ⁢f∥Ly2sσ⁢∥f∥Ly∞sσ¯-sσ⁢∥f∥Ly∞μ-sσ¯

(A.7)≤C⁢∥f∥Hyσsσ⁢∥f∥Ly∞sσ¯-sσ⁢∥f∥Ly∞μ-sσ¯.

Therefore (A.5) in connection with (A.7), recalling again the definition of the Hyσ-norm, yields

(A.8)∥∇y⁡f∥Ly2⁢σ⁢∥|f|μ∥Wys,2⁢σs≤C⁢∥f∥Hyσ⁢∥f∥Ly∞μ.

Finally, combining (A.3), (A.4) and (A.8), one arrives at (A.2). It remains to consider estimate (A.6). Since the function |f|μ is Hölder continuous of order 0<μ<1, one is in a position to apply Lemma A.1 with q=q2=2⁢σs, q1=∞ and σ=σ¯ getting

∥|∇y|s⁢|f|μ∥Ly2⁢σs≤C⁢∥|∇y|σ¯⁢f∥Ly2⁢σs⁢sσ¯sσ¯⁢∥|f|μ-sσ¯∥Ly∞,

that is, the desired (A.6). ∎

Corollary A.1.

The same conclusions of Proposition A.1 remain valid if one replaces the function f⁢|f|μ by |f|1+μ.

Acknowledgements

The author is grateful to Nicola Visciglia for interesting and helpful discussions concerning Theorem 1.2.

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Received: 2016-3-31

Revised: 2017-3-4

Accepted: 2017-4-17

Published Online: 2017-5-18

Published in Print: 2017-8-1

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https://doi.org/10.1515/anly-2016-0013

Keywords for this article

Nonlinear Schrödinger equation on product manifolds; well-posedness; scattering theory; energy spaces; non-isotropic Sobolev spaces; Strichartz estimates