On the Global Well-posedness for the Boussinesq System with Horizontal Dissipation
By Changxing Miao, Xiaoxin Zheng. 2012
130 citations recorded by the source corpus.
In this paper, we investigate the Cauchy problem for the tridimensional Boussinesq equations with horizontal dissipation. Under the assumption that the initial data is axisymmetric without swirl, we prove the global well-posedness for this system. In the absence of vertical dissipation, there is no smoothing effect on the vertical derivatives. To make up this shortcoming, we first establish a magic relationship between $${\frac{u^{r}}{r}}$$ and $${\frac{\omega_\theta}{r}}$$ by taking full advantage of the structure of the axisymmetric fluid without swirl and some tricks in harmonic analysis. This together with the structure of the coupling of (1.2) entails the desired regularity.
Subfields: Applied Mathematics, Mathematical Physics, Oceanography
View Paper: https://www.semanticscholar.org/paper/6305df2a3f8dd6f6b6ccd61c6e5a9b9312af0214
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In Applied Mathematics
In Mathematical Physics
In Oceanography
Citations in the corpus
- Regularity Criterion to the axially symmetric Navier-Stokes Equations (2015, background): “ositive constant whose meaning may change from line to line. If |f|2 is axially symmetric, we will denote kfk 2 L2 = Z |f| rdrdz, dx= rdrdz. The following estimate will be used very often. (see [5][8][13]) Lemma 2.1. ∇ ur r 2 L2 ≤ kΩk2 L2, ∇2 ur r 2 L2 ≤ k∂zΩk 2 L2 . 4 Proof. By virtue of ∂r(rur) + ”
- Criticality of the Axially Symmetric Navier-Stokes Equations (2015, methodology): “Alternatively, one may also use the magic formula given by Miao and Zheng in [19] to prove it.”