A well-posedness result for viscous compressible fluids with only bounded density

Abstract

We are concerned with the existence and uniqueness of solutions with only bounded density for the barotropic compressible Navier-Stokes equations. Assuming that the initial velocity has slightly sub-critical regularity and that the initial density is a small perturbation (in the L ∞ norm) of a positive constant, we prove the existence of local-in-time solutions. In the case where the density takes two constant values across a smooth interface (or, more generally, has striated regularity with respect to some nondegenerate family of vector-fields), we get uniqueness. This latter result supplements the work by D. Hoff in [26] with a uniqueness statement, and is valid in any dimension d ≥ 2 and for general pressure laws.2010 Mathematics Subject Classification: 35Q35 (primary); 35B65, 76N10, 35B30, 35A02 (secondary).

Lorentz spaces in action on pressureless systems arising from models of collective behavior

“…p = 2d and r = 7/6. It is shown in [11] that those solutions satisfy the energy balance and (3.8). Since the computations of the previous step just follow from the properties of the heat flow and basic functional analysis, each (ρ n , u n ) satisfies the estimates therein.…”

“…Then, we observe that the very same arguments leading to the control of div u also allow to bound ∇u in L 1 (R + ; L ∞ (R 2 )). From that point, we follow the energy method of [11,Sec. 4] going to Lagrangian coordinates in order to prove uniqueness, and the rigorous proof of existence is obtained by compactness arguments, after constructing a sequence of smoother solutions (see the next section).…”

“…The idea is to smooth out the data, and to solve (3.1) supplemented with those data, according to the local-in-time existence result of Danchin et al [11] (that just requires the initial velocity to be smooth enough, and the initial density to be close to 1 in L ∞ ). Then, the previous steps provide uniform bounds that allow to show that those smoother solutions are actually global, and one can eventually pass to the limit.…”

“…As a first, we truncate ρ 0 and smooth out u 0 to meet the conditions of the local-intime existence theorem of [11]: we fix a sequence (ρ n 0 , u n 0 ) n∈N that converges weakly to (ρ 0 , u 0 ) and satisfy the conditions therein. Let (ρ n , u n ) n∈N be the corresponding sequence of maximal solutions, defined on [0, T n ) × R d and belonging for all T < T n to the classical maximal regularity spacė…”

“…From this stage, passing to the limit in the slightly larger (but reflexive) spacė W 2,1 5/2 (R 3 × R + ) ∩Ẇ 2,1 10/7 (R 3 × R + ) or inẆ 2,1 4/3 (R 2 × R + ) for the velocity can be done as in [11] (passing to the limit directly in the nonreflexive spacesẆ 2,1 p,( p,1) (R d × R + ) would require more care). The mass conservation equation may be handled according to Di Perna and Lions' theory [19] (see details in [11]) and the momentum equation does not present any difficulty compared to works on global weak solutions, since a lot of regularity is available on the velocity and there is no pressure term.…”