Nonlinear partial differential equations (PDEs) have their origin in the quest to describe nature by mathematics. However, several key questions have not been answered, or only been partly answered. Indeed, central issues in my research are: (1) Do the PDEs have a solution? (2) Given that the PDE has a solution, is it unique? (3) Is the solution stable with respect to changes in the input data? (4) If the equation has a unique and stable solution, how can one compute the solution by numerical algorithms? Variations of these questions for selected classes of PDEs and stochastic PDEs (SPDEs) constitute the main focus of my research, including the interplay between theoretical results (mathematical properties of the solution) and numerical computations (how to compute the solution). Most of the activity is basic mathematical research, but the PDE models are motivated by applications, including porous media flow (oil recovery), solid-liquid separation processes (sedimentation), traffic flow, water waves, finance, biology & medicine.
SIAM Journal on Mathematical Analysis (2021–)
SIAM Journal on Numerical Analysis (2007–)
Journal of Hyperbolic Differential Equations (2004–)
Networks and Heterogeneous Media (2006– / EiC from 2017)
AIMS Book Series: Applied Mathematics (2011–)
Advances in Numerical Analysis (2008-2018)
Advances in Applied Mathematics and Mechanics (2009-2018)