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On a one-dimensional nonlocal flux with fractional dissipation. (English) Zbl 1231.35172
From the authors’ abstract: In the paper under review the authors study a class of one-dimensional conservation laws with nonlocal flux and fractional dissipation, namely: \[ \partial_t \theta - (\theta H \theta)_x =-\nu (- \partial_{xx})^{\gamma/2}\theta, \] where \(H\) is the Hilbert transform. In the regime \(\nu>0\) and \(1< \gamma \leq 2\), the authors prove local existence and regularity of solutions regardless of the sign of the initial data. For values \(\nu \geq 0\) and \(0 \leq \gamma \leq 2\), the authors construct a certain class of positive smooth initial data with sufficiently localized mass, such that corresponding solutions blow up in finite time. The paper extends recent results of A. Castro and D. Córdoba [Adv. Math. 219, No. 6, 1916–1936 (2008; Zbl 1186.35002)].

MSC:
35Q35 PDEs in connection with fluid mechanics
35Q86 PDEs in connection with geophysics
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