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. This work is a continuation of [6]. We study the time-asymptotic behavior of solutionsto the general Navier-Stokes equations in odd multi-dimensions. Through the pointwise estimates ofthe Green’s function of the linearized system, we obtain explicit expressions of the time-asymptoticbehavior of the solutions and exhibit the weak Huygen’s principle.Key words. compressible ow, conservation laws, general Navier-Stokes equation, multi-dimensions, Green’s function, pointwise estimateAMS subject classications. 35L65, 76N15 1. Introduction. In this paper we derive a detailed description of the asymp-totic behavior of solutions to the Cauchy problem for the general Navier-Stokes sys-tems of conservation laws in several space dimensionsρ t +div (ρv) = 0(ρv j ) t +div (ρv j v)+P(ρ,e) x j = e△v j +ηdivv x j ,(j = 1,··· ,n),(ρE) t +div(ρEv +P(ρ,e)v) = △(κT(e)+ 12 e|v| 2 )+ediv((∇v)v) +(η −e)div((divv)v),(1.1)with smooth initial data close to a constant state. Here ρ(x,t),v(x,t),e(x,t),P =P(ρ,e) and T(e) represent, respectively, the uid density, velocity, specic internalenergy, pressure and normalized temperature, and E = e +

The Pointwise Estimates of Solutions for General Navier-Stokes Systems in Odd Multi-Dimensions