Reference request: Spectral analysis of advection diffusion PDE

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.

• 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 ? – András Bátkai Jul 31 '13 at 21:26
• @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. – Piyush Grover Jul 31 '13 at 22:28
• What do you mean by spectral properties? – timur Aug 4 '13 at 0:21
• @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. – Piyush Grover Aug 4 '13 at 23:23

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.
{"ft_lang_label":"__label__en","ft_lang_prob":0.9327161,"math_prob":0.85510445,"size":991,"snap":"2019-43-2019-47","text_gpt3_token_len":203,"char_repetition_ratio":0.1023303,"word_repetition_ratio":0.02631579,"special_character_ratio":0.18365288,"punctuation_ratio":0.101123594,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9514273,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-10-17T06:42:41Z\",\"WARC-Record-ID\":\"<urn:uuid:cbe6e13a-4096-4729-a989-c47f48d0f6a4>\",\"Content-Length\":\"121670\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:954ca22a-f9d0-4939-8085-e44af94479da>\",\"WARC-Concurrent-To\":\"<urn:uuid:1aad8f95-9a86-42f5-9db7-b2f36802b190>\",\"WARC-IP-Address\":\"151.101.193.69\",\"WARC-Target-URI\":\"https://mathoverflow.net/questions/138273/reference-request-spectral-analysis-of-advection-diffusion-pde\",\"WARC-Payload-Digest\":\"sha1:SDE537JJDQACD7MSBU3T4IHJBZH667TZ\",\"WARC-Block-Digest\":\"sha1:C2OI4SEOHYZFJOZL2QOF5TM4SZF3BHUV\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-43/CC-MAIN-2019-43_segments_1570986672723.50_warc_CC-MAIN-20191017045957-20191017073457-00194.warc.gz\"}"}