Classical solutions of the Dirichlet problem for the Darcy-Forchheimer-Brinkman system

Dagmar Medková; Dagmar Medková

doi:10.3934/math.2019.6.1540

AIMS Mathematics

2019, Volume 4, Issue 6: 1540-1553. doi: 10.3934/math.2019.6.1540

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Classical solutions of the Dirichlet problem for the Darcy-Forchheimer-Brinkman system

Dagmar Medková ^,

Institute of Mathematics of the Czech Academy of Sciences, Žitná 25, 115 67 Praha 1, Czech Republic

Received: 15 May 2019 Accepted: 05 September 2019 Published: 24 September 2019
MSC : 35Q35

We study solutions of the Dirichlet problem for the Brinkman system and for the Darcy-Forchheimer-Brinkman system in the spaces of functions ${\mathcal C}^{k, \alpha }(\overline \Omega; {\mathbb R}^m)\times {\mathcal C}^{k-1, \alpha } (\overline \Omega)$, where $\Omega \subset {\mathbb R}^m$ is a bounded domain.
- Brinkman system,
- Darcy-Forchheimer-Brinkman system,
- Dirichlet problem,
- classical solution,
- regularity
Citation: Dagmar Medková. Classical solutions of the Dirichlet problem for the Darcy-Forchheimer-Brinkman system[J]. AIMS Mathematics, 2019, 4(6): 1540-1553. doi: 10.3934/math.2019.6.1540

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Abstract

We study solutions of the Dirichlet problem for the Brinkman system and for the Darcy-Forchheimer-Brinkman system in the spaces of functions ${\mathcal C}^{k, \alpha }(\overline \Omega; {\mathbb R}^m)\times {\mathcal C}^{k-1, \alpha } (\overline \Omega)$, where $\Omega \subset {\mathbb R}^m$ is a bounded domain.

References

[1]	D. A. Nield, A. Bejan, Convection in Porous Media, 4 Eds., New York: Springer, 2013.
[2]	M. Kohr, M. L. de Cristoforis, S. E. Mikhailov, et al. Integral potential method for a transmission problem with Lipschitz interface in R3 for the Stokes and Darcy-Forchheimer-Brinkman PDE systems, Z. Angew. Math. Phys., 67 (2016), 116.
[3]	M. Kohr, M. L. de Cristoforis, W. L. Wendland, On the Robin-transmission boundary value problems for the nonlinear Darcy-Forchheimer-Brinkman and Navier-Stokes systems, J. Math. Fluid Mech., 18 (2016), 293-329. doi: 10.1007/s00021-015-0236-3
[4]	M. Kohr, M. L. de Cristoforis, W. L. Wendland, Boundary value problems of Robin type for the Brinkman and Darcy-Forchheimer-Brinkman systems in Lipschitz domains, J. Math. Fluid Mech., 16 (2014), 595-630. doi: 10.1007/s00021-014-0176-3
[5]	R. Gutt, M. Kohr, S. E. Mikhailov, et al. On the mixed problem for the semilinear DarcyForchheimer-Brinkman PDE system in Besov spaces on creased Lipschitz domains, Math. Meth. Appl. Sci., 40 (2017), 7780-7829. doi: 10.1002/mma.4562
[6]	T. Grosan, M. Kohr, W. L. Wendland, Dirichlet problem for a nonlinear generalized DarcyForchheimer-Brinkman system in Lipschitz domains, Math. Meth. Appl. Sci., 38 (2015), 3615-3628. doi: 10.1002/mma.3302
[7]	R. Gutt, T. Grosan, On the lid-driven problem in a porous cavity: A theoretical and numerical approach, Appl. Math. Comput., 266 (2015), 1070-1082.
[8]	M. Kohr, M. L. de Cristoforis, W. L. Wendland, Poisson problems for semilinear Brinkman systems on Lipschitz domains in Rⁿ, Z. Angew. Math. Phys., 66 (2015), 833-864. doi: 10.1007/s00033-014-0439-0
[9]	M. Kohr, W. L. Wendland, Variational approach for the Stokes and Navier-Stokes systems with nonsmooth coefficients in Lipschitz domains on compact Riemannian manifolds, Calc. Var. Partial Differ. Eq., 57 (2018), 165.
[10]	D. Medková, Bounded solutions of the Dirichlet problem for the Stokes resolvent system, Complex Var. Elliptic Eq., 61 (2016), 1689-1715. doi: 10.1080/17476933.2016.1200565
[11]	A. Kufner, O. John, S. Fučík, Function Spaces, Prague: Academia, 1977.
[12]	E. M. Stein, Singular Integrals and Differentiability of Functions, New Jersey: Princeton University Press, 1970.
[13]	W. Varnhorn, The Stokes Equations, Berlin: Akademie Verlag, 1994.
[14]	D. Medková, The Laplace Equation, Springer, 2018.
[15]	V. A. Solonnikov, General boundary value problems for Douglas-Nirenberg elliptic systems Ⅱ, Proc. Steklov Inst. Math., 92 (1966), 212-272.
[16]	P. Maremonti, R. Russo, G. Starita, On the Stokes equations: The boundary value problem, In: P. Maremonti, Advances in Fluid Dynamics, Dipartimento di Matematica, 1999, 69-140.
[17]	C. Pozrikidis, Boundary Integral and Singularity Methods for Linearized Viscous Flow, New York: Cambridge University Press, 1992.
[18]	G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Steady State Problems, 2 Eds., Springer, 2011.
[19]	D. Gilberg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Berlin Heidelberg: Springer, 2001.
[20]	R. A. Adams, J. L. Fournier, Sobolev Spaces, 2 Eds., Oxford: Elsevier, 2003.
[21]	M. Dobrowolski, Angewandte Functionanalysis. Functionanalysis, Sobolev-Räume und elliptische Differentialgleichungen, 2 Eds., Berlin Heidelberg: Springer, 2010.
[22]	G. P. Galdi, G. Simader, H. Sohr, On the Stokes problem in Lipschitz domains, Ann. Mat. Pura Appl., 167 (1994), 147-163. doi: 10.1007/BF01760332
[23]	M. Mitrea, M. Wright, Boundary Value Problems for the Stokes System in Arbitrary Lipschitz Domains, Paris: Société mathématique de France, 2012.
[24]	C. Amrouche, V. Girault, Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension, Czech. Math. J., 44 (1994), 109-140.
[25]	D. R. Smart, Fixed Point Theorems, Cambridge: Cambridge University Press, 1974.

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