Global existence of weak solution to Navier–Stokes equations with large external force and general pressure

Abstract

We study the 3‐D compressible Navier–Stokes equations with an external potential force and a general pressure. We prove the global‐in‐time existence of weak solutions with small‐energy initial data and with densities being positive and essentially bounded. No smallness assumption is made on the external force. Copyright © 2013 John Wiley & Sons, Ltd.

Existence and a blow-up criterion of solution to the 3D compressible Navier-Stokes-Poisson equations with finite energy

“…In this present work, we try to deepen our understanding on the compressible NSP system by addressing the solutions to (1) from another new perspective, in the sense that the initial data (2) is assumed to be small in some weaker norms (L 2 ) with nonnegative and essentially bounded initial densities, and no further smallness assumption is imposed on the higher-regularity norms of the initial data. Such idea was first initiated by Hoff [6]- [8] in studying compressible Navier-Stokes system which was later extended by Suen [17]- [18] for compressible Naiver-Stokes system with potential forces as well as by Suen-Hoff [20] and Suen [19] for compressible magnetohydrodynamics (MHD). The weak solutions obtained in the present work are known as the intermediate weak solutions which enjoy the following properties:…”

“…Using (18), the first integral on the right of (63) is bounded in the same way as in Lemma 4.2 of Hoff [6]…”

“…The proof consists of a maximum-principle argument applied along particle trajectories of u which is similar to the one given in [19] pg. 48-50 for the corresponding magnetohydrodynamics system as well as those in [17]- [18] with slight modifications. We choose a positive number b ′ satisfyinḡ…”

Some Serrin type blow‐up criteria for the three‐dimensional viscous compressible flows with large external potential force

“…In the case of large initial data, Lions 5 obtained the existence of global-in-time finite energy weak solutions, yet the problem of uniqueness for those weak solutions remains completely open. In between the two types of solutions as mentioned above, a type of "intermediate weak" solutions was first suggested by Hoff in previous studies [6][7][8] and later generalized by Matsumura and Yamagata, 9 Cheung and Suen, [10][11][12][13] and other systems which include compressible magnetohydrodynamics (MHD), [14][15][16] compressible Navier-Stokes-Poisson system, 17 and chemotaxis systems. 18 Solutions as obtained in this intermediate class are less regular than those small-smooth type solutions obtained by Matsumura and Nishida 3 and Danchin 4 in such a way that the density and velocity gradient may be discontinuous across some hypersurfaces in R 3 .…”

Refined blow-up criteria for the three-dimensional viscous compressible flows with large external potential force and general pressure

“…For the case of large initial data, Lions in his seminal work [9] obtained the existence of global-in-time finite energy weak solutions, yet the problem of uniqueness for those weak solutions remains completely open. In between the two types of solutions as mentioned above, a type of "intermediate weak" solutions were suggested by Hoff [6][7][8] and later generalized by Matsumura and Yamagata [13], Suen in [3,18,19] and other systems which include compressible magnetohydrodynamics (MHD) [15,16,22], compressible Navier-Stokes-Poisson system [21] and chemotaxis systems [11]. The initial data for intermediate weak solutions are of small energy, and in general they are not necessary smooth compared with classical solutions.…”