Monotonicity formula methods for nonlinear PDEs

It is known that for linear elliptic and parabolic equations of second order one can construct monotone functions from the solution. A typical example is the mean value integral of harmonic function over a ball. In this case the mean value integral is monotone function of the radius of the ball. There are more complex examples of this sort such as Almgren's frequency formula which, among other things, helps to identify the structure of the zero set of harmonic function. The aim of this project is to construct monotone functions for the solutions of some nonlinear equations. The choice of this type of operators is adequate since there are various physical problems where the nonlinear equations emerge. For instance, the flow of non-Newtonian fluids with power law dependence of the shear tensor from the velocity, the flow of gas in porous media in turbulent regime, the quantum field theory and the interaction of two biological groups without self-limiting. We aim to construct monotone functions for three free boundary problems with nonlinear governing equations and point out some applications in stochastic game theory (Tug-of-War model), Chemical Kinetics and Combustion (smouldering of cigarettes and flame propagation).

The beneficiaries are;
1) The Community of Researchers working in the field of nonlinear elliptic and parabolic partial differential equations of second order and scientists whose research is relevant to the equations of this sort.
2) The BRE Centre for Fire Safety Engineering in the UK. The results, that we expect to obtain for the smouldering model, may be used to improve the design of reduced ignition propensity cigarettes which have the self-extinguishing property.
3) The wider public will benefit, through a more detailed understanding of the risks (in-flight safety of aeroplanes, peat fires pollution) caused by smouldering fires and having effective ways of preventing and extinguishing them.

Due to the various applications discussed in the Case of Support, we think that launching this project will have broader impact. This will be achieved in several steps; by training graduate students and postdocs in interdisciplinary way with an emphasis on applications; by improving the understanding of the phenomena that these models reflect and writing one or two expository papers that will avoid the technicalities and explain the results of the proposed research in a simple and heuristic way so that PhD students, postdocs and researchers from related fields can easily access. Finally there will be a visitor programme hosting up to four researchers a year to maintain effective collaboration.
The proposed research has a potential for fostering a world leading research activity and contributing into scientific competitiveness of the United Kingdom.