##
D.E. Pelinovsky and A.R. Giniyatullin

#
Finite-time singularities in the dynamical evolution of contact lines

###
Bulletin of the Moscow State Regional University (Physics and Mathematics) 2012 N.3, 14-24 (2012)

**Abstract:**

We study finite-time singularities in the linear advection–diffusion equation with a variable
speed on a semi-infinite line. The variable speed is determined by an additional condition at
the boundary, which models the dynamics of a contact line of a hydrodynamic flow at a 180
contact angle. Using apriori energy estimates, we derive conditions on variable speed that
guarantee that a sufficiently smooth solution of the linear advection–diffusion equation blows
up in a finite time. Using the class of self-similar solutions to the linear advection–diffusion
equation, we find the blow-up rate of singularity formation. This blow-up rate does not agree
with previous numerical simulations of the model problem.

**Keywords**:

linear advection-diffusion equation, variable speed, contact line,
apriori energy estimates, self-similar solutions.