According to customary wisdom, ever since Functional Analysis, and specifically ever since Sobolev spaces have been used in solving PDEs starting in the 1930s, the all pervading and overriding mentality is that PDEs and even the linear ones can only be solved by particular methods which must be tuned to the particular types of the respective PDEs, and which types prove to be rather countless. Thus a general enough and type independent solution method for PDEs is considered to be simply impossible. Is that indeed the situation? Or it is only the situation so far, since there are no well organized attempts to set up general, type independent solution methods for PDEs? Seemingly, for more than two decades now, there have been two such rather different attempts at general, type independent solution methods which, perhaps, should be worth subjecting to a critical analysis. Details can be found in arxiv;math/0407026, and in reference [5] in arxiv:math/0703515.
The theory of differential equations deals with one independent variable(ordinary) and with greater than one independent variable (partial)
The question is not about ordinary differential equations which only have one single independent variable. No, the question is about ARBITRARY partial differential equations which can any finite number of independent variables. So that, so sorry, I do not quite understand what you write ...
Interesting question! For clarification, are you referring to a general theory on whether or not a solution exists for an arbitrary PDE or a broad notion of type-independent conditions by which one may characterize the set of solutions to many PDEs?
The closest thing to a type-independent theory is in Lars Hormander's Analysis of Linear Partial Differential Operators I–IV. Even there, there is a distinction between elliptic, hypo-elliptic and hyperbolic PDE's which do require different approaches. In this way, the local ODE existence/uniqueness theory is simpler.
To Jason : I mean a nonlinear type independent theory for the existence and regularity of solutions for PDEs. An example in the particular analytic case is the classical Cauchy-Kovalevskaia theorem. This theorem is for all nonlinear systems of PDEs. However, it is far from being general enough since it is only for analytic PDEs., Also, it does not give in general a solution on the whole domain of definition of the respective PDEs.
To David : Please, read carefully the question : it not only about linear PDEs, which is the case in the book of Hormander you mention. No, the question is about as large a class of nonlinear systems of PDEs, with possibly associated initial and boundary value problems.
From what I have seen in your paper and your mention of the Cauchy-Kovalevskaia theorem, I take it that initial and/or boundary data would be assumed in a general theory. One result that might be of interest to you is arxiv.org/abs/1109.2184. It shows that one may determine if a semigroup (closely related to a possibly nonlinear IVP) is global or local by considering a linear eigenvalue problem involving the Lie generator. In the case of local semigroups a stop time function related to the maximal domain of existence for given initial data is also discussed.
Another approach is to not restrict the theory to initial/boundary conditions. This goes back to my original question. I view the problem of characterizing the set of solutions as that of finding the right conditions, possibly involving non-boundary data, for a given domain, as opposed to finding the domain for which given conditions have a solution. Whether that is a different problem or a different perspective goes back to your suggestion:
"...there are no well organized attempts to set up general, type independent solution methods for PDEs."
What is the precise class of PDEs to be considered? Are any additional conditions assumed? When is a function deemed to be a solution? The quest for a general type-independent theory depends on how these questions are answered, and I do -not- consider it to be a pointless endeavor.
Dear Jason, Thank you for your reply. I happen to know personally that author of arxiv.org/abs/1109.2184, Prof. Neuberger, and keep up an occasional contact with him. His mentioned paper is not intended, and it is not about as general PDEs as possible. He did some time ago such a theory, however, even that one was not general enough, since it used Hilbert spaces.
Amusingly, it appears that the very issue of "general, type independent, nonlinear systems of PDEs" is hard to understand after more than two centuries of only and only solving highly particular PDEs, and doing so with highly particular methods :-) :-) :-)
So that, first perhaps, it should be clarified what can one mean by "general, type independent, nonlinear systems of PDEs" ???
Back in 1994, together with a junior collaborator, we published the research monograph "Solution of Order Continuous Nonlinear PDEs through Order Completion", North Holland Mathematics Studies, Vol. 181. The mentioned general method of Neuberger is, so far, the only other such general method I happen to know about.
But to be precise, the results in the mentioned book have been massively extended by my former doctoral student and present colleague Dr. Jan-Harm van der Walt in a number of his papers published in the last decade.
Now, precisely due to the fact that even the very issue at hand seems not easy to grasp, one cannot be surprised if there are no other such attempts, or even if they are, well, they are simply lost in the immensity of publications ... :-) :-) :-)
This is why I try to bring up the issue : perhaps, someone may jump up and say that, well, there are other such attempts as well ... :-) :-) :-)
Anyhow, I can be contacted at [email protected]
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