Local well-posedness for the Zakharov system in dimension d ≤ 3

Akansha Sanwal

doi:10.3934/dcds.2021147

Article Contents

2022, Volume 42, Issue 3: 1067-1103. Doi: 10.3934/dcds.2021147

This issue Previous Article Periodic and asymptotically periodic fourth-order Schrödinger equations with critical and subcritical growth Next Article Zero-dimensional and symbolic extensions of topological flows

Local well-posedness for the Zakharov system in dimension d ≤ 3

Akansha Sanwal

Universität Bielefeld, Fakultät für Mathematik, Postfach 10 01 31, 33501 Bielefeld, Germany

Received: March 31, 2021

Revised: July 31, 2021

Early access: October 2021

Published: March 2022

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Abstract

The Zakharov system in dimension $ d\leqslant 3 $ is shown to be locally well-posed in Sobolev spaces $ H^s \times H^l $, extending the previously known result. We construct new solution spaces by modifying the $ X^{s,b} $ spaces, specifically by introducing temporal weights. We use the contraction mapping principle to prove local well-posedness in the same. The result obtained is sharp up to endpoints.

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Mathematics Subject Classification: Primary: 35Q55.

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Figure 1. Region of well-posedness for $ d = 3 $ in [2]

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Figure 2. New region of well-posedness for $ d\leq 3 $

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Table 1. $ H \times L\rightarrow L $

$ \|\tau_1\| $	$ \|\tau_2\| $	$ \|\tau_0\| $	Conclusion
$ \lesssim \lambda_1 $	$ \lesssim \lambda_1 $	$ \lesssim \lambda_1 $	$ \lambda_0^2\lesssim \lambda_1 $
$ \lesssim \lambda_1 $	$ \lambda_1 \ll \cdot \ll \lambda_1^2 $	$ \lambda_1 \ll \cdot \ll \lambda_1^2 $	$ \|{\tau _W}\|\sim\|{\tau _0}\|,\;\;{\lambda _1} \ll \lambda _0^2 $
$ \lambda_1\ll\cdot \ll \lambda_1^2 $	$ \lesssim \lambda_1 $	$ \lambda_1 \ll \cdot \ll \lambda_1^2 $	$ \|\tau_S\|\sim \|\tau_0\|, \lambda_1 \ll \lambda_0^2 $
$ \lambda_1 \ll\cdot \ll \lambda_1^2 $	$ \lambda_1 \ll \cdot \ll \lambda_1^2 $	$ \lambda_1 \ll \cdot \ll \lambda_1^2 $	$ \|\tau_S\|,\|\tau_W\|\sim \lambda_0^2, \lambda_1 \ll \lambda_0^2 $
$ \lambda_1 \ll\cdot \ll \lambda_1^2 $	$ \lambda_1 \ll \cdot \ll \lambda_1^2 $	$ \ll \lambda_1 $	$ \lambda_0^2 \ll \lambda_1 $

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