Global Solvability of the Generalized Boussinesq System with Linear or Nonlinear Buoyancy Force (Contemporary Mathematics, 2025)
This paper is devoted to the global solvability of the Boussinesq system with fractional Laplacian (−∆)α in  for n ≥ 3, where the buoyancy force has the form |θ|m−1θen with m ≥ 1. By establishing estimates for the difference  |θ1|m−1θ1 − |θ2|m−1θ2 in Besov spaces and employing the maximal regularity property of (−∆)α in Lorentz spaces, we prove the following results: under some reasonable assumptions on the exponents α, m, p, r and ρ, if the small initial data of velocity and temperature (or salinity) fall in  (where p1 = p for 1 < p < n, and p2 = p/2 for n ≤ p < 2n) when m = 1, and in  when m > 1, then the generalized Boussinesq system admits a unique global strong solution (u, θ) in ×  (with i = 1, 2 cor-responding to the definition of p1, p2) for m = 1 and in ×  for m > 1, respectively.
Keywords: global solvability; Boussinesq system; fractional Laplacian; linear or nonlinear buoyancy force