| Popis: |
This work establishes a sufficient condition for the regularity criterion of the Boussinesq equation in terms of the derivative of the pressure in one direction. It is shown that if the partial derivative of the pressure $${\partial _{3}\pi }$$ satisfies the logarithmical Serrin-type condition $$\int_{0}^{T}\frac{\left\Vert \partial _{3}\pi (s)\right\Vert_{L^{\lambda }}^{q}}{1+\ln (1+\left\Vert \theta \right\Vert_{L^{4}})} {d}s < \infty \quad \text{with}\quad\frac{2}{q}+\frac{3}{\lambda }=\frac{7}{4}\quad \text{and}\quad\frac{12}{7} < \lambda \leq \infty,$$ then the solution $${(u,\theta )}$$ remains smooth on $${\left[0,T\right]}$$ . Compared to the Navier–Stokes result, there is a logarithmic correction involving $${\theta}$$ in the denominator. |
