Well-posedness analysis of a stationary Navier–Stokes hemivariational inequality

Fixed Point Theory and Algorithms for Sciences and Engineering - 2021
Min Ling1, Weimin Han2
1School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, Shaanxi 710049, China
2Department of Mathematics, University of Iowa, Iowa City, IA 52242-1410, USA

Tóm tắt

AbstractThis paper provides a well-posedness analysis for a hemivariational inequality of the stationary Navier-Stokes equations by arguments of convex minimization and the Banach fixed point. The hemivariational inequality describes a stationary incompressible fluid flow subject to a nonslip boundary condition and a Clarke subdifferential relation between the total pressure and the normal component of the velocity. Auxiliary Stokes hemivariational inequalities that are useful in proving the solution existence and uniqueness of the Navier–Stokes hemivariational inequality are introduced and analyzed. This treatment naturally leads to a convergent iteration method for solving the Navier–Stokes hemivariational inequality through a sequence of Stokes hemivariational inequalities. Equivalent minimization principles are presented for the auxiliary Stokes hemivariational inequalities which will be useful in developing numerical algorithms.

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Tài liệu tham khảo

Atkinson, K., Han, W.: Theoretical Numerical Analysis: A Functional Analysis Framework, 3rd edn. Springer, New York (2009)

Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley-Interscience, New York (1983)

Fan, L., Liu, S., Gao, S.: Generalized monotonicity and convexity of non-differentiable functions. J. Math. Anal. Appl. 279, 276–289 (2003)

Fang, C., Czuprynski, K., Han, W., Cheng, X.L., Dai, X.: Finite element method for a stationary Stokes hemivariational inequality with slip boundary condition. IMA J. Numer. Anal. 40, 2696–2716 (2020)

Fang, C., Han, W.: Well-posedness and optimal control of a hemivariational inequality for nonstationary Stokes fluid flow. Discrete Contin. Dyn. Syst. 36, 5369–5386 (2016)

Fang, C., Han, W., Migórski, S., Sofonea, M.: A class of hemivariational inequalities for nonstationary Navier-Stokes equations. Nonlinear Anal., Real World Appl. 31, 257–276 (2016)

Girault, V., Raviart, P.A.: Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms. Springer, Berlin (1986)

Han, W.: Minimization principles for elliptic hemivariational inequalities. Nonlinear Anal., Real World Appl. 54, article number 103114 (2020)

Han, W.: A revisit of elliptic variational-hemivariational inequalities. Numer. Funct. Anal. Optim. 42, 371–395 (2021)

Han, W., Czuprynski, K., Jing, F.: Mixed finite element method for a hemivariational inequality of stationary Navier-Stokes equations. J. Sci. Comput. 89, article number 8 (2021)

Han, W., Sofonea, M.: Numerical analysis of hemivariational inequalities in contact mechanics. Acta Numer. 28, 175–286 (2019)

Haslinger, J., Miettinen, M., Panagiotopoulos, P.D.: Finite Element Method for Hemivariational Inequalities: Theory, Methods and Applications. Kluwer Academic, Boston (1999)

Lamb, H.: On the conditions for steady motion of a fluid. Proc. Lond. Math. Soc. s1-9(1), 91–93 (1877)

Ling, M., Han, W.: Minimization principle in study of a Stokes hemivariational inequality. Appl. Math. Lett. 121, article number 107401 (2021)

Ling, M., Wang, F., Han, W.: The nonconforming virtual element method for a stationary Stokes hemivariational inequality with slip boundary condition. J. Sci. Comput. 85, article number 56 (2020)

Migórski, S.: Hemivariational inequalities modeling viscous incompressible fluids. J. Nonlinear Convex Anal. 5, 217–227 (2004)

Migórski, S., Ochal, A.: Hemivariational inequalities for stationary Navier-Stokes equations. J. Math. Anal. Appl. 306, 197–217 (2005)

Migórski, S., Ochal, A.: Navier-Stokes problems modeled by evolution hemivariational inequalities. Discrete Contin. Dyn. Syst. 2007, 731–740 (2007)

Migórski, S., Ochal, A., Sofonea, M.: Nonlinear Inclusions and Hemivariational Inequalities: Models and Analysis of Contact Problems. Springer, New York (2013)

Naniewicz, Z., Panagiotopoulos, P.D.: Mathematical Theory of Hemivariational Inequalities and Applications. Dekker, New York (1995)

Panagiotopoulos, P.D.: Hemivariational Inequalities, Applications in Mechanics and Engineering. Springer, Berlin (1993)

Sofonea, M., Migórski, S.: Variational-Hemivariational Inequalities with Applications. Pure and Applied Mathematics. Chapman & Hall, Boca Raton (2018)

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Fixed Point Theory and Algorithms for Sciences and Engineering

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