Partial regularity of the solutions to a turbulent problem in porous media
HTML articles powered by AMS MathViewer
- by H. B. de Oliveira and A. Paiva PDF
- Proc. Amer. Math. Soc. 147 (2019), 3961-3981 Request permission
Abstract:
A one-equation turbulent model that is being used with success in the applications to model turbulent flows through porous media is studied in this work. We consider the classical Navier–Stokes equations, with feedback forces fields, coupled with the equation for the turbulent kinetic energy (TKE) through the turbulence production term and through the turbulent and the diffusion viscosities. Under suitable growth conditions on the feedback functions involved in the model, we prove the local higher integrability of the gradient solutions to the steady version of this problem.References
- Emilio Acerbi and Giuseppe Mingione, Regularity results for stationary electro-rheological fluids, Arch. Ration. Mech. Anal. 164 (2002), no. 3, 213–259. MR 1930392, DOI 10.1007/s00205-002-0208-7
- Alain Bensoussan and Jens Frehse, Regularity results for nonlinear elliptic systems and applications, Applied Mathematical Sciences, vol. 151, Springer-Verlag, Berlin, 2002. MR 1917320, DOI 10.1007/978-3-662-12905-0
- M. E. Bogovskiĭ, Solutions of some problems of vector analysis, associated with the operators $\textrm {div}$ and $\textrm {grad}$, Theory of cubature formulas and the application of functional analysis to problems of mathematical physics, Trudy Sem. S. L. Soboleva, No. 1, vol. 1980, Akad. Nauk SSSR Sibirsk. Otdel., Inst. Mat., Novosibirsk, 1980, pp. 5–40, 149 (Russian). MR 631691
- Françoise Brossier and Roger Lewandowski, Impact of the variations of the mixing length in a first order turbulent closure system, M2AN Math. Model. Numer. Anal. 36 (2002), no. 2, 345–372. MR 1906822, DOI 10.1051/m2an:2002016
- L. Consiglieri and T. Shilkin, Regularity to stationary weak solutions in the theory of generalized Newtonian fluids with energy transfer, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 271 (2000), no. Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 31, 122–150, 316 (English, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 115 (2003), no. 6, 2771–2788. MR 1810613, DOI 10.1023/A:1023369819312
- Tomás Chacón Rebollo and Roger Lewandowski, Mathematical and numerical foundations of turbulence models and applications, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser/Springer, New York, 2014. MR 3288092, DOI 10.1007/978-1-4939-0455-6
- P. Dreyfuss, Results for a turbulent system with unbounded viscosities: weak formulations, existence of solutions, boundedness and smoothness, Nonlinear Anal. 68 (2008), no. 6, 1462–1478. MR 2388827, DOI 10.1016/j.na.2006.12.040
- Pierre-Étienne Druet and Joachim Naumann, On the existence of weak solutions to a stationary one-equation RANS model with unbounded eddy viscosities, Ann. Univ. Ferrara Sez. VII Sci. Mat. 55 (2009), no. 1, 67–87. MR 2506064, DOI 10.1007/s11565-009-0062-8
- G. P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations, 2nd ed., Springer Monographs in Mathematics, Springer, New York, 2011. Steady-state problems. MR 2808162, DOI 10.1007/978-0-387-09620-9
- T. Gallouët, J. Lederer, R. Lewandowski, F. Murat, and L. Tartar, On a turbulent system with unbounded eddy viscosities, Nonlinear Anal. 52 (2003), no. 4, 1051–1068. MR 1941245, DOI 10.1016/S0362-546X(01)00890-2
- F. W. Gehring, The $L^{p}$-integrability of the partial derivatives of a quasiconformal mapping, Acta Math. 130 (1973), 265–277. MR 402038, DOI 10.1007/BF02392268
- M. Giaquinta and G. Modica, Regularity results for some classes of higher order nonlinear elliptic systems, J. Reine Angew. Math. 311(312) (1979), 145–169. MR 549962
- Mariano Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies, vol. 105, Princeton University Press, Princeton, NJ, 1983. MR 717034
- Enrico Giusti, Direct methods in the calculus of variations, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. MR 1962933, DOI 10.1142/9789812795557
- J. Lederer and R. Lewandowski, A RANS 3D model with unbounded eddy viscosities, Ann. Inst. H. Poincaré C Anal. Non Linéaire 24 (2007), no. 3, 413–441 (English, with English and French summaries). MR 2321200, DOI 10.1016/j.anihpc.2006.03.011
- M.J.S. de Lemos, Turbulence in Porous Media, Second Edition, Elsevier, Waltham, MA, 2012.
- Roger Lewandowski, The mathematical analysis of the coupling of a turbulent kinetic energy equation to the Navier-Stokes equation with an eddy viscosity, Nonlinear Anal. 28 (1997), no. 2, 393–417. MR 1418142, DOI 10.1016/0362-546X(95)00149-P
- J. Málek, J. Nečas, M. Rokyta, and M. Růžička, Weak and measure-valued solutions to evolutionary PDEs, Applied Mathematics and Mathematical Computation, vol. 13, Chapman & Hall, London, 1996. MR 1409366, DOI 10.1007/978-1-4899-6824-1
- Norman G. Meyers and Alan Elcrat, Some results on regularity for solutions of non-linear elliptic systems and quasi-regular functions, Duke Math. J. 42 (1975), 121–136. MR 417568
- B. Mohammadi and O. Pironneau, Analysis of the $k$-epsilon turbulence model, RAM: Research in Applied Mathematics, Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994. MR 1296252
- Joachim Naumann and Jörg Wolf, On Prandtl’s turbulence model: existence of weak solutions to the equations of stationary turbulent pipe-flow, Discrete Contin. Dyn. Syst. Ser. S 6 (2013), no. 5, 1371–1390. MR 3039704, DOI 10.3934/dcdss.2013.6.1371
- Hermenegildo Borges de Oliveira, A note on the existence for a model of turbulent flows through porous media, Differential and difference equations with applications, Springer Proc. Math. Stat., vol. 230, Springer, Cham, 2018, pp. 21–38. MR 3829883, DOI 10.1007/978-3-319-75647-9_{2}9
- H. B. de Oliveira and A. Paiva, On a one-equation turbulent model with feedbacks, Differential and difference equations with applications, Springer Proc. Math. Stat., vol. 164, Springer, [Cham], 2016, pp. 51–61. MR 3571713, DOI 10.1007/978-3-319-32857-7_{5}
- H. B. de Oliveira and A. Paiva, A stationary one-equation turbulent model with applications in porous media, J. Math. Fluid Mech. 20 (2018), no. 2, 263–287. MR 3808571, DOI 10.1007/s00021-017-0325-6
- H. B. de Oliveira and A. Paiva, Existence for a one-equation turbulent model with strong nonlinearities, J. Elliptic Parabol. Equ. 3 (2017), no. 1-2, 65–91. MR 3736848, DOI 10.1007/s41808-017-0005-y
- E. W. Stredulinsky, Higher integrability from reverse Hölder inequalities, Indiana Univ. Math. J. 29 (1980), no. 3, 407–413. MR 570689, DOI 10.1512/iumj.1980.29.29029
Additional Information
- H. B. de Oliveira
- Affiliation: FCT - Universidade do Algarve, Faro, Portugal; and CMAFCIO - Universidade de Lisboa, Portugal
- MR Author ID: 747453
- Email: holivei@ualg.pt
- A. Paiva
- Affiliation: FCT - Universidade do Algarve, Faro, Portugal
- MR Author ID: 1204242
- Received by editor(s): June 8, 2018
- Received by editor(s) in revised form: January 10, 2019
- Published electronically: June 14, 2019
- Additional Notes: The first author was partially supported by Grant SFRH/BSAB/135242/2017 and by the Project UID/MAT/04561/2013, both from the Portuguese Foundation for Science and Technology (FCT), Portugal.
- Communicated by: Catherine Sulem
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 3961-3981
- MSC (2010): Primary 76F60, 76S05, 35J57, 35B65, 76D03
- DOI: https://doi.org/10.1090/proc/14545
- MathSciNet review: 3993789