Partial differential equations (PDEs) appear in the modeling of a large number of natural phenomena, such as the flow of air in the atmosphere; water waves propagating over the oceans; electrical signals in a circuit; and even the collective behavior of large groups of animals such as birds, fish or humans. Given data about the current state of the system, the solution of the PDE informs us of the future behavior of the system. For a given PDE and some class of input data, the mathematician's task is to determine whether the solution exists, whether it is unique, what its qualitative properties are, and whether it can be computed, either exactly or approximately.
It is known that many important PDEs suffer from non-uniqueness -- the PDE can predict several different outcomes for the same input data. A selection principle is then required to single out the physically correct solution from the multitude of solution. Such selection principles have been very successful for many classes of PDEs, but for others (perhaps most importantly, for the Euler equations for modeling gas flow), no known selection principle is able to systematically single out only one "correct" solution.
The goal of the project Irregularity and Noise In Continuity Equations (INICE) is to further the understanding of selection principles for PDEs, more specifically so-called continuity equations. In the first phase of the project we have studied the zero noise limit for stochastic (ordinary) differential equations and found simple conditions that enable us to identify the zero noise limit, what solutions are selected by this selection procedure, and how each of them are weighted.
Partial differential equations (PDEs) are ubiquitous in engineering and the applied sciences, and a major mathematical challenge lies in proving that they are well-posed -- that solutions exist and are unique. However, a large class of PDEs are ill-posed, and selection criteria are required in order to single out a unique, physically reasonable solution. For important equations for gas dynamics such as the Euler and Navier-Stokes equations, no such selection criteria are known. This situation is in contrast with many stochastic PDEs (SPDEs) -- equations that also include the effect of noise -- which are often well-posed under very mild conditions. The general idea that a PDE can be well-posed with noise, but ill-posed without, opens up the possibility of using noise as a selection mechanism for deterministic PDEs by adding a small amount of noise to the deterministic PDE and studying the zero-noise limit of the resulting SPDE.
The main goal of the project Irregularity and Noise In Continuity Equations (INICE) is to characterize the zero-noise limit for several linear and nonlinear PDEs, including certain linear transport/continuity equations, nonlinear hyperbolic conservation laws and nonlinear, nonlocal conservation laws. In the process we develop techniques that in the long run can be applied to problems including mixing of scalar tracers in irregular velocity fields and deterministic, turbulent flows such as the compressible Euler equations. We will develop, analyze and implement efficient numerical methods which can approximate small-noise stochastic PDEs and the zero-noise limit. These methods will be applied both to complement theoretical results and to analyze equations that are outside the reach of theoretical tools.