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Low regularity analysis of the Zakharov--Kuznetsov equation on $\mathbb{R} \times \mathbb{T}$

Gonzalo Cao-Labora

TL;DR

This work analyzes the Zakharov--Kuznetsov equation on the cylinder $\mathbb{R}\times\mathbb{T}$, establishing sharp local wellposedness results at low regularity and a probabilistic counterpart below the deterministic thresholds. The key methodological advance is a resonant-set analysis that exploits geometric properties of the resonances, coupled with a novel hybrid function space $Z^{s,b}$ that blends the $L^2$-based $X^{s,b}$ norm with a lower-regularity, $L^\infty$-based term to tame highly concentrated mass in frequency space. The deterministic theory yields LWP for $s>\tfrac{3}{4}$ unconditionally and for $s>\tfrac{1}{2}$ under a low-frequency condition, with optimality (up to endpoints) shown via counterexamples. The probabilistic part shows almost-sure LWP for generic data in $H^{s'-}$ with $s'>-\tfrac{1}{26}$ by subtracting the linear evolution and leveraging greater control on the randomness, suggesting a pathway toward Gibbs-type measures. Overall, the paper bridges deterministic and probabilistic regimes by combining resonant analysis, refined bilinear estimates, and probabilistic smoothing techniques that may extend to other dispersive problems lacking full dispersion in one variable.

Abstract

We consider the Cauchy problem for the Zakharov-Kuznetsov equation in the cylinder. We improve the local wellposedness to spaces of regularity $s > 1/2$. The result is optimal in terms of the corresponding bilinear estimate or Picard iteration. Our method is based on an improvement of the understanding of the resonant set, identifying and exploiting its particular geometric properties. We also consider the problem under randomization of the initial data, in which case we obtain solutions for generic data in $H^{s}$ for some $s < 0$. To do so, we consider a novel approach based on lower regularity modifications of the classical $X^{s, b}$ spaces that allow to control concentration of mass in small sets of frequencies.

Low regularity analysis of the Zakharov--Kuznetsov equation on $\mathbb{R} \times \mathbb{T}$

TL;DR

This work analyzes the Zakharov--Kuznetsov equation on the cylinder , establishing sharp local wellposedness results at low regularity and a probabilistic counterpart below the deterministic thresholds. The key methodological advance is a resonant-set analysis that exploits geometric properties of the resonances, coupled with a novel hybrid function space that blends the -based norm with a lower-regularity, -based term to tame highly concentrated mass in frequency space. The deterministic theory yields LWP for unconditionally and for under a low-frequency condition, with optimality (up to endpoints) shown via counterexamples. The probabilistic part shows almost-sure LWP for generic data in with by subtracting the linear evolution and leveraging greater control on the randomness, suggesting a pathway toward Gibbs-type measures. Overall, the paper bridges deterministic and probabilistic regimes by combining resonant analysis, refined bilinear estimates, and probabilistic smoothing techniques that may extend to other dispersive problems lacking full dispersion in one variable.

Abstract

We consider the Cauchy problem for the Zakharov-Kuznetsov equation in the cylinder. We improve the local wellposedness to spaces of regularity . The result is optimal in terms of the corresponding bilinear estimate or Picard iteration. Our method is based on an improvement of the understanding of the resonant set, identifying and exploiting its particular geometric properties. We also consider the problem under randomization of the initial data, in which case we obtain solutions for generic data in for some . To do so, we consider a novel approach based on lower regularity modifications of the classical spaces that allow to control concentration of mass in small sets of frequencies.

Paper Structure

This paper contains 41 sections, 17 theorems, 247 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Historical background
  3. Deterministic result
  4. Probabilistic result
  5. Organization of the paper
  6. Notation
  7. Proof of the Theorems \ref{['th:deterministic_Xs_space']}, \ref{['th:deterministic_Ys_space']} and \ref{['th:det_fail']}
  8. Reduction of the proof of \ref{['eq:one1']}
  9. Reduction to the case of comparable frequencies
  10. Case \ref{['case:low']}
  11. Case \ref{['case:Llarge']}
  12. Case \ref{['case:Ndif']}
  13. Bilinear estimates and frequency decomposition
  14. New bilinear estimates
  15. Analysis of the resonant set
  16. ...and 26 more sections

Key Result

Theorem 1.3

Let $s > 3/4$ and fix $u_0 \in H^s (\mathbb R \times \mathbb T )$. There exist $T$ and $b > 1/2$ such that the Zakharov--Kuznetsov equation eq:ZK is locally wellposed in $X^{s, b}_T$. That is, for any $u_0 \in H^s (\mathbb R \times \mathbb T)$ there exists a unique local solution $u \in X^{s, b}_{T}

Figures (4)

  • Figure 1: Diagram showing frequencies in $S_{\rm{bad}}$, which corresponds to the main resonance. Each frequency vector should lie in the corresponding box, and the picture is up to rearrangement of the frequencies. In such configuration, we have that $|\Delta | = |\phi (\nu, k) + \phi (\zeta, m) + \phi (\xi, n) |\lesssim 1$. The resonance occurs around the triple cancellation $\Delta = \partial_\nu \Delta = \partial_\zeta \Delta = 0$.
  • Figure 2: Frequency space representation of Proposition \ref{['prop:refined']}.
  • Figure 3: Illustration of Lemma \ref{['lemma:localization']}. The vectors $w_1$, $w_2$ and $w_3$ (shown in blue, red and green respectively) add up to zero and have approximately equal lengths. $w_2$ and $w_3$ must lie in the dashed balls centered at $R_{\pm 2\pi/3}(w_1)$.
  • Figure 4: The region $\mathcal{R}_M$ (shaded) consists of $O(M)$-neighborhoods of three lines $(0,t)$, $(-t/2,-t/2)$, and $(t/2,-t/2)$, restricted to the annulus between radii $N_{\rm{min}}$ and $8N_{\rm{min}}$. We also have covered $\mathcal{R}_M$ with squares of size $cM$. In order to improve the clarity of the picture, it is not done at scale, and the covering in squares of length $cM$ is only done for one of the six connected components of $\mathcal{R}_M$

Theorems & Definitions (36)

  • Remark 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Definition 1.7
  • Theorem 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 26 more