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In which kinds of PDEs are the more interesting function spaces required? I am thinking of spaces such as Besov and Triebel spaces, and their weighted versions.

For example, Sobolev spaces $L^2(0,T;H^1)$ can be used in linear parabolic PDEs. If we weaken this to consider certain parabolic PDEs with monotone operators then we can need to use spaces like $L^p(0,T;W^{1,q}).$ For degenerate problems, one can use weighted Sobolev spaces.

Where next? And what to read to get a good understand of the next steps? I would be grateful for any answers.

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Spatial weights would be relevant in non-homogeneous settings in which one expects the behaviour at different regions of space to be different. For instance, if there is an obstacle or a boundary, a weight that depends on the distance to the boundary would be natural in order to capture boundary effects. If the initial data is originally assumed to be concentrated at the origin, then weights involving the distance $|x|$ to the origin are also natural. Similarly, weights involving time $t$ are sometimes natural in evolution equations, particularly if one is trying to describe decay or blowup in time.

More generally, if there is a natural singular set in physical space or frequency space, then it is natural to weight one's spaces around that set. The $X^{s.b}$ spaces mentioned in Willie's answer are a good example of this in the frequency domain (and Sobolev spaces themselves reflect the privileged nature of the frequency origin for many PDE, as the zero set for the symbol of the underlying linear operator (e.g. the Laplacian)).

If one needs to prevent the solution from concentrating all its mass or energy into a ball, then Morrey or Campanato spaces are occasionally useful.

As for the frequency-based refinements to Sobolev spaces (e.g. Besov and Triebel-Lizorkin, but also Hardy spaces, BMO, BV, etc.), these are "within logarithms" of Sobolev spaces, in the sense that if the ratio between the finest and coarsest spatial scale of interest (or equivalently, the ratio between the highest and lowest frequency scale of interest) is comparable to $N$, then the ratio between a Besov or Triebel-Lizorkin norm and its Sobolev counterpart (as plotted for instance on this type diagram: http://terrytao.wordpress.com/2010/03/11/a-type-diagram-for-function-spaces/ ) is at most a power of $\log N$. Because of this, Sobolev spaces generally suffice for all "subcritical" or "non-endpoint" situations in which one does not have to contend with a logarithmic pileup of contributions from each scale. If one is working in a critical setting (which is more or less the same thing as a scale-invariant or a dimensionless setting), these refinements can often be necessary to stop the logarithmic divergences caused by such things as the failure of the endpoint Sobolev inequality, e.g. $H^{n/2}({\bf R}^n) \not \subset L^\infty({\bf R}^n)$. (In this particular case, one can sometimes replace the Sobolev space $H^{n/2}$ with the smaller Besov space $B^{n/2}_1$ to recover the endpoint embedding, although there is no free lunch here and this will likely make some other estimate in one's analysis harder to prove.)

In general, unless one is perturbing off of an existing method, one does not proceed by randomly picking function spaces and hoping that one's argument closes. Often the function spaces one ends up using are dictated by trying to directly estimate solutions (or approximations to solutions). For instance, if one is trying to establish a local well-posedness result for a semilinear evolution equation in some standard space, e.g. $H^s({\bf R}^n)$, one can try to expand the $H^s$ norm of that solution using a basic formula such as the Duhamel formula or the energy inequality. In trying to estimate the terms arising from that formula by harmonic analysis methods (e.g. Holder inequality, Sobolev embedding, etc.), one is naturally led to the need to control the solution in other norms as well. If all goes well, all the norms on the right-hand side can be controlled by what already has on the left-hand side plus the initial data, and then one has a good chance of closing an argument; if not, one often has to tweak the argument by either strengthening or weakening the norms one is trying to control, as dictated by what the harmonic analysis is telling you. The final norms one uses to close the argument often arise from a lengthy iteration of this procedure (which unfortunately is often hidden from view in the published version of the paper, which usually focuses on the final choice of spaces that worked, rather than the initial guesses which didn't quite work but needed to be perturbed into the final choice).

Ultimately, in PDE one is usually more interested in the functions themselves, rather than the function spaces (though there are exceptions, e.g. if one is taking a dynamical systems perspective, or is relying on a fixed-point theorem exploiting the global topology of the function space). The reason that function spaces appear so prominently in PDE arguments is that functions have an infinite number of degrees of freedom, and the basic physical features of such functions (e.g. amplitude, frequency, location) are not easy to define directly in a precise and rigorous fashion. Function space norms serve as mathematically rigorous proxies for these physical statistics, but in the end they are only formal tools (with the exception of some physically natural norms or norm-like quantities, such as the mass or energy) and one should really be thinking about the physical features of the solution to the PDE directly. I discuss this point at http://terrytao.wordpress.com/2010/04/02/amplitude-frequency-dynamics-for-semilinear-dispersive-equations/ in the setting of semilinear dispersive equations (but there are similar perspectives for other PDE also).

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Is "occasionally useful" used in the sense of Whitehead and Russel? :-) Anyway, I really like the description of Besov spaces; it really helps clarify some ideas I had in the back of my head but could never put into words. –  Willie Wong Sep 23 '13 at 9:14
    
Thank you! A great answer. –  user35613 Sep 30 '13 at 15:56
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In many aspects of dispersive PDEs, the "optimal" function spaces are those adapted to the symbol of the linear evolution. They were introduced by Bourgain for the nonlinear Schroedinger equation (and also the KdV equation in part II of that paper) and Klainerman-Machedon for the nonlinear wave equations. In the literature you will usually see them called Bourgain spaces or $X^{s,b}$ spaces in the case of NLS or KdV or other dispersive equations, or $H^{s,\delta}$ spaces or wave-Sobolev spaces in the case of the wave equation. These types of spaces are particularly well adapted for proving multilinear product estimates are are very useful of the study of semilinear PDEs. See this paper of D'Ancona-Foschi-Selberg for the type of things one can do with these spaces.

(The rough idea is the following: Suppose we have a constant coefficient partial differential operator $P(D)$ (where $P$ is a polynomial), and we are interested in solving the equation $P(D) u = f$. Formally we can take the Fourier transform on both sides and study $P(i\xi) \hat{u} = \hat{f}$ and to solve we just "divide": $\hat{u} = \hat{f} / P(i\xi)$. From just this simple notion we see that to relate regularity/integrability properties of $\hat{u}$ to that of $\hat{f}$, we need to understand how $\hat{f}$ behaves on the singular set $\{ P(i\xi) = 0 \}$ and its neighborhoods. The Bourgain and wave-Sobolev spaces are adapted in the sense that they are functions spaces that are weighted in the Fourier side with weights capturing the singular behaviour of $P(i\xi)$. [In a very rough sense we can say that the classical homogeneous Sobolev spaces $\dot{H}^s$ are "adapted" to the Laplacian for which $P(i\xi) = - |\xi|^2$ vanishes only at the origin.])

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Thank you Willie! –  user35613 Sep 30 '13 at 15:58
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For example:

  • In the search for 'optimal' function spaces to prove (long time) existence for Navier-Stokes equations, Besov spaces are one of the directions which has been / is currently explored with positive answers and negative answers for a nested family of such spaces, see e.g. Tataru's work.
  • In regularity theory for systems of second order elliptic equations (such as linear elasticity in inhomogeneous media) where the maximum principle cannot be used, Sobolev based ideas arguments are too rough to prove regularity via bootstrap. Morrey-Campanato spaces are better suited, see e.g. Troianiello, Elliptic di fferential equations and obstacle problems 1987.
  • Finer Sobolev embedding theorems can be written in Besov spaces, see e.g. Tartar, An Introduction to Sobolev Spaces and Interpolation Spaces 2000.
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Thanks very much! –  user35613 Sep 30 '13 at 16:03
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I've heard that regularity theory for $p$-Laplacian equations, i.e. equations of the form $$ \mathrm{div}(c_1|\nabla u|^{p-2}\nabla u) + c_2 |u|^{p-2} = f $$ with $c_1,c_2 > 0$, $p>2$ need Besov spaces to describe the maximal regularity. See Jacques Simon, Régularité de la solution d'une équation non linéaire dans $\mathbb{R}^N$.

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Thanks a lot!!! –  user35613 Sep 30 '13 at 16:05
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