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Global and singular solutions to the generalized Proudman-Johnson equation. (English) Zbl 1204.35067
Summary: We show that there is a class of solutions to the generalized Proudman-Johnson equation which exist globally for all parameters \(a\) having the form \(-\frac {n+3}{n+1}\) for \(n \in \mathbb N\), thereby extending a result of A. Bressan and A. Constantin [SIAM J. Math. Anal. 37, No. 3, 996–1026 (2005; Zbl 1108.35024)]. Furthermore, we present new proofs for the existence of solutions developing spontaneous singularities and compute the corresponding blow-up rates.

MSC:
35D30 Weak solutions to PDEs
74H35 Singularities, blow-up, stress concentrations for dynamical problems in solid mechanics
35Q35 PDEs in connection with fluid mechanics
35B44 Blow-up in context of PDEs
Keywords:
blow-up rate
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