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Register a new userIsolated singularities for the n-Liouville equation
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Authors
Pierpaolo Esposito
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In dimension n isolated singularities -- at a finite point or at infinity -- for solutions of finite total mass to the n-Liouville equation are of logarithmic type. As a consequence, we simplify the classification argument in arXiv:1609.03608 and establish a quantization result for entire solutions of the singular n-Liouville equation.
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In this paper, we consider a 1D periodic transport equation with nonlocal flux and fractional dissipation $$ u_{t}-(Hu)_{x}u_{x}+kappaLambda^{alpha}u=0,quad (t,x)in R^{+}times S, $$ where $kappageq0$, $0<alphaleq1$ and $S=[-pi,pi]$. We first establish the local-in-time well-posedness for this transport equation in $H^{3}(S)$. In the case of $kappa=0$, we deduce that the solution, starting from the smooth and odd initial data, will develop into singularity in finite time. If adding a weak dissipation term $kappaLambda^{alpha}u$, we also prove that the finite time blowup would occur.
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