Mathematical Sciences: Nonlinear Partial Differential Equations

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Abstract Sverak 9622795 Sverak's will study a variety of questions from system of non-linear partial differential equations. These include: (1) The study of general multiple variational integrals for vector valued functions, their Euler-Lagrange equations, the closely related Morrey's quasi-convexity condition and other notions of ellipticity, together with certain natural objects (semi-convex hulls) associated with these ellipticity conditions. These investigation should help to understand global properties of solutions of a number of non-linear systems. (2) The study of Liouville-type theorems for the three-dimensional incompressible Navier-Stokes equations and their applications to the study of (potential) singularities of solutions of these equations. The motivation for the study of variational integrals comes from the very general and extremely important classical principle that many equations arising in engineering, physics, and also pure mathematics can be directly linked to minimization of (suitably defined) energy. For example, the basic equations describing deformations of materials under stress are very closely related to this principle. Although our mathematical understanding of minimization is quite good in many important cases, there are also many situations of considerable practical importance where our understanding is rather poor. The above mentioned equations for deformations of materials give a good example of such situation (except for the case when the stress in the material is relatively small). I expect that the study of the first circle of problems proposed above will shed some light on open questions surrounding energy minimization. It turns out that results related to these questions can also help to elucidate other mathematical problems which at first do not seem to be directly related to energy minimization (such as non-linear oscillations). However, a closer analysis shows that there are some unexpected connections. The understanding of mathematic al problems surrounding energy minimization is also quite important from the practical point of view: if we simulate minimization on a computer, then non-trivial mathematical facts we know can potentially save us a considerable amount of computations we need to do. In fact, in many cases a good theoretical analysis can enable one to attack problems which at first seem out of reach of even the fastest computers. One can put forward similar arguments to explain the motivation behind the second circle of problems proposed above. In fact, in this case the need for good mathematical analysis is perhaps even more apparent: it is well known that in many cases of great practical importance, there is simplyno known method to reliably calculate fluid flows (describedby the Navier-Stokes equations), even with the help of the fastest computers. I hope that the proposed investigations will increase our understanding of the behavior of solutions of these equations.