Global regularity of solutions to the Boussinesq equations with fractional diffusion (Q378924)

The authors consider the \(d\)-dimensional (\(d \geq 3\)) incompressible Boussinesq equations: NEWLINE\[NEWLINE \partial_t u + u \cdot \nabla u + \nabla \pi + \nu \mathcal{L}_1^2 u = \theta e_d, NEWLINE\]NEWLINE NEWLINE\[NEWLINE \partial_t \theta + u \cdot \nabla \theta + \kappa \mathcal{L}_2^2 \theta = 0, NEWLINE\]NEWLINE NEWLINE\[NEWLINE \text{div}\, u = 0, NEWLINE\]NEWLINE NEWLINE\[NEWLINE u(x, 0) = u_0(x), \, \theta(x, 0) = \theta_0(x). NEWLINE\]NEWLINE where \(\mathcal{L}_1\) and \(\mathcal{L}_2\) are Fourier multipliers with symbols \(m_1\) and \(m_2\), respectively, that satisfy NEWLINE\[NEWLINE m_1(\xi) \geq \frac{|\xi|^{\alpha}}{g_1(\xi)}, \quad m_2(\xi) \geq \frac{|\xi|^{\beta}}{g_2(\xi)}. NEWLINE\]NEWLINE The global existence of the classical solution for this problem is proved in the two following cases:NEWLINENEWLINE(i) \(\alpha \geq \tfrac{1}{2} + \tfrac{d}{4}\), \(\beta \geq 0\), \((\alpha, \beta) \neq (\tfrac{1}{2} + \tfrac{d}{4}, 0)\), (ii) \(g_1 \geq 1\) and \(g_2 \geq 1\) are nondecreasing functions such that NEWLINE\[NEWLINE \int_1^{\infty} \frac{d \tau}{\tau g_11^2(\tau)(g_1^2(\tau))} = \infty NEWLINE\]NEWLINE and \(\alpha \geq \tfrac{1}{2} + \tfrac{d}{4}\), \(\beta > 0\) and \(\alpha + \beta \geq 1 + \tfrac{d}{2}\).NEWLINENEWLINEIn parallel, the authors obtain similar regularity results for the Benard equation.NEWLINENEWLINEThe proof technique is based on energy methods and the approach developed previously for the 2D Boussinesq equations. The main tools are a priori estimates, the Fourier localization technique, and Bony's paraproduct decomposition.