Leray-Hopf solutions to a viscoelastoplastic fluid model with nonsmooth stress-strain relation. (English) Zbl 1481.35310
Summary: We consider a fluid model including viscoelastic and viscoplastic effects. The state is given by the fluid velocity and an internal stress tensor that is transported along the flow with the Zaremba-Jaumann derivative. Moreover, the stress tensor obeys a nonlinear and nonsmooth dissipation law as well as stress diffusion. We prove the existence of global-in-time weak solutions satisfying an energy inequality under general Dirichlet conditions for the velocity field and Neumann conditions for the stress tensor.
MSC:
| 35Q30 | Navier-Stokes equations |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
| 76A10 | Viscoelastic fluids |
| 35Q35 | PDEs in connection with fluid mechanics |
| 76D03 | Existence, uniqueness, and regularity theory for incompressible viscous fluids |
Keywords:
viscoelastic fluid; stress diffusion; viscoplasticity; inhomogeneous time-dependent boundary values; weak solutions; energy inequalityReferences:
| [1] | Moresi, L.; Dufour, F.; Mühlhaus, H.-B., Mantle convection modeling with viscoelastic/brittle lithosphere: Numerical methodology and plate tectonic modeling, Pure Appl. Geophys., 159, 2335-2356 (2002) |
| [2] | Gerya, T.; Yuen, D. A., Robust characteristics method for modelling multiphase visco-elasto-plastic thermo-mechanical problems, Phys. Earth Plan. Inter., 163, 1-4, 83-105 (2007) |
| [3] | Herrendörfer, R.; Gerya, T.; van Dinther, Y., An invariant rate- and state-dependent friction formulation for viscoeastoplastic earthquake cycle simulations, J. Geophys. Research: Solid Earth, 123, 5018-5051 (2017) |
| [4] | Preuss, S.; Herrendörfer, R.; Gerya, J.-P.; Ampuero, Taras.; van Dinther, Y., Seismic and aseismic fault growth lead to different fault orientations, J. Geophys. Res. Solid Earth, 124, 8867-8889 (2019) |
| [5] | Joseph, D. D.; Renardy, M.; Saut, J.-C., Hyperbolicity and change of type in the flow of viscoelastic fluids, Arch. Ration. Mech. Anal., 87, 213-251 (1985) · Zbl 0572.76011 |
| [6] | Renardy, M.; Renardy, Y., Linear stability of place Couette flow of an upper convected Maxwell fluid, J. Non-Newton. Fluid Mech., 22, 23-33 (1986) · Zbl 0608.76006 |
| [7] | Renardy, M.; Hrusa, W. J.; Nohel, J. A., Mathematical Problems in Viscoelasticity (1987), Longman Sci. & Techn. J. Wiley & Sons, Inc. · Zbl 0719.73013 |
| [8] | Cook, L. P.; Schleiniger, G., The inlet layer in the flow of viscoelastic fluids, J. Non-Newton. Fluid Mech., 40, 3, 307-321 (1991) · Zbl 0743.76012 |
| [9] | Renardy, M., Mathematical analysis of viscoelastic flows, CBMS-NSF Reg, (Conf. Ser. Appl. Math. 73 (2000), SIAM) · Zbl 0956.76001 |
| [10] | Lions, P. L.; Masmoudi, N., Global solutions for some oldroyd models of non-Newtonian flows, Chinese Ann. Math. Ser. B, 21, 2, 131-146 (2000) · Zbl 0957.35109 |
| [11] | Cai, Y.; Lei, Z.; Lin, F.; Masmoudi, N., Vanishing viscosity limit for incompressible viscoelasticity in two dimensions, Comm. Pure Appl. Math., 72, 10, 2063-2120 (2019) · Zbl 1439.35391 |
| [12] | Bulíček, M.; Gwiazda, P.; Málek, J.; Świerczewska-Gwiazda, A., On unsteady flows of implicitly constituted incompressible fluids, SIAM J. Math. Anal., 44, 4, 2756-2801 (2012) · Zbl 1256.35074 |
| [13] | Blechta, J.; Málek, J.; Rajagopal, K. R., On the classification of incompressible fluids and a mathematical analysis of the equations that govern their motion, SIAM J. Math. Anal., 52, 2, 1232-1289 (2020) · Zbl 1432.76075 |
| [14] | Bulíček, M.; Málek, J.; Průša, V.; Süli, E., PDE analysis of a class of thermodynamically compatible viscoelastic rate-type fluids with stress-diffusion, (Mathematical Analysis in Fluid Mechanics—Selected Recent Results, 710 of Contemp. Math (2018), Amer. Math. Soc.), 25-51 · Zbl 1404.35346 |
| [15] | Málek, J.; Průša, V.; Skřivan, T.; Süli, E., Thermodynamics of viscoelastic rate-type fluids with stress diffusion, Phys. Fluids, 30, 1-23 (2018), 023101 |
| [16] | Bathory, M.; Bulíček, M.; Málek, J., Large data existence theory for three-dimensional unsteady flows of rate-type viscoelastic fluids with stress diffusion, Adv. Nonlin. Analysis, 10, 1, 501-521 (2021) · Zbl 1464.35207 |
| [17] | Galdi, G. P., An Introduction To the Mathematical Theory of the Navier-Stokes Equations. Steady-State Problems (2011), Springer: Springer New York · Zbl 1245.35002 |
| [18] | Roubíček, T., (Nonlinear Partial Differential Equations with Applications. Nonlinear Partial Differential Equations with Applications, 153 of International Series of Numerical Mathematics (2013), Birkhäuser/Springer Basel AG: Birkhäuser/Springer Basel AG Basel) |
| [19] | Bauschke, H. H.; Combettes, P. L., Convex analysis and monotone operator theory in Hilbert spaces, (Attouch, Hédy, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC (2017), Springer: Springer Cham) · Zbl 1359.26003 |
| [20] | Antman, S. S., Physically unacceptable viscous stresses, Z. Angew. Math. Phys., 49, 6, 980-988 (1998) · Zbl 0928.74007 |
| [21] | Farwig, R.; Galdi, G. P.; Sohr, H., A new class of weak solutions of the Navier-Stokes equations with nonhomogeneous data, J. Math. Fluid Mech., 8, 3, 423-444 (2006) · Zbl 1104.35032 |
| [22] | Farwig, R.; Kozono, H.; Sohr, H., Global weak solutions of the Navier-Stokes system with nonzero boundary conditions, Funkcial. Ekvac., 53, 2, 231-247 (2010) · Zbl 1291.35030 |
| [23] | Farwig, R.; Kozono, H.; Sohr, H., Global leray-hopf weak solutions of the Navier-Stokes equations with nonzero time-dependent boundary values, (Parabolic Problems, 80 of Progr. Nonlinear Differential Equations Appl (2011), Birkhäuser/Springer Basel AG: Birkhäuser/Springer Basel AG Basel), 211-232 · Zbl 1247.35090 |
| [24] | Farwig, R.; Kozono, H.; Sohr, H., Global weak solutions of the Navier-Stokes equations with nonhomogeneous boundary data and divergence, Rend. Semin. Mat. Univ. Padova, 125, 51-70 (2011) · Zbl 1236.35103 |
| [25] | Galdi, G. P., An introduction to the Navier-Stokes initial-boundary value problem, (Fundamental Directions in Mathematical Fluid Mechanics. Fundamental Directions in Mathematical Fluid Mechanics, Adv. Math. Fluid Mech (2000), Birkhäuser: Birkhäuser Basel), 1-70 · Zbl 1108.35133 |
| [26] | Grubb, G., Nonhomogeneous Dirichlet Navier-Stokes problems in low regularity \(L_p\) Sobolev spaces, J. Math. Fluid Mech., 3, 1, 57-81 (2001) · Zbl 0992.35065 |
| [27] | Fursikov, A.; Gunzburger, M.; Hou, L., Trace theorems for three-dimensional, time-dependent solenoidal vector fields and their applications, Trans. Amer. Math. Soc., 354, 3, 1079-1116 (2002) · Zbl 0988.46024 |
| [28] | Raymond, J.-P., Stokes and Navier-Stokes equations with nonhomogeneous boundary conditions, Ann. Inst. H. PoincarÉ Anal. Non Linéaire, 24, 6, 921-951 (2007) · Zbl 1136.35070 |
| [29] | Sohr, H., The Navier-STokes Equations, BirkhäUser Advanced Texts: Basler Lehrbücher. [BirkhäUser Advanced Texts: Basel Textbooks] (2001), Birkhäuser Verlag: Birkhäuser Verlag Basel · Zbl 0983.35004 |
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