On the global existence for the axisymmetric Euler-Boussinesq system in critical Besov spaces (Q2883250)

The Cauchy problem to the Euler-Boussinesq system is studied in the paper. NEWLINE\[NEWLINE \begin{cases} \frac{\partial v}{\partial t}+(v\cdot\nabla)v+\nabla p=\rho\, e_z,\quad & x\in \mathbb{R}^3,\quad t>0, \\ \frac{\partial \rho}{\partial t}+v\cdot\nabla \rho -\Delta \rho=0,\quad \text{div}\,v=0,\quad & x\in \mathbb{R}^3,\quad t>0, \\ v(x,0)=v_0(x),\quad \rho(x,0)=\rho_0(x),\quad & x\in \mathbb{R}^3. \end{cases} NEWLINE\]NEWLINE Here \(v=(v_1,v_2,v_3)\) is the velocity, \(p\) is the pressure, \(\rho\) is the density, \(e_z=(0,0,1)\).NEWLINENEWLINEThe main result is the following theorem: Let \(v_0\in B^{5/2}_{2,1}\) be an axisymmetric vector field with zero divergence without swirl and \(\rho_0\in B^{1/2}_{2,1}\cap L^q\) be an axisymmetric function with \(q>6\); then there exists a unique global solution \((v,\rho)\) to the problem such that NEWLINE\[NEWLINE v\in C(\mathbb{R}_+;B^{5/2}_{2,1}),\quad \rho\in C(\mathbb{R}_+;B^{1/2}_{2,1}\cap L^q), NEWLINE\]NEWLINE where \(B^{\alpha}_{2,1}\) is Besov space. The assumption of axial symmetry plays a crucial role in a proof.