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The subject of moving curves (and surfaces) in three dimensional space (3-D) is a fascinating topic not only because it represents typical nonlinear dynamical systems in classical mechanics, but also finds important applications in a…

We establish new results for path-dependent Hamilton-Jacobi equations with nonlinear monotone, and coercive operators on Hilbert space, which were initially studied in Bayraktar and Keller [J. Funct. Anal., 275 (8) (2018), pp. 2096-2161].…

We consider the free boundary problems of degenerate elliptic equations that describe the level set formulation of the interface motion evolved by anisotropic forced mean curvature flows. The type of free boundary problems in this paper was…

The steady motion of a viscous incompressible fluid in a junction of unbounded channels with sources and sinks is modeled through the Navier-Stokes equations under inhomogeneous Dirichlet boundary conditions. In contrast to many previous…

In this note we study the singular vanishing-viscosity limit of a gradient flow set in a finite-dimensional Hilbert space and driven by a smooth, but possibly non convex, time-dependent energy functional. We resort to ideas and techniques…

In this paper we study a finite-depth layer of viscous incompressible fluid in dimension $n \ge 2$, modeled by the Navier-Stokes equations. The fluid is assumed to be bounded below by a flat rigid surface and above by a free, moving…

The ``no-slip'' boundary condition, i.e., zero fluid velocity relative to the solid at the fluid-solid interface, has been very successful in describing many macroscopic flows. A problem of principle arises when the no-slip boundary…

Computational fluid dynamics and aerodynamics, which complement more expensive empirical approaches, are critical for developing aerospace vehicles. During the past three decades, computational aerodynamics capability has improved…

Consider an exterior space-time domain where the incompressible Navier-Stokes equation and continuity equation hold with no bodies or force fields present, and smooth velocity at initial time. This is equivalent to the velocity being…

This paper considers the existence of local and global-in-time strong solutions to the advection-diffusion equation with variable coefficients on an evolving surface with a boundary. We apply both the maximal $L^p$-in-time regularity for…

The level set method commonly requires a reinitialization of the level set function due to interface motion and deformation. We extend the traditional technique for reinitializing the level set function to a method that preserves the…

The numerical modelling of convection dominated high density ratio two-phase flow poses several challenges, amongst which is resolving the relatively thin shear layer at the interface. To this end we propose a sharp discretisation of the…

Full-physics modeling of multiphase flow in porous media, e.g., for carbon storage and groundwater management, requires the nonlinear coupling of various physical processes. Industry standard nonlinear solvers, typically of Newton-type, are…

Highly concentrated patterns have been observed in a spatially heterogeneous, nonlocal, model of BGK type implementing a velocity-jump process. We study both a linear and a nonlinear case and describe the concentration profile. In…

Bilevel programming has recently received a great deal of attention due to its abundant applications in many areas. The optimal value function approach provides a useful reformulation of the bilevel problem, but its utility is often limited…

Building upon the dynamic programming principle for set-valued functions arising from many applications, in this paper we propose a new notion of set-valued PDEs. The key component in the theory is a set-valued It\^{o} formula,…

We consider here a two-dimensional incompressible fluid in a periodic channel, whose density is advected by pure transport, and whose velocity is given by the Stokes equation with gravity source term. Dirichlet boundary conditions are taken…

This paper considers and proposes some algorithms to compute the mean curvature flow under topological changes. Instead of solving the fully nonlinear partial differential equations based on the level set approach, we propose some…

This paper presents a streamfunction-vorticity formulation for the Navier--Stokes and Euler equations on general surfaces. Notably, this includes non-simply connected surfaces, on which the harmonic components of the velocity field play a…

This paper studies Hamilton-Jacobi equations of evolution type defined in a general metric space. We give a notion of a solution through optimal principles and establish a unique existence theorem of the solution for initial value problems.…