Nonlinear Partial Differential Equations and Applications

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The modeling of many phenomena in the physical and social sciences and engineering, such as porous media, composite materials, turbulence and combustion, traffic models, spread of crime, agent models and others, involve heterogeneous media described by partial differential equations. These typically depend upon many parameters and vary randomly on a small scale. In addition, often the available information (e.g., data used in weather prediction) is not exact (deterministic) but statistical (random), with large fluctuations. On macroscopic scales that are much larger than the ones of the heterogeneities, the models often exhibit an effective deterministic behavior, which is much simpler than the original one. The process of averaging such data is known as homogenization. Mathematically, this means that the original random problem is replaced by a deterministic one. When this averaging is not possible, which is typically the case when the fluctuations are too strong (wild), it is necessary to deal with so-called stochastic media (stochastic partial differential equations), which have rather singular behavior in space and time. The mathematical study of both the stochastic averaging and the stochastic partial differential equations requires original ideas and the development of new methodologies, since both topics fall outside the traditional theories of averaging and partial differential equations. Another burgeoning area of research in which similar issues surface is mathematical biology, where experiments at the molecular scale, as well as theoretical advances, have led to new, sophisticated mathematical models. Novel tools and ideas are needed to study these problems further and to validate all the relevant regimes/scales of the parameters affecting the experimentally observed and theoretically conjectured behaviors. This project is directed at the development of general methodologies to study random homogenization, nonlinear stochastic partial differential equations, and applications to front propagation, phase transitions, and mathematical biology. Random environments are much more general than periodic ones. The latter basically involve fixed translations of a certain equation, whereas the former can be thought of as involving all possible (relevant) equations. This leads to considerable issues concerning the lack of compactness. It is therefore necessary to develop novel arguments that combine both the differential and random structures of the media under scrutiny. In this setting, the equation is the random variable and the special dependence signifies the location in space where the equation is observed. The principal investigator and his collaborators were the first to consider stochastic homogenization in stationary ergodic environments. A large part of the project is dedicated to further development of the theory. Stochastic partial differential equations have coefficients with very singular (Brownian) behavior. In the linear context, this can usually be handled by known methods, such as the classical martingale approach. This method is based on the linear character of the higher order part of the equation and thus cannot be used for nonlinear problems, where it is necessary to find appropriate alternative notions of solutions. These, in the context of first- and second-order nonlinear equations, are the stochastic viscosity and pathwise entropy solutions that have been introduced by the principal investigator and his collaborators. A part of the project is the study of the qualitative behavior/properties of these solutions. In the context of mathematical biology, the principal investigator plans to work on models of adaptation/selection as well as on models of the biology of development. The former concerns questions related to the adaptation of species to global change, the resistance of insects to pesticides, etc. The latter aims at developing models to study how positional information is provided to proliferating cells, the main questions being the formation and location of sharp and precise boundaries.