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As the title says, I am looking for a authoritative reference/monograph on this topic. My interest is in spectral properties of this PDE, and NOT on existence/uniqueness etc. which is usually the focus of most theoretical PDE texts. NOT on numerical methods to solve the PDE, which is usually the focus of engineering/physics PDE texts.

My motivation is to understand the various functional analytic aspects of this PDE under varying conditions: e.g. change in the spectrum when changing the advective component from something "regular" to chaotic, dependence of spectrum on bounded Vs unbounded flows, dependence on diffusivity etc.

I have been trying to read recent literature on such equations, and I haven't found a single solid source which has all the basic results on spectral properties of such equations.

Is there a recent book or monograph which focuses on spectral theory of (linear or nonlinear) parabolic PDEs in general, or advection-diffusion equations in particular ?

Thanks.

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  • $\begingroup$ Your question is way too broad. Do you mean something like this: rd.springer.com/chapter/10.1007%2F978-3-0348-8545-4_10 or rd.springer.com/article/10.1007%2FBF01211592 ? $\endgroup$
    – András Bátkai
    Commented Jul 31, 2013 at 21:26
  • $\begingroup$ @Adras: I am looking for compilation of results similar to what you have linked to, but for advection-diffusion equations (ADE) (as applied in fluid mechanics). I guess there is whole lot more literature on Schrodinger's equation, than the "true" ADE with real coefficients. $\endgroup$
    – Piyush Grover
    Commented Jul 31, 2013 at 22:28
  • $\begingroup$ What do you mean by spectral properties? $\endgroup$
    – timur
    Commented Aug 4, 2013 at 0:21
  • $\begingroup$ @Timur: By spectral properties, I mean the various inferences that can be made about the spectrum of the advection-diffusion operator (i.e.L(f)= $u(x).\nabla f(x) + K\nabla^2 f(x) $) on the suitable function space such as $L^2(\Omega)$ where $\Omega\in R^n, x\in \Omega$, $u(x)$ is the advective velocity field. $\endgroup$
    – Piyush Grover
    Commented Aug 4, 2013 at 23:23

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I am not sure whether there has been a comprehensive study of spectral properties of such equations, but you may be interested in the paper ``Diffusion and Mixing in Fluid Flow'' by Constantin, Kiselev, Ryzhik and Zlatos and the references therein.

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.76.6407

An online version can be found here:

http://math.stanford.edu/~ryzhik/ckrz.pdf

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  • $\begingroup$ Thanks for this reference. I haven't this one before, and it seems very few of the papers from the applied fluid mixing community have referred to it. $\endgroup$
    – Piyush Grover
    Commented Aug 4, 2013 at 23:29
  • $\begingroup$ After skimming through this paper, and spending some time tracking the references, I think I now have plenty of material on topic. Very helpful lead. $\endgroup$
    – Piyush Grover
    Commented Aug 5, 2013 at 21:15

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