Hölder continuous weak solutions of the 3D Boussinesq equation with thermal diffusion

2025/06/03 by Zipeng Chen, Zhaoyang Yin, Chen, Zipeng +1
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2506.02927

openalex publication_date 2025/06/03 · openalex created_date 2025/10/14 · openalex updated_date 2026/07/28

Abstract

In this paper, we show the existence of Hölder continuous periodic weak solutions of the 3D Boussinesq equation with thermal diffusion, which apprroximate the Onsager's critical spatial regularity and satisfy the prescribed kinetic energy. More precisely, for any smooth e(t):[0,T]→ ℝ₊ and β∈ (0, (1)/(3)), there exist v∈ C^β([0,T]× \mathbbT ³) and θ∈ C_t^1,\fracβ2C_x^2,β([0,T]× \mathbbT ³) which solve (\refe:boussinesq equation) in the sense of distribution and satisfy e(t)=∫_\mathbbT ³|v(t,x)|²dx, ∀ t∈ [0,T].

Citations

• Finite energy weak solution of 2d Boussinesq equation with diffusive temperature
• Nonuniqueness of weak solutions to the Navier-Stokes equation
• Onsager's conjecture for admissible weak solutions
• Nonuniqueness of weak solutions to the SQG equation
• A Proof of Onsager's Conjecture
• Dissipative Euler flows with Onsager-critical spatial regularity
• On Nonperiodic Euler Flows with Hölder Regularity
• Onsager's conjecture almost everywhere in time
• Hölder Continuous Euler Flows in Three Dimensions with Compact Support in Time
• Dissipative Euler Flows and Onsager's Conjecture

Related

• Hölder continuous weak solution of 2d Boussinesq equation with diffusive temperature
• A regularity criterion in weak spaces to Boussinesq equations
• A new approach for the regularity of weak solutions of the 3D Boussinesq system
• On The Continuous Periodic Weak Slution of Boussinesq Equations
• Existence of blowup solutions to Boussinesq equations on ℝ³ with dissipative temperature