## tisdag 20 maj 2014

### Wellposedness vs the Clay Navier-Stokes Problem?

In a sequence of posts I have argued that the omission of wellposedness in the Official Description of the Clay Navier-Stokes Prize Problem by Charles Fefferman, makes the problem meaningless. To support this I quote from Wellposedness and Physical Possibility by B. Gyenis:

Well posedness is widely held to be an essential feature of physical theories. Consider the following remarks of Mikhail M. Lavrentiev, Alan Rendall, and Robert M. Wald – leading experts in their respective fields of physics – intended as motivations for the continuous dependence condition:
• One should remember that the main goal of solving mathematical problems is to describe certain physical processes in mathematical terms. In this case the initial data are obtained experimentally; and since measurements cannot be absolutely precise, the data contain mea- surement errors. For a mathematical model to describe a real physical process, the problem should be supplemented with some additional requirements reflecting, in a physical sense, the fact that the solution should have only small variations under slight changes of initial data or, to put it conventionally, the stability of the solution under small perturbations in the data. (Lavrentiev et al.; 2003, p. 6)
• The condition of continuity is sometimes called Cauchy stability. The reason for including it is as follows. If PDE are to be applied to model phenomena in the natural world it must be remembered that measurements are never exact but always associated with some error. As a consequence it is impossible to know initial data for a problem exactly and so if solutions depend on the initial data in an uncontrollable way the model cannot make useful predictions. Cauchy stability guarantees that this does not happen and thus represents a necessary condition for the application of PDE to the real world. (Rendall; 2008, p. 134)
• If a theory can be formulated so that “appropriate initial data” may be specified (possibly subject to constraints) such that the subsequent dynamical evolution of the system is uniquely determined, we say that the theory possesses an initial value formulation. How- ever, even if such a formulation exists, there remain further properties that a physically viable theory should satisfy. First, in an appropriate sense, “small changes” in initial data should produce only correspondingly “small changes” in the solution over any fixed compact region of spacetime. If this property were not satisfied, the theory would lose essentially all predictive power, since initial conditions can be measured only to a finite accuracy. It is generally assumed that the pathological behavior which would result from the failure of this property does not occur in physics. [...]2 (Wald; 1984, p. 224)
These remarks express a sentiment widely shared among physicists: wellposedness is a necessary condition for models to describe real physical processes. Lack of wellposedness would be pathological and it “does not occur in physics,” at least not in describing forward time propagation of physical processes.

OK, so leading experts of physics consider wellposedness to a necessary requirement for a mathematical model of some physical phenomena to be meaningful. The Navier-Stokes equations is the basic model of fluid mechanics, and as such requires some form of wellposedness to be meaningful.

The leading mathematical expert Charles Fefferman formulates the Clay Navier-Stokes problem without reference to wellposedness and thus apparently considers wellposedness to not be a central aspect. But doing so Fefferman separates the mathematics of Navier-Stokes equations from physics, which goes against the reason of formulating a Prize Problems about a mathematical model of fundamental importance in physics.

When I ask The Clay Institute and Fefferman to give a comment concerning these facts, I get zero response. I think my viewpoints are reasonable and essential and thus worthy of some form of answer.