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Register a new userRelaxation to equilibrium in the one-dimensional thin-film equation with partial wetting
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We investigate the large time behavior of compactly supported solutions for a one-dimensional thin-film equation with linear mobility in the regime of partial wetting. We show the stability of steady state solutions. The proof uses the Lagrangian coordinates. Our method is to establish and exploit differential relations between the energy and the dissipation as well as some interpolation inequalities. Our result is different from earlier results because here we consider solutions with finite mass.
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In this technical report, we consider a nonlinear 4th-order degenerate parabolic partial differential equation that arises in modelling the dynamics of an incompressible thin liquid film on the outer surface of a rotating horizontal cylinder in the presence of gravity. The parameters involved determine a rich variety of qualitatively different flows. Depending on the initial data and the parameter values, we prove the existence of nonnegative periodic weak solutions. In addition, we prove that these solutions and their gradients cannot grow any faster than linearly in time; there cannot be a finite-time blow-up. Finally, we present numerical simulations of solutions.
We consider a nonlinear 4th-order degenerate parabolic partial differential equation that arises in modelling the dynamics of an incompressible thin liquid film on the outer surface of a rotating horizontal cylinder in the presence of gravity. The parameters involved determine a rich variety of qualitatively different flows. Depending on the initial data and the parameter values, we prove the existence of nonnegative periodic weak solutions. In addition, we prove that these solutions and their gradients cannot grow any faster than linearly in time; there cannot be a finite-time blow-up. Finally, we present numerical simulations of solutions.
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This paper analyzes inverse scattering for the one-dimensional Helmholtz equation in the case where the wave speed is piecewise constant. Scattering data recorded for an arbitrarily small interval of frequencies is shown to determine the wave speed uniquely, and a direct reconstruction algorithm is presented. The algorithm is exact provided data is recorded for a sufficiently wide range of frequencies and the jump points of the wave speed are equally spaced with respect to travel time. Numerical examples show that the algorithm works also in the general case of arbitrary wave speed (either with jumps or continuously varying etc.) giving progressively more accurate approximations as the range of recorded frequencies increases. A key underlying theoretical insight is to associate scattering data to compositions of automorphisms of the unit disk, which are in turn related to orthogonal polynomials on the unit circle. The algorithm exploits the three-term recurrence of orthogonal polynomials to reduce the required computation.
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