Robustness of strong solutions to the compressible Navier-Stokes system

Abstract

We consider the Navier-Stokes system describing the time evolution of a compressible barotropic fluid confined to a bounded spatial domain in the 3-D physical space, supplemented with the Navier's slip boundary conditions. It is shown that the class of global in time strong solutions is robust with respect to small perturbations of the initial data. Explicit qualitative estimates are given also in terms of the shape of the underlying physical domain, with applications to problems posed on thin cylinders.

Mass transport in multicomponent compressible fluids: Local and global well-posedness in classes of strong solutions for general class-one models

“…In the latter approach one uses the conservation law structure of the system to derive a priori bounds for higher derivatives of the solution in L 2 . For several variants of the method in the case of Navier-Stokes or Navier-Stokes-Fourier, see [MN83,Val82,Val83,VZ86], [Sol95], or also [CCK04,Hof12,BFJ15] and references. Usually, somewhat more regularity of the domain and the coefficients is demanded because it is necessary to differentiate the equations several times.…”

“…• Robustness estimates on bounded time intervals are to be found in Theorem 1 of [BFJ15], as well as more recent references in the stability discussion. • The additional regularity required in Theorem 2.4 for the data is sufficient, but might be not optimal.…”

Necessary conditions for distributed optimal control of linearized compressible Navier-Stokes equations with state constraint

“…A number of authors (cf. [2,5]) have observed that often more regularity is needed for the base state of the linearization. Collis et al [8] similarly observe that "...the solution of the linearized state equation is less regular than the state .…”

“…2 ) in order to absorb the estimates (60) into the left-hand side of (59). Having chosen this ε, inserting the estimates (60) into (59) we get|∂ t u| 2 + |divS(∇u)| 2 dxds…”