Global Well‐Posedness and Large Time Behavior of Boussinesq Equations With Fractional Dissipation

ABSTRACT This paper is devoted to the global well‐posedness for small initial data and the large‐time behavior of solutions to the ‐dimensional () incompressible Boussinesq equations with fractional dissipation. We first establish the asymptotic stability of the system by proving that the ‐norm of the solutions decays to zero over time. Subsequently, we prove the local existence of solutions via a mollification approach and the Picard theorem, and then establish a series of a priori estimates that allow us to extend these solutions globally in time using a continuity argument. Furthermore, for initial data lying in negative Sobolev spaces, we demonstrate the global well‐posedness and propagation of regularity in these spaces. A key contribution of this work is the detailed analysis of the large‐time behavior, where we derive both upper and lower bounds for the decay rates of the solutions and their higher‐order derivatives. The fact that these bounds coincide establishes the sharpness (optimality) of the decay rates. To the best of our knowledge, this work provides the first comprehensive study on the stability and large‐time dynamics of the multi‐dimensional Boussinesq equations with general fractional dissipation. By introducing novel techniques in Fourier analysis and energy methods, we not only extend several previous results to the ‐dimensional case but also improve upon others, particularly by relaxing the restrictions on the fractional exponents and the initial data.