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Self-similar solutions for a transport equation with non-local flux. (English) Zbl 1186.35154
Summary: The authors construct self-similar solutions for an $$N$$-dimensional transport equation, where the velocity is given by the Riesz transform. These solutions imply nonuniqueness of weak solution. In addition, self-similar solution for a one-dimensional conservative equation involving the Hilbert transform is obtained.

##### MSC:
 35Q35 PDEs in connection with fluid mechanics 76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids 35A22 Transform methods (e.g., integral transforms) applied to PDEs 35L67 Shocks and singularities for hyperbolic equations
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