Existence of suitable weak solutions to the Navier-Stokes equations in time varying domains. (English) Zbl 1386.35145
Summary: We consider suitable weak solutions to the Navier-Stokes equations in time varying domains. We develop Schauder theory for the approximate Stokes equations in time varying domains whose solutions satisfy a uniform localized energy estimate including boundary. Existence of suitable weak solutions in time varying domains follows from compactness in Lebesgue and Sobolev spaces.
MSC:
| 35K20 | Initial-boundary value problems for second-order parabolic equations |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35D30 | Weak solutions to PDEs |
References:
| [1] | Bergmann, M.; Hovnanian, J.; Iollo, A., An accurate Cartesian method for incompressible flows with moving boundaries, Commun. Comput. Phys., 15, 5, 1266-1290 (2014) · Zbl 1373.76156 |
| [2] | Caffarelli, L.; Kohn, R.; Nirenberg, L., Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math., 35, 771-831 (1982) · Zbl 0509.35067 |
| [3] | Choe, H. J.; Lewis, J., On the singular set in the Navier-Stokes equations, J. Funct. Anal., 175, 348-369 (2000) · Zbl 0965.35115 |
| [4] | Conca, C.; Carlos; San Martn, J.; Tucsnak, M., Existence of solutions for the equations modelling the motion of a rigid body in a viscous fluid, Comm. Partial Differential Equations, 25, 5-6, 1019-1042 (2000) · Zbl 0954.35135 |
| [5] | Desjardins, B.; Esteban, M. J., Existence of weak solutions for the motion of rigid bodies in a viscous fluid, Arch. Ration. Mech. Anal., 146, 1, 59-71 (1999) · Zbl 0943.35063 |
| [6] | Duarte, F.; Gormaz, R.; Natesan, S., Arbitrary Lagrangian-Eulerian method for Navier-Stokes equations with moving boundaries, Comput. Methods Appl. Mech. Engrg., 193, 45-47, 4819-4836 (2004) · Zbl 1112.76388 |
| [7] | Feireisl, E., On the motion of rigid bodies in a viscous incompressible fluid, J. Evol. Equ., 3, 3, 419-441 (2003) · Zbl 1039.76071 |
| [8] | Fujita, H.; Sauer, N., On existence of weak solutions of the Navier-Stokes equations in regions with moving boundaries, J. Fac. Sci. Univ. Tokyo Sect. I, 17, 403-420 (1970) · Zbl 0206.39702 |
| [9] | Giga, Y.; Sohr, H., Abstract \(L^P\) estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains, J. Funct. Anal., 102, 72-94 (1991) · Zbl 0739.35067 |
| [10] | Gilbarg, D.; Trudinger, N. S., Elliptic Partial Differetial Equations of Second Order (2001), Springer-Verlag: Springer-Verlag Berlin · Zbl 0691.35001 |
| [11] | Gunzburger, M. D.; Lee, H. C.; Seregin, G. A., Global existence of weak solutions for viscous incompressible flows around a moving rigid body in three dimensions, J. Math. Fluid Mech., 2, 3, 219-266 (2000) · Zbl 0970.35096 |
| [12] | Gustafson, S.; Kang, K.; Tsai, T. P., Regularity criteria for suitable weak solutions of the Navier-Stokes equations near the boundary, J. Differential Equations, 226, 2, 594-618 (2006) · Zbl 1159.35396 |
| [13] | Gustafson, S.; Kang, K.; Tsai, T. P., Interior regularity criteria for suitable weak solutions of the Navier-Stokes equations, Comm. Math. Phys., 273, 1, 161-176 (2007) · Zbl 1126.35042 |
| [14] | Ladyzenskaja, O. A., An initial-boundary value problem for the Navier-Stokes equations in domains with boundary changing in time, Zap. Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 11, 97-128 (1968), (in Russian) · Zbl 0205.39201 |
| [15] | Ladyzhenskaya, O. A.; Seregin, G. A., On partial regularity of suitable weak solutions to the three-dimensional Navier-Stokes equations, J. Math. Fluid Mech., 1, 4, 356-387 (1999) · Zbl 0954.35129 |
| [16] | Legendre, G.; Takahashi, T., Convergence of a Lagrange-Galerkin method for a fluid-rigid body system in ALE formulation, M2AN Math. Model. Numer. Anal., 42, 4, 609-644 (2008) · Zbl 1142.76032 |
| [17] | Lin, F., A new proof of the Caffarelli, Kohn and Nirenberg theorem, Comm. Pure Appl. Math., 51, 241-257 (1998) · Zbl 0958.35102 |
| [18] | Neustupa, J., Existence of a weak solution to the Navier-Stokes equation in a general time varying domain by the Rothe method, Math. Methods Appl. Sci., 32, 6, 653-683 (2009) · Zbl 1160.35494 |
| [19] | Scheffer, V., Hausdorff measure and the Navier-Stokes equations, Comm. Math. Phys., 55, 97-112 (1977) · Zbl 0357.35071 |
| [20] | Seregin, G. A., Local regularity of suitable weak solutions to the Navier-Stokes equations near the boundary, J. Math. Fluid Mech., 4, 1, 1-29 (2002) · Zbl 0997.35044 |
| [21] | Seregin, G. A.; Shilkin, T. N.; Solonnikov, V. A., Boundary partial regularity for the Navier-Stokes equations, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 310 (2004), Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 35 [34], 158-190, 228 ; reprinted in J Math. Sci. (NY) 132 (2006), no. 3, 339-358 · Zbl 1095.35031 |
| [22] | Sohr, H.; von Wahl, W., On the regularity of the pressure of weak solutions of Navier-Stokes equations, Arch. Math., 46, 428-439 (1986) · Zbl 0574.35070 |
| [23] | Solonnikov, V. A., Estimates of the solutions of the nonstationary Navier-Stokes system, (Russian) Boundary value problems of mathematical physics and related questions in the theory of functions, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 38, 153-231 (1973) · Zbl 0346.35083 |
| [24] | Starovoǐtov, V. N., On the nonuniqueness of the solution of the problem of the motion of a rigid body in a viscous incompressible fluid, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 306 (2003), Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funktsii., 34 (2005), 199-209, 231-232 (in Russian); translation in J. Math. Sci. (N.Y.) 130 (4) (2005) 4893-4898 · Zbl 1148.35347 |
| [25] | Takahashi, T., Analysis of strong solutions for the equations modeling the motion of a rigid-fluid system in a bounded domain, Adv. Differential Equations, 8, 12, 1499-1532 (2003) · Zbl 1101.35356 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
