Existence of weak solutions for unsteady motions of generalized Newtonian fluids

Diening, Lars; Růžička, Michael; Wolf, Jörg

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 9 (2010) no. 1, pp. 1-46.

We prove the existence of weak solutions $𝐮:Q_T\to {ℝ}^n$ of the equations of unsteady motion of an incompressible fluid with shear-dependent viscosity in a cylinder $Q_T=\Omega \times (0,T)$, where $\Omega \subset {ℝ}^n$ denotes a bounded domain. Under the assumption that the extra stress tensor $𝐒$ possesses a $q$-structure with $q>\frac{2n}{n+2}$, we are able to construct a weak solution $𝐮\in L^q(0,T;W_0^{1,q}\left(\Omega \right))\cap C_w([0,T];L^2\left(\Omega \right))$ with $div𝐮=0$. Our approach is based on the Lipschitz truncation method, which is new in this context.

Classification : 76D03, 35D05, 35D46, 34A34

[1] E. Acerbi and N. Fusco, Semicontinuity problems in the calculus of variations, Arch. Ration. Mech. Anal. 86 (1984), 125–145. | MR | Zbl

[2] H. Amann, Stability of the rest state of a viscous incompressible fluid, Arch. Ration. Mech. Anal. 126 (1994), 231–242. | MR | Zbl

[3] G. Astarita and G. Marucci, “Principles of non–Newtonian Fluid Mechanics”, McGraw-Hill, London, 1974.

[4] G. K. Batchelor, “An Introduction to Fluid Dynamics”, Cambridge University Press, Cambridge, 1967. | MR

[5] H. Bellout, F. Bloom and J. Nečas, Young measure-valued solutions for non-Newtonian incompressible fluids, Comm. Partial Differential Equations 19 (1994), 1763–1803. | MR | Zbl

[6] R. B. Bird, R. C. Armstrong and O. Hassager, “Dynamic of Polymer Liquids”, John Wiley, 1987, 2nd edition.

[7] D. Bothe and J. Prüss, $L^p$-theory for a class of non-Newtonian fluids, SIAM J. Math. Anal. 39 (2007), 379–421. | MR | Zbl

[8] D. Cioranescu, Quelques exemples de fluides newtoniens generalisés, In: “Mathematical Topics in Fluid Mechanics” (Lisbon, 1991) (Harlow), Pitman Res. Notes Math. Ser., Vol. 274, Longman Sci. Tech., Harlow, 1992, 1–31.

[9] R. R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), 569–645. | MR | Zbl

[10] H. Beirão Da Veiga, On the regularity of flows with Ladyzhenskaya shear-dependent viscosity and slip or nonslip boundary conditions, Comm. Pure Appl. Math. 58 (2005), 552–577. | MR | Zbl

[11] H. Beirão Da Veiga, P. Kaplický and M. Růžička, Boundary regularity of shear-thickening flows, J. Math. Fluid Mech. (2010), to appear. | MR | Zbl

[12] M. De Guzmán, “Differentiation of Integrals in $R^n$”, Springer-Verlag, Berlin, 1975, with appendices by Antonio Córdoba, and Robert Fefferman, and two by Roberto Moriyón, Lecture Notes in Mathematics, Vol. 481. | MR

[13] L. Diening, J. Málek and M. Steinhauer, On Lipschitz truncations of Sobolev functions (with variable exponent) and their selected applications, ESAIM Control Optim. Calc. Var. 14 (2008), 211–232, DOI: 10.1051/cocv:2007049. | EuDML | Numdam | MR | Zbl

[14] L. Diening and M. Růžička, Strong solutions for generalized Newtonian fluids, J. Math. Fluid Mech. 7 (2005), 413–450. | MR | Zbl

[15] J. Frehse, J. Málek and M. Steinhauer, An existence result for fluids with shear dependent viscosity – steady flows, Nonlinear Anal. 30 (1997), 3041–3049. | MR | Zbl

[16] J. Frehse, J. Málek and M. Steinhauer, On existence result for fluids with shear dependent viscosity – unsteady flows, In: “Partial Differential Equations”, W. Jäger, J. Nečas, O. John, K. Najzar and J. Stará (eds.), Chapman and Hall, 2000, 121–129. | Zbl

[17] J. Frehse, J. Málek and M. Steinhauer, On analysis of steady flows of fluids with shear-dependent viscosity based on the Lipschitz truncation method, SIAM J. Math. Anal. 34 (2003), 1064–1083 (electronic). | MR | Zbl

[18] M. Fuchs, On stationary incompressible Norton fluids and some extensions of Korn’s inequality, Z. Anal. Anwendungen 13 (1994), 191–197. | MR | Zbl

[19] G. P. Galdi, “An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Linearized Steady Problems”, Tracts in Natural Philosophy, Vol. 38, Springer, New York, 1994. | MR | Zbl

[20] G. P. Galdi, C. G. Simader and H. Sohr, On the Stokes problem in Lipschitz domains, Ann. Mat. Pura Appl. (4) 167 (1994), 147–163. | MR | Zbl

[21] D. Gilbarg and N. S. Trudinger, “Elliptic Partial Differential Equations of Second Order”, Springer, Berlin, 2001, Reprint of the 1998 edition. | MR | Zbl

[22] E. Hopf, Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichugen, Math. Nachr. 4 (1951), 213–231. | MR | Zbl

[23] R. R. Huilgol, “Continuum Mechanics of Viscoelastic Liquids”, Hindustan Publishing Corporation, Delhi, 1975. | MR | Zbl

[24] J. Kinnunen and J. L. Lewis, Very weak solutions of parabolic systems of $p$-Laplacian type, Ark. Mat. 40 (2002), 105–132. | MR | Zbl

[25] O. A. Ladyzhenskaya, New equations for the description of motion of viscous incompressible fluids and solvability in the large of boundary value problems for them, Proc. Steklov Inst. Math. 102 (1967), 95–118.

[26] O. A. Ladyzhenskaya, Modifications of the Navier-Stokes equations for large gradients of the velocities, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 7 (1968), 126–154.

[27] O. A. Ladyzhenskaya, On some modifications of the Navier-Stokes equations for large gradients of velocity, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI) 7 (1968), 126–154. | MR

[28] O. A. Ladyzhenskaya, “The Mathematical Theory of Viscous Incompressible Flow”, Gordon and Breach, New York, 1969, 2nd edition. | MR | Zbl

[29] J. Leray, Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Math. 63 (1934), 193–248. | MR

[30] J. L. Lions, “Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires”, Dunod, Paris, 1969. | MR | Zbl

[31] J. Málek, J. Nečas and A. Novotný, Measure-valued solutions and asymptotic behavior of a multipolar model of a boundary layer, Czechoslovak Math. J. 42 (117) (1992), 549–576. | EuDML | MR | Zbl

[32] J. Málek, J. Nečas, M. Rokyta and M. Růžička, “Weak and Measure-valued Solutions to Evolutionary PDEs”, Applied Mathematics and Mathematical Computations, Vol. 13, Chapman & Hall, London, 1996. | MR | Zbl

[33] J. Málek, J. Nečas and M. Růžička, On the non-Newtonian incompressible fluids, Math. Models Methods Appl. Sci. 3 (1993), 35–63. | MR | Zbl

[34] J. Málek, J. Nečas and M. Růžička, On Weak Solutions to a Class of Non-Newtonian Incompressible Fluids in Bounded Three–dimensional Domains. The Case $p\ge 2$, Adv. Differential Equations 6 (2001), 257–302. | MR | Zbl

[35] J. Málek and K. R. Rajagopal, Mathematical issues concerning the Navier-Stokes equations and some of its generalizations, Handb. Differential Equations, Elsevier/North-Holland, Amsterdam, 2005, 371–459. | MR | Zbl

[36] J. Málek, K. R. Rajagopal and M. Růžička, Existence and regularity of solutions and the stability of the rest state for fluids with shear dependent viscosity, Math. Models Methods Appl. Sci. 5 (1995), 789–812. | MR | Zbl

[37] J. Malý and W. P. Ziemer, “Fine Regularity of Solutions of Elliptic Partial Differential Equations”, Mathematical Surveys and Monographs, Vol. 51, American Mathematical Society, Providence, RI, 1997. | MR | Zbl

[38] J. Naumann and M. Wolff, Interior integral estimates on weak solutions of nonlinear parabolic systems, 1994, preprint Humboldt University.

[39] J. Nečas, Sur le normes équivalentes dans $W_p^k\left(\Omega \right)$ et sur la coercivité des formes formellement positives, Séminaire Equations aux Dérivées Partielles, Montreal 317 (1966), 102–128.

[40] J. Nečas, “Les Méthodes Directes en la Thèorie des Equations Elliptiques”, Academia, Praha, 1967. | MR

[41] J. Nečas, Theory of multipolar viscous fluids, In: “The Mathematics of Finite Elements and Applications”, VII (Uxbridge, 1990) (London), Academic Press, London, 1991, 233–244. | MR | Zbl

[42] C. Parés, Existence, uniqueness and regularity of solution of the equations of a turbulence model for incompressible fluids, Appl. Anal. 43 (1992), 245–296. | MR | Zbl

[43] G. Da Prato, Spazi ${ℒ}^{(p,\theta )}(\Omega ,\delta )$ e loro proprietà, Ann. Mat. Pura Appl. (4) 69 (1965), 383–392. | MR | Zbl

[44] K. R. Rajagopal, Mechanics of Non-Newtonian Fluids, In: “Recent Developments in Theoretical Fluid Mechanics”, G. P. Galdi and J. Nečas (eds.), Research Notes in Mathematics Series, Vol. 291, Longman, 1993, 129–162. | MR | Zbl

[45] M. Růžička, Flow of shear dependent electrorheological fluids: unsteady space periodic case, In: “Applied Nonlinear Analysis”, A. Sequeira (ed.), Kluwer/Plenum, New York, 1999, 485–504. | MR | Zbl

[46] M. Růžička, Modeling, mathematical and numerical analysis of electrorheological fluids, Appl. Math. 49 (2004), 565–609. | EuDML | MR | Zbl

[47] P. Shvartsman, On extensions of Sobolev functions defined on regular subsets of metric measure spaces, J. Approx. Theory 144 (2007), 139–161. | MR | Zbl

[48] E. M. Stein, “Singular Integrals and Differentiability Properties of Functions”, Princeton University Press, Princeton, N.J., 1970. | MR | Zbl

[49] M. Struwe, On the Hölder continuity of bounded weak solutions of quasilinear parabolic systems, Manuscripta Math. 35 (1981), 125–145. | EuDML | MR | Zbl

[50] C. Truesdell and W. Noll, “The Non-Linear Field Theories of Mechanics”, Handbuch der Physik, Vol. III/3, Springer, New York, 1965. | MR | Zbl

[51] J. Wolf, “Regularität schwacher Lösungen elliptischer und parabolischer Systeme partieller Differentialgleichungen mit Entartung. Der Fall $1<p<2$”, Dissertation, Humboldt Universität, 2001.

[52] J. Wolf, Existence of weak solutions to the equations of nonstationary motion of non-Newtonian fluids with shear-dependent viscosity, J. Math. Fluid Mech. 9 (2007), 104–138. | MR | Zbl