Regularity, Stability, and Uniqueness Questions for Certain Non-Linear Partial Differential Equations

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Partial Differential Equations (PDEs) are at the basis of many mathematical models used in science and engineering. Examples include the equations for fluid flows or the equation describing the distribution of stress in various structures. In practice, the equations are often solved with the use of computers and a good theoretical understanding of the equations is important for finding effective algorithms. At present, our theoretical understanding of many PDEs is incomplete. There is an important difference between linear models (for which our understanding is better) and non-linear models. In linear models, the reaction of the system to a disturbance is, roughly speaking, proportional to the disturbance. In non-linear models, this is not the case, and many of the mathematical difficulties can be traced to this effect. Linear regimes are often relevant for small disturbances from equilibria, whereas large disturbances are often governed by non-linear phenomena. This project will focus on the non-linear phenomena. In particular, one of the most serious effects in the class of equations which will be investigated is the formation of singularities and the related loss of predictive power of the equations. This will be studied in the context of fundamental equations (such as the equations of incompressible fluid mechanics) and also for various model equations, which can provide suitable stepping stones towards making progress on difficult open problems. This project provides research training opportunities for graduate students.At a more technical level, the project focuses on the following areas: (i) One-dimensional models exhibiting features of PDEs of fluid mechanics. These include the De Gregorio model (which can be thought of as an extension of the Constantin-Lax-Majda model), equations modeling boundary behavior of 2d systems, and vector-valued Burgers-type equations. In spite of their simplicity, such models can be a good source of ideas and their improved understanding can lead to progress on the fundamental equations. In fact, ideas going back to these models have already proved important in the context of the full three-dimensional incompressible Euler equations; (ii) Equations arising in physics and geometry for which we have a satisfactory chance of obtaining a fairly good understanding. These include the Complex Ginzburg-Landau equation, the 2d harmonic map heat flow, and some classical non-linear elliptic systems arising from multi-dimensional variational integrals for vector-valued functions. (The last theme has connections to non-linear elasticity.) All these are important equations in their own right and the PI believes that some of the long-standing open problems related to them can be successfully addressed; (iii) Approachable aspects of the full equations of the incompressible fluid dynamics (the Navier-Stokes and Euler equations). This includes investigations of the solvability of equations describing generalized self-similar singularities, relations between possible non-uniqueness of the Leray-Hopf solutions and questions about instabilities, the stability of the double exponential growth for 2d Euler near the boundaries, and other issues.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.