Global wellposedness for the energy-critical Zakharov system below the ground state

Candy, Timothy; Herr, Sebastian; Nakanishi, Kenji

doi:10.1016/j.aim.2021.107746

Cited by 6 publications

(32 citation statements)

References 20 publications

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“…However recent work has shown that radial solutions to (1.1) with energy below the ground state (Q, −Q 2 ) are global and scatter [16]. On the other hand, in the non-radial case, it has recently been shown [9] that all solutions with data below the ground state are global in time. As we require this result later, we recall the precise statement from [9].…”

Section: Introductionmentioning

confidence: 99%

“…On the other hand, in the non-radial case, it has recently been shown [9] that all solutions with data below the ground state are global in time. As we require this result later, we recall the precise statement from [9]. Theorem 1.1 (GWP below ground state [9]).…”

Section: Introductionmentioning

confidence: 99%

“…As we require this result later, we recall the precise statement from [9]. Theorem 1.1 (GWP below ground state [9]). Let (f, g) ∈ H 1 × L 2 with…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, obtaining such a profile decomposition seems extremely difficult. Consequently the small data theory in [9,10] which required smallness in L 2 t L 4…”

Section: Introductionmentioning

confidence: 99%

“…The wave data g ∈ L 2 in Theorem 1.7 is not small, and consequently we need to absorb the free wave into the left hand side of the equation. This issue was resolved in [9] by proving a uniform Strichartz estimate for the Schrödinger equation with a free wave potential, and we exploit this estimate here. As a simple toy model, consider the equation…”

Section: Introductionmentioning

confidence: 99%

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Minimal non-scattering solutions for the Zakharov system

Candy¹

2022

Preprint

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We consider the Zakharov system in the energy critical dimension d = 4 with energy below the ground state. It is known that below the ground state solutions exist globally in time, and scatter in the radial case. Scattering below the ground state in the non-radial case is an open question. We show that if scattering fails, then there exists a minimal energy non-scattering solution below the ground state. Moreover the orbit of this solution is precompact modulo translations. The proof follows by a concentration compactness argument, together with a refined small data theory for energy dispersed solutions.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…As we require this result later, we recall the precise statement from [9]. Theorem 1.1 (GWP below ground state [9]). Let (f, g) ∈ H 1 × L 2 with…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, obtaining such a profile decomposition seems extremely difficult. Consequently the small data theory in [9,10] which required smallness in L 2 t L 4…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Minimal non-scattering solutions for the Zakharov system

Candy¹

2022

Preprint

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Decoupling Inequality for Paraboloid Under Shell Type Restriction and Its Application to the Periodic Zakharov System

Kinoshita,

Nakamura,

Sanwal

2023

Commun. Math. Phys.

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In this paper, we establish local well-posedness for the Zakharov system on $${\mathbb {T}}^d$$ T d , $$d\ge 3$$ d ≥ 3 in a low regularity setting. Our result improves the work of Kishimoto (J Anal Math 119:213–253, 2013). Moreover, the result is sharp up to $$\varepsilon $$ ε -loss of regularity when $$d=3$$ d = 3 and $$d\ge 5$$ d ≥ 5 as long as one utilizes the iteration argument. We introduce ideas from recent developments of the Fourier restriction theory. The key element in the proof of our well-posedness result is a new trilinear discrete Fourier restriction estimate involving paraboloid and cone. We prove this trilinear estimate by improving Bourgain–Demeter’s range of exponent for the linear decoupling inequality for paraboloid (Bourgain and Demeter in Ann Math 182:351–389) under the constraint that the input space-time function f satisfies $$\textrm{supp}\, {\hat{f}} \subset \{ (\xi ,\tau ) \in {\mathbb {R}}^{d+1}: 1- \frac{1}{N} \le |\xi | \le 1 + \frac{1}{N},\; |\tau - |\xi |^2| \le \frac{1}{N^{2}} \} $$ supp f ^ ⊂ { ( ξ , τ ) ∈ R d + 1 : 1 - 1 N ≤ | ξ | ≤ 1 + 1 N , | τ - | ξ | 2 | ≤ 1 N 2 } for large $$N\ge 1$$ N ≥ 1 .

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Global well-posedness and scattering for the Zakharov system at the critical space in three spatial dimensions with small and radial initial data

Kato

Kinoshita²

2023

Journal of Mathematical Analysis and Applications

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Global wellposedness for the energy-critical Zakharov system below the ground state

Cited by 6 publications

References 20 publications

Minimal non-scattering solutions for the Zakharov system

Minimal non-scattering solutions for the Zakharov system

Decoupling Inequality for Paraboloid Under Shell Type Restriction and Its Application to the Periodic Zakharov System

Global well-posedness and scattering for the Zakharov system at the critical space in three spatial dimensions with small and radial initial data

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