Non-uniform dependence for the inviscid Boussinesq equations in Besov spaces

Yuanyuan Zhang; Yuanyuan Zhang

doi:10.3934/math.20251135

AIMS Mathematics

2025, Volume 10, Issue 11: 25624-25638. doi: 10.3934/math.20251135

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Research article

Non-uniform dependence for the inviscid Boussinesq equations in Besov spaces

Yuanyuan Zhang ^,

School of Mathematics, Nanjing University, Nanjing 210093, China

Received: 18 August 2025 Revised: 20 October 2025 Accepted: 28 October 2025 Published: 06 November 2025
MSC : 35B30, 35Q35, 35S30

This paper addresses the initial value problem to the inviscid Boussinesq equations in $ \mathbb R^2 $. We rigorously show that the data-to-solution map fails to be uniformly continuous within a broad class of nonhomogeneous Besov spaces $ B^s_{p, r}(\mathbb R^2) $ cited in ^[20] (i.e., $ s > 1+\frac{2}{p}, 1 < p < \infty, 1\leq r \leq \infty $ or $ s = 1+ \frac{2}{p}, 1 < p < \infty, r = 1 $). This result partially extends the nowhere uniform continuity previously demonstrated by Inci^[9] in the Sobolev spaces $ H^m(\mathbb R^2) $ with $ m > 2 $. Our proof leverages the interaction between terms of low and high frequencies. Besides, the linearized system to the inviscid Boussinesq equations plays a pivotal role in the construction of appropriate approximate solutions.
- Boussinesq equations,
- Cauchy problem,
- non-uniform continuous dependence,
- Besov spaces,
- approximate solutions
Citation: Yuanyuan Zhang. Non-uniform dependence for the inviscid Boussinesq equations in Besov spaces[J]. AIMS Mathematics, 2025, 10(11): 25624-25638. doi: 10.3934/math.20251135

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Abstract

This paper addresses the initial value problem to the inviscid Boussinesq equations in $ \mathbb R^2 $. We rigorously show that the data-to-solution map fails to be uniformly continuous within a broad class of nonhomogeneous Besov spaces $ B^s_{p, r}(\mathbb R^2) $ cited in ^[20] (i.e., $ s > 1+\frac{2}{p}, 1 < p < \infty, 1\leq r \leq \infty $ or $ s = 1+ \frac{2}{p}, 1 < p < \infty, r = 1 $). This result partially extends the nowhere uniform continuity previously demonstrated by Inci^[9] in the Sobolev spaces $ H^m(\mathbb R^2) $ with $ m > 2 $. Our proof leverages the interaction between terms of low and high frequencies. Besides, the linearized system to the inviscid Boussinesq equations plays a pivotal role in the construction of appropriate approximate solutions.

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