Before running a simulation for turbulence (e.g Rayleigh-Benard Convection), how do we check for well-posedness of Navier-Stokes with other equations for a given boundary condition??
Can someone point to any reference to read on this topic??
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1$\begingroup$ What do you mean by "other equations". $\endgroup$– VikramAug 28, 2019 at 9:55
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$\begingroup$ I meant energy and other advection-diffusion equations. I only want to get the general idea wrt how this kind of initial analysis is done. $\endgroup$– user162281Aug 28, 2019 at 10:11
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$\begingroup$ Compressible or incompressible? $\endgroup$– David KetchesonAug 28, 2019 at 18:37
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$\begingroup$ I imagine you know that you'll get a million dollars if you can show well-posedness of just the N-S equation by itself. Adding anything else will only make things more complicated! $\endgroup$– Wolfgang BangerthAug 28, 2019 at 21:04
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$\begingroup$ @DavidKetcheson Incompressible $\endgroup$– user162281Aug 29, 2019 at 8:44
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1 Answer
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The following paper gives a good overview of well-posedness for the compressible Navier Stokes equations.
https://link.springer.com/content/pdf/10.1007%2Fs10915-016-0303-9.pdf
The basic building block is the energy method, where you linearize the equations, then left multiply with the primitive variables, and then integrate. This gives you a norm which you check for stability. The number and form of boundary conditions can be determined from the eigenvalues of the boundary operator.
This gives a necessary condition for well-posedness, and other equations can be similarly treated. However, I am not sure if this is sufficient. There are often other norms that can be used such as entropy, but the above approach seems to be a robust recipe.
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$\begingroup$ Thank you, sir. This is what I was looking for. Thanks again. Can you share more literature with more deeper analysis relating to well-posedness problem? I would really appreciate it. $\endgroup$ Aug 29, 2019 at 6:09
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$\begingroup$ Deeper analysis is often is much more mathematical. Maybe that question would be more suitable for math stackexchange. $\endgroup$– VikramAug 29, 2019 at 8:46
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$\begingroup$ Regarding sufficiency, under the assumption of smooth solutions this approach is sufficient to ensure boundedness of solutions by the localisation, linearisation and Duhamel principles. The energy estimate alone is not sufficient to establish existence and uniqueness globally and for all times however $\endgroup$– ekkilopSep 12, 2019 at 12:26