Abstract
This paper is dedicated to the study of viscous compressible barotropic fluids in dimension N ≧ 2. We address the question of the global existence of strong solutions for initial data close to a constant state having critical Besov regularity. First, this article shows the recent results of Charve and Danchin (Arch Ration Mech Anal 198(1):233–271, 2010) and Chen et al. (Commun Pure Appl Math 63:1173–1224, 2010) with a new proof. Our result relies on a new a priori estimate for the velocity that we derive via the intermediary of the effective velocity, which allows us to cancel out the coupling between the density and the velocity as in Haspot (Well-posedness in critical spaces for barotropic viscous fluids, 2009). Second, we improve the results of Charve and Danchin (2010) and Chen et al. (2010) by adding as in Charve and Danchin (2010) some regularity on the initial data in low frequencies. In this case we obtain global strong solutions for a class of large initial data which rely on the results of Hoff (Arch Rational Mech Anal 139:303–354, 1997), Hoff (Commun Pure Appl Math 55(11):1365–1407, 2002), and Hoff (J Math Fluid Mech 7(3):315–338, 2005) and those of Charve and Danchin (2010) and Chen et al. (2010). We conclude by generalizing these results for general viscosity coefficients.
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Abidi H., Paicu M.: Équation de Navier–Stokes avec densité et viscosité variables dans l’espace critique.. Annales de l’institut Fourier 57(3), 883–917 (2007)
Bahouri H., Chemin J.-Y.: Équations d’ondes quasilinéaires et estimation de Strichartz. Am. J. Math. 121, 1337–1377 (1999)
Bahouri, H., Chemin, J.-Y., Danchin, R.: Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der mathematischen Wissenschaften, vol. 343. Springer, Berlin (2011)
Bony J.-M.: Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires. Annales Scientifiques de l’école Normale Supérieure. 14, 209–246 (1981)
Bresch D., Desjardins B.: Existence of global weak solutions for a 2D Viscous shallow water equations and convergence to the quasi-geostrophic model. Commun. Math. Phys. 238(1–2), 211–223 (2003)
Bresch D., Desjardins B.: Sur un modèle de Saint-Venant visqueux et sa limite quasi-géostrophique. C. R. Math. Acad. Sci. Paris 335(12), 1079–1084 (2002)
Charve F., Danchin R.: A global existence result for the compressible Navier–Stokes equations in the critical Lp framework. Arch. Ration. Mech. Anal. 198(1), 233–271 (2010)
Chemin J.-Y.: Théorèmes d’unicité pour le système de Navier–Stokes tridimensionnel. J. Anal. Math. 77, 27–50 (1999)
Chemin, J.-Y.: About Navier–Stokes system. Prépublication du Laboratoire d’Analyse Numérique de Paris 6, R96023 (1996)
Chemin J.-Y., Lerner N.: Flot de champs de vecteurs non lipschitziens et équations de Navier–Stokes. J. Differ. Equ. 121, 314–328 (1992)
Chen Q., Miao C., Zhang Z.: Global well-posedness for the compressible Navier–Stokes equations with the highly oscillating initial velocity. Commun. Pure Appl. Math. 63, 1173–1224 (2010)
Danchin, R.: Fourier analysis methods for PDE’s (Preprint 2005)
Danchin R.: Local Theory in critical Spaces for compressible viscous and heat-conductive gases. Commun. Partial Differ. Equ. 26(78), 1183–1233 (2001)
Danchin R.: Global existence in critical spaces for compressible Navier–Stokes equations. Invent. Math. 141, 579–614 (2000)
Danchin R.: Global existence in critical spaces for compressible viscous and heat-conductive gases. Archiv. Ration. Mech. Anal. 160, 1–39 (2001)
Danchin R.: On the uniqueness in critical spaces for compressible Navier–Stokes equations. Nonlinear Differ. Equ. Appl. 12(1), 111–128 (2005)
Danchin R.: Well-posedness in critical spaces for barotropic viscous fluids with truly not constant density. Commun. Partial Differ. Equ. 32(9), 1373–1397 (2007)
Haspot B.: Cauchy problem for Navier–Stokes system with a specific term of capillarity. M3AS 20(7), 1–39 (2010)
Haspot, B.: Cauchy problem for viscous shallow water equations with a term of capillarity. Hyperbolic problems: theory, numerics and applications, pp. 625–634. In: Proceedings of Symposia in Applied Mathematics, vol. 67, Part 2. American Mathematical Society, Providence, 2009
Haspot, B.: Local well-posedness for density-dependent incompressible fluids with non-Lipschitz velocity. Annales de l’Institut Fourier (to appear)
Haspot, B.: Well-posedness in critical spaces for the system of compressible Navier–Stokes in larger spaces. J. Differ. Equ. (to appear)
Haspot, B.: Regularity of weak solution for compressible barotropic Navier–Stokes equations. Arxiv (January 2010) and preprint
Hoff D.: Global existence for 1D, compressible, isentropic Navier–Stokes equations with large initial data. Trans. Am. Math. Soc. 303(1), 169–181 (1987)
Hoff, D.: Uniqueness of weak solutions of the Navier–Stokes equations of multidimensional, compressible flow. SIAM J. Math. Anal. 37(6), (2006)
Hoff D.: Discontinuous solutions of the Navier–Stokes equations for multidimensional flows of the heat conducting fluids. Arch. Ration. Mech. Anal. 139, 303–354 (1997)
Hoff D.: Global solutions of the Navier–Stokes equations for multidimensional compressible flow with discontinuous initial data. J. Differ. Equ. 120(1), 215–254 (1995)
Hoff D.: Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations of state and discontinuous initial data. Arch. Rational Mech. Anal. 132(1), 1–14 (1995)
Hoff D.: Dynamics of singularity surfaces for compressible, viscous flows in two space dimensions. Commun. Pure Appl. Math. 55(11), 1365–1407 (2002)
Hoff D.: Compressible flow in a half-space with Navier-boundary conditions. J. Math. Fluid Mech. 7(3), 315–338 (2005)
Hoff D., Santos M.: Lagrangean structure and propagation of singularities in multidimensional compressible flow. Arch. Ration. Mech. Anal. 188(3), 509–543 (2008)
Hoff D., Zumbrun K.: Multi-dimensional diffusion waves for the Navier–Stokes equations of compressible flow. Indian. Univ. Math. J. 44, 603–676 (1995)
Kazhikov A.V.: The equation of potential flows of a compressible viscous fluid for small Reynolds numbers: existence, uniqueness and stabilization of solutions. Sibirsk. Mat. Zh. 34(3), 70–80 (1993)
Kazhikov A.V., Shelukhin V.V.: Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas. Prikl. Mat. Meh. 41(2), 282–291 (1977)
Lions, P.-L.: Mathematical Topics in Fluid Mechanics, Compressible models, vol. 2. Oxford University Press, Oxford, 1998
Matsumura A., Nishida T.: The initial value problem for the equations of motion of compressible viscous and heat-conductive gases. J. Math. Kyoto Univ. 20(1), 67–104 (1980)
Matsumura A., Nishida T.: The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids. Proc. Japan Acad. Ser. A Math. Sci. 55(9), 337–342 (1979)
Meyer, Y.: Wavelets, paraproducts, and Navier–Stokes equation. In: Current Developments in Mathematics 1996 (Cambridge, MA), pp. 105–212. Int. Press, Boston, 1997
Nash J.: Le problème de Cauchy pour les équations différentielles d’un fluide général. Bull. Soc. Math. France 90, 487–497 (1962)
Runst, T., Sickel, W.: Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations. de Gruyter Series in Nonlinear Analysis and Applications, 3. Walter de Gruyter and Co., Berlin 1996
Serre D.: Solutions faibles globales des équations de Navier–Stokes pour un fluide compressible. Comptes rendus de l’Académie des sciences Série 1 303(13), 639–642 (1986)
Solonnikov V.A.: Estimates for solutions of nonstationary Navier–Stokes systems. Zap. Nauchn. Sem. LOMI 38, 153–231 (1973)
Solonnikov V.A.: Estimates for solutions of nonstationary Navier–Stokes systems. J. Soviet Math. 8, 467–529 (1977)
Valli V., Zajaczkowski W.: Navier–Stokes equations for compressible fluids: global existence and qualitative properties of the solutions in the general case. Commun. Math. Phys. 103(2), 259–296 (1986)
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Communicated by C. Le Bris
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Haspot, B. Existence of Global Strong Solutions in Critical Spaces for Barotropic Viscous Fluids. Arch Rational Mech Anal 202, 427–460 (2011). https://doi.org/10.1007/s00205-011-0430-2
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DOI: https://doi.org/10.1007/s00205-011-0430-2