## Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations of state and discontinuous initial data.(English)Zbl 0836.76082

Summary: We extend to general polytropic pressures $$P (\rho) = K \rho^\gamma$$, $$\gamma > 1$$, the existence theory for isothermal $$(\gamma = 1)$$ flows of Navier-Stokes fluids in two and three space dimensions, with fairly general initial data. Specifically, we require that the initial density is close to a constant in $$L^2$$ and $$L^\infty$$, and that the initial velocity is small in $$L^2$$ and bounded in $$L^{2^n}$$ (in two dimensions the $$L^2$$ norms must be weighted slightly). Solutions are obtained as limits of approximate solutions corresponding to mollified initial data. The key point is that the approximate densities are shown to converge strongly, so that nonlinear pressures can be accommodated, even in the absence of any uniform regularity information for the approximate densities.

### MSC:

 76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics 35Q30 Navier-Stokes equations

### Keywords:

existence; Navier-Stokes fluids; approximate densities
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### References:

 [1] R. J. DiPerna & P.-L. Lions, Ordinary differential equations, transport theory, and Sobolev spaces, Inventories Math. 98 (1989), 511-547. · Zbl 0696.34049 [2] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 1977. · Zbl 0361.35003 [3] D. Hoff, Construction of solutions for compressible, isentropic Navier-Stokes equations in one space dimension with nonsmooth initial data, Proc. Roy. Soc. Edinburgh A 103 (1986), 301-315. · Zbl 0635.35074 [4] D. Hoff, Global existence for 1D, compressible, isentropic Navier-Stokes equations with large initial data, Trans. Amer. Math. Soc. 303 (1987), 169-181. · Zbl 0656.76064 [5] D. Hoff, Discontinuous solutions of the Navier-Stokes equations for compressible flow, Arch. Rational Mech. Anal. 114 (1991), 15-46. · Zbl 0732.35071 [6] D. Hoff, Global wellposedness of the Cauchy problem for nonisentropic gas dynamics with discontinuous initial data, J. Diff. Eqs. 95 (1992), 33-73. · Zbl 0762.35085 [7] D. Hoff, Spherically symmetric solutions of the Navier-Stokes equations for compressible, isothermal flow with large, discontinuous initial data, Indiana Univ. Math. J. 41 (1992), 1225-1302. · Zbl 0765.35033 [8] D. Hoff, Global solutions of the Navier-Stokes equations for multidimensional, compressible flow with discontinuous initial data, to appear in J. Diff. Eqs. · Zbl 0836.35120 [9] P.-L. Lions, Existence globale de solutions pour les équations de Navier-Stokes compressibles isentropiques, C. R. Acad. Sci. Paris 316 (1993), 1335-1340. · Zbl 0778.76086 [10] D. Serre, Solutions faibles globales des équations de Navier-Stokes pour un fluide compressible, C.R. Acad. Sci. Paris 303 (1986), 639-642. · Zbl 0597.76067 [11] D. Serre, Sur l’équation monodimensionnelle d’un fluide visqueux, compressible et conducteur de chaleur, C. R. Acad. Sci. Paris 303 (1986), 703-706. · Zbl 0611.35070 [12] D. Serre, Variations de grande amplitude pour la densité d’un fluide visqueux compressible, Physica D 48 (1991), 113-128. · Zbl 0739.35071 [13] V. V. Shelukhin, On the structure of generalized solutions of the one dimensional equations of a polytropic viscous gas, Prikl. Mat. Mekh. 48 (1984), 912-920. · Zbl 0591.76128
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