Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Exponential Decay for Lions-Feireisl's Weak Solutions to the Barotropic
  Compressible Navier-Stokes Equations in 3D Bounded Domains   For barotropic compressible Navier-Stokes equations in three-dimensional (3D)
bounded domains, we prove that any finite energy weak solution obtained by
Lions [Mathematical topics in fluid mechanics, Vol. 2. Compressible
models(1998)] and Feireisl-Novotn\'{y}-Petzeltov\'{a} [J. Math. Fluid Mech.
3(2001), 358-392] decays exponentially to the equilibrium state. This result is
established by both using the extra integrability of the density due to Lions
and constructing a suitable Lyapunov functional just under the framework of
Lions-Feireisl's weak solutions.

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The categories of this paper are: hep-ex
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The categories of this paper are: q-bio
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