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Local well-posedness for the quasilinear Schrödinger equations via the generalized energy method

Jie Shao, Yi Zhou

TL;DR

The paper addresses the Cauchy problem for general quasilinear Schrödinger equations by introducing a generalized energy method that fuses momentum-type and energy estimates and obtains local well-posedness through viscosity methods. It handles both small- and large-data scenarios, deriving low-regularity results for quadratic interactions and recovering low-regularity outcomes for small-data cubic interactions, by leveraging momentum identities, weighted estimates, and pseudo-differential tools under non-trapping conditions. The framework unifies and extends prior results from Kenig–Ponce–Vega and Marzuola–Metcalfe–Tataru across elliptic and ultrahyperbolic settings, highlighting the role of commutator bounds and half-derivative techniques in overcoming derivative loss. Overall, the generalized energy approach provides a cohesive, viscosity-based pathway to local well-posedness for quasilinear Schrödinger equations with quadratic and cubic nonlinearities, with implications for related dispersive flows.

Abstract

We study the Cauchy problem of quasilinear Schrödinger equations, for which Kenig et al. (Invent Math, 2004; Adv Math, 2006) obtained large data local well-posedness by pseudo-differential techniques and viscosity methods, while Marzuola et al. (Adv Math, 2012; Kyoto J Math, 2014; Arch Ration Mech Anal, 2021) and Ben et al. (Arch Ration Mech Anal, 2024) improved the results by dispersive arguments. In this paper, we introduce a generalized energy method that combines momentum and energy estimates to close the bounds, thereby obtaining our results through viscosity methods. If the data is small, the proof relies mainly on integration by parts and Sobolev embeddings, much like the classical local existence theory for semilinear Schrödinger equations. For large data, the framework remains applicable with the incorporation of certain pseudo-differential tools. In the case of quadratic interactions, we establish low regularity local well-posedness for both small and large data in the same function spaces as in works of Kenig et al. For cubic interactions with small initial data, we recover the low regularity results obtained by Marzuola et al. (Kyoto J Math, 2014).

Local well-posedness for the quasilinear Schrödinger equations via the generalized energy method

TL;DR

The paper addresses the Cauchy problem for general quasilinear Schrödinger equations by introducing a generalized energy method that fuses momentum-type and energy estimates and obtains local well-posedness through viscosity methods. It handles both small- and large-data scenarios, deriving low-regularity results for quadratic interactions and recovering low-regularity outcomes for small-data cubic interactions, by leveraging momentum identities, weighted estimates, and pseudo-differential tools under non-trapping conditions. The framework unifies and extends prior results from Kenig–Ponce–Vega and Marzuola–Metcalfe–Tataru across elliptic and ultrahyperbolic settings, highlighting the role of commutator bounds and half-derivative techniques in overcoming derivative loss. Overall, the generalized energy approach provides a cohesive, viscosity-based pathway to local well-posedness for quasilinear Schrödinger equations with quadratic and cubic nonlinearities, with implications for related dispersive flows.

Abstract

We study the Cauchy problem of quasilinear Schrödinger equations, for which Kenig et al. (Invent Math, 2004; Adv Math, 2006) obtained large data local well-posedness by pseudo-differential techniques and viscosity methods, while Marzuola et al. (Adv Math, 2012; Kyoto J Math, 2014; Arch Ration Mech Anal, 2021) and Ben et al. (Arch Ration Mech Anal, 2024) improved the results by dispersive arguments. In this paper, we introduce a generalized energy method that combines momentum and energy estimates to close the bounds, thereby obtaining our results through viscosity methods. If the data is small, the proof relies mainly on integration by parts and Sobolev embeddings, much like the classical local existence theory for semilinear Schrödinger equations. For large data, the framework remains applicable with the incorporation of certain pseudo-differential tools. In the case of quadratic interactions, we establish low regularity local well-posedness for both small and large data in the same function spaces as in works of Kenig et al. For cubic interactions with small initial data, we recover the low regularity results obtained by Marzuola et al. (Kyoto J Math, 2014).
Paper Structure (16 sections, 15 theorems, 315 equations)

This paper contains 16 sections, 15 theorems, 315 equations.

Table of Contents

  1. Introduction
  2. Notations and Preliminaries
  3. Notations
  4. Preliminaries
  5. Momentum balance equalities
  6. Small data and quadratic interactions
  7. Momentum type estimates
  8. Energy estimates and proof of the main result
  9. Large data and quadratic interactions
  10. Momentum type estimates
  11. Energy estimates
  12. Smallness of w
  13. Small data and cubic interactions
  14. Momentum type estimates
  15. Energy estimates and proof of main results
  16. ...and 1 more sections

Key Result

Theorem 1.1

$(a)$ Suppose that $s>\frac{d}{2}+3, d \geqslant 1$, qNLS is quadratic interaction problem and for some integer $N$ that depends only on $d, s$. Then the ultrahyperbolic equation qNLS under condition g0 is locally wellposed in $H^{s} (\mathbb{R}^d) \cap L^2(\langle x\rangle^N \mathrm{d}x)$ for $t\in [0,1]$. $(b)$ The same results holds for qNLS0 with $s>\frac{d}{2}+4, d\geqslant 1.$

Theorems & Definitions (24)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.4
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 3.1
  • ...and 14 more