Geodesic
Estimates of the distance to the solution of an evolutionary problem obtained by linearization of the Navier--Stokes equation
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 48, Tome 489 (2020), pp. 67-80

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The paper is concerned with a linearization of the Navier–Stokes equation in the space-time cylinder $Q_T$. The main goal is to deduce computable estimates of the distance between the exact solution and a function in the energy admissible class of vector valued functions. First, the estimates are derived for the case, where this class contains only divergence free (solenoidal) functions. In the next section, estimates of the distance to sets of divergence free functions depending on the space and time variables are considered. These results are used to extend earlier derived estimates to non–solenoidal approximations. The corresponding estimates contain an additional term, which can be viewed as a penalty for the violation of the divergence free condition. 
@article{ZNSL_2020_489_a3,
     author = {S. Repin},
     title = {Estimates of the distance to the solution of an evolutionary problem obtained by linearization of the {Navier--Stokes} equation},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {67--80},
     publisher = {mathdoc},
     volume = {489},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2020_489_a3/}
}
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S. Repin. Estimates of the distance to the solution of an evolutionary problem obtained by linearization of the Navier--Stokes equation. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 48, Tome 489 (2020), pp. 67-80. http://geodesic.mathdoc.fr/item/ZNSL_2020_489_a3/