| Abstract: |
| This paper talk I will discuss on unique solvability of some inverse source problems for the Navier-Stokes equations. The inverse problem consists in recovering a time-dependent source coefficient (intensity of external forces) from additional information on the velocity field given in an integral form over a spatial domain. We investigate this problem in three settings: the linear case, the nonlinear case, and a special case of the right-hand side. For the linear problem, we establish global-in-time existence, uniqueness, and stability results for both weak and strong solutions under suitable assumptions on the data. For the nonlinear problem, we consider this in two- and three-dimensional cases and prove the local-in-time (for sufficiently small data) existence and uniqueness of strong solutions. In addition, we analyze a special case of the right-hand side that allows to improve these results, in particular, in the two-dimensional case we prove the global-in-time unique solvability of the nonlinear problem. |
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