Finite time singularity for the modified SQG patch equation (Q338426)

The authors of the present paper consider the question of finite time singularity formation for the \textit{modified SQG patch equation} in the half-plane.NEWLINENEWLINEIt is well-known that the 2D incompressible Euler equations have globally regular solutions. For a related system, the inviscid surface quasi-geostrophic (SQG) equation, the question of global regularity for its solutions is still open. The modified SQG equation can be seen as an interpolation between the two previous models, hence posing the same question is natural and important.NEWLINENEWLINEThe modified SQG equation can be described via the equation NEWLINE\[NEWLINE\partial_t\omega+(u\cdot\nabla)\omega=0NEWLINE\]NEWLINE with the initial condition \(\omega(\cdot,0)=\omega_0\) and the Biot-Savart law for the velocity NEWLINE\[NEWLINEu:=\nabla^\perp(-\Delta)^{-1+\alpha}\omega,NEWLINE\]NEWLINE where \(\alpha\in[0,1/2]\) is a parameter. The values \(\alpha=0\) and \(\alpha=1/2\) correspond to the 2D Euler (in vorticity formulation) and to the SQG equations, respectively.NEWLINENEWLINEThe authors work with a special class of solutions, called \textit{vortex patches}. These can be written as NEWLINE\[NEWLINE\omega(x,t)=\sum_k\theta_k\chi_{\Omega_k(t)}(x),NEWLINE\]NEWLINE where \(\theta_j\) are constants and \(\Omega_j(t)\) are time-evolving open sets with nonzero mutual distances and smooth boundaries.NEWLINENEWLINEThe main results of the authors in the half-plane read as follows: {\parindent=6mm \begin{itemize}\item[1.] For \(\alpha=0\) (i.e., Euler case) and any \(\gamma\in(0,1]\), for each \(C^{1,\gamma}\) patch initial data \(\omega_0,\) there exists a unique global in time \(C^{1,\gamma}\) patch solution of the previous system with \(\omega(\cdot,0)=\omega_0\). \item[2.] For any \(\alpha\in(0,1/24)\) there are \(H^3\) patch-like initial data \(\omega_0\) for which the unique global \(H^3\) patch solution of the above system with \(\omega(\cdot,0)=\omega_0\) becomes singular in finite time. NEWLINENEWLINE\end{itemize}} This well-written paper adds an important and beautiful contribution to the theory of fluid dynamics and PDEs in particular.