Well-posedness of Isentropic Compressible Navier-stokes Equations with Low Regularity
DescriptionAs a fundamental model in classical continuum mechanics, the compressible Navier- Stokes equation is always a hot research subject in the mathematics and physics communities. The history of study to describe rigorously the ow of a compressible fluid covered a long time beginning from observations of Euler in the middle of the 18th century and of Navier and Stokes in the first half of the 19th century, and continue up to now. Despite of its importance in physics and engineering, on one hand, the existence theories of weak solutions of isentropic compressible Navier-Stokes equations asgÎ[1, n/2] remain as challenging open problems in mathematics, even though the global weak solution had been established pioneerly by Lions in his monograph and had also been extended to allowg> n/2 by Feireisl-Novotný-Petzeltová two decades ago. On the other hand, the global wellposedness near an equilibrium of compressible Navier-Stokes equations with low reg- ularity in critical spaces remains open, and hence attracts a lot of attentions in the past decades. The first topic in this proposal focuses on the concentration phenomenon and the existence of weak solutions of isentropic compressible Navier-Stokes equations in three dimensional spaces with large initial data as the adiabatic constantgÎ [1, 6/5]. This proposed problem is interesting and challenging because of the lack of aprioriestimates, and the non-compatibility between the weak convergence and the nonlinearity. Precisely due to the lack of aprioriestimates, the concentration of density would appear and it makes the compactness of the pressure hard to be established. This proposal will study the concentration of density in term of the regularity of velocity. As a consequence, an improvement on the integrability of density will be established. Moreover the “executive viscous flux" will be applied to deal with the possible oscillation of approximating solutions. The success of this proposed problem needs a careful analysis of both concentration and oscillation phenomena with the help of the relation between Riesz' potentials of density and momentum, and will in turn shed lights on understanding better the weak convergence method in compressible fluids. The second topic in this proposal aims to address the global wellposedness of compressible Navier-Stokes equations near an equilibrium with low regularity in critical spaces. Away from the initial layer, the velocity is expected to be bounded; while the density is expected to be bounded for all time. The decay mechanisms in the linearised hyperbolic- parabolic coupling system provide the necessary global control on the growth of norms of solutions. The success of this proposed problem relies on a better understanding of the smoothing effect of velocity and the damping effect of density with the help of “effective viscous fluxes", and will pave the way to the global wellposedness of other compressible models with low regular data in the critical spaces.
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