Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Optimal distributed control of two-dimensional nonlocal
  Cahn-Hilliard-Navier-Stokes systems with degenerate mobility and singular
  potential   In this paper, we consider a two-dimensional diffuse interface model for the
phase separation of an incompressible and isothermal binary fluid mixture with
matched densities. This model consists of the Navier--Stokes equations,
nonlinearly coupled with a convective nonlocal Cahn--Hilliard equation. The
system rules the evolution of the volume-averaged velocity $\uvec$ of the
mixture and the (relative) concentration difference $\varphi$ of the two
phases. The aim of this work is to study an optimal control problem for such a
system, the control being a time-dependent external force acting on the fluid.
We first prove the existence of an optimal control for a given tracking type
cost functional. Then we study the differentiability properties of the
control-to-state map $\vvec \mapsto [\uvec,\varphi]$, and we establish
first-order necessary optimality conditions. These results generalize the ones
obtained by the first and the third authors jointly with E.~Rocca in [19].
There the authors assumed a constant mobility and a regular potential with
polynomially controlled growth. Here, we analyze the physically more relevant
case of a degenerate mobility and a singular (e.g., logarithmic) potential.
This is made possible by the existence of a unique strong solution which was
recently proved by the authors and C.\,G.~Gal in [14].

20
The categories of this paper are: acc-phys
22
The categories of this paper are: chao-dyn
13
The categories of this paper are: quant-ph
30
The categories of this paper are: q-alg
25
The categories of this paper are: patt-sol
6
The categories of this paper are: astro-ph
27
The categories of this paper are: comp-gas
0
The categories of this paper are: hep-ph
7
The categories of this paper are: hep-th
24
The categories of this paper are: dg-ga
29
The categories of this paper are: funct-an
18
The categories of this paper are: econ
12
The categories of this paper are: nucl-th
19
The categories of this paper are: eess
26
The categories of this paper are: bayes-an
23
The categories of this paper are: solv-int