Thermodynamically consistent modeling for dissolution/growth of bubbles in an incompressible solvent

We derive mathematical models of the elementary process of dissolution/growth of bubbles in a liquid under pressure control. The modeling starts with a fully compressible version, both for the liquid and the gas phase so that the entropy principle can be easily evaluated. This yields a full PDE system for a compressible two-phase fluid with mass transfer of the gaseous species. Then the passage to an incompressible solvent in the liquid phase is discussed, where a carefully chosen equation of state for the liquid mixture pressure allows for a limit in which the solvent density is constant. We finally provide a simplification of the PDE system in case of a dilute solution

Finite difference methods for linear transport equations

DiPerna-Lions (Invent. Math. 1989) established the existence and uniqueness results for linear transport equations with Sobolev velocity fields. This paper provides mathematical analysis on two simple finite difference methods applied to linear transport equations on a bounded domain with divergence-free (unbounded) Sobolev velocity fields. The first method is based on a Lax-Friedrichs type explicit scheme with a generalized hyperbolic scale, where truncation of an unbounded velocity field and its measure estimate are implemented to ensure the monotonicity of the scheme; the method is $L^p$-strongly convergent. The second method is based on an implicit scheme with $L^2$-estimates, where the discrete Helmholtz-Hodge decomposition for discretized velocity fields plays an important role to ensure the divergence-free constraint in the discrete problem; the method is scale-free and $L^2$-strongly convergent. The key point for both of our methods is to obtain fine $L^2$-bounds of approximate solutions that tend to the norm of the exact solution given by DiPerna-Lions. Finally, the explicit scheme is applied to the case with smooth velocity fields from the viewpoint of the level-set method involving transport equations, where rigorous discrete approximation of geometric quantities of level sets is discussed

A remark on Tonelli's calculus of variations

This paper provides a quite simple method of Tonelli's calculus of variations with positive definite and superlinear Lagrangians. The result complements the classical literature of calculus of variations before Tonelli's modern approach. Inspired by Euler's spirit, the proposed method employs finite dimensional approximation of the exact action functional, whose minimizer is easily found as a solution of Euler's discretization of the exact Euler-Lagrange equation. The Euler-Cauchy polygonal line generated by the approximate minimizer converges to an exact smooth minimizing curve. This framework yields an elementary proof of the existence and regularity of minimizers within the family of smooth curves and hence, with a minor additional step, within the family of Lipschitz curves, without using modern functional analysis on absolutely continuous curves and lower semicontinuity of action functionals