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doi: 10.3934/dcdsb.2021051
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Existence of global weak solutions of $ p $-Navier-Stokes equations

Jian-Guo Liu 1, and  Zhaoyun Zhang 2,,

1. 

Department of Mathematics and Department of Physics, Duke University, Durham, NC 27708, USA
2. 

School of Mathematics (Zhuhai), Sun Yat-sen University, Zhuhai 519082, China, and, Department of Physics, Duke University, Durham, NC 27708, USA

* Corresponding author: Zhaoyun Zhang

Received  November 2020 Revised  January 2021 Early access February 2021

This paper investigates the global existence of weak solutions for the incompressible $ p $-Navier-Stokes equations in $ \mathbb{R}^d $ $ (2\leq d\leq p) $. The $ p $-Navier-Stokes equations are obtained by adding viscosity term to the $ p $-Euler equations. The diffusion added is represented by the $ p $-Laplacian of velocity and the $ p $-Euler equations are derived as the Euler-Lagrange equations for the action represented by the Benamou-Brenier characterization of Wasserstein-$ p $ distances with constraint density to be characteristic functions.

Keywords: p-Navier-Stokes equations, global weak solutions, p-Laplacian, monotonic operator.
Mathematics Subject Classification: Primary: 35K92, 76D03, 47J30.
Citation: Jian-Guo Liu, Zhaoyun Zhang. Existence of global weak solutions of $ p $-Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021051
References:
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D. Breit, Existence theory for stochastic power law fluids, J. Math. Fluid. Mech., 17 (2015), 295-326.  doi: 10.1007/s00021-015-0203-z.  Google Scholar

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F. E. Browder, Nonlinear elliptic boundary value problems, Bull. Amer. Math. Soc., 69 (1963), 862-874.  doi: 10.1090/S0002-9904-1963-11068-X.  Google Scholar

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K. Cheng, C. Wang and S. M. Wise, An energy stable BDF2 Fourier pseudo-spectral numerical scheme for the square phase field crystal equation, Commun. Comput. Phys., 26 (2019), 1335-1364.  doi: 10.4208/cicp.2019.js60.10.  Google Scholar

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W. Cong and J.-G. Liu, A degenerate $p$-Laplacian Keller-Segel model, Kinet. Relat. Models., 9 (2016), 687-714.  doi: 10.3934/krm.2016012.  Google Scholar

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W. Feng, A. J. Salgado, C. Wang and S. M. Wise, Preconditioned steepest descent methods for some nonlinear elliptic equations involving p-Laplacian terms, J. Comput. Phys., 334 (2017), 45-67.  doi: 10.1016/j.jcp.2016.12.046.  Google Scholar

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P. Hartman and G. Stampacchia, On some non-linear elliptic differential-functional equations, Acta Math., 115 (1966), 271-310.  doi: 10.1007/BF02392210.  Google Scholar

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L. Li and J.-G. Liu, $p$-Euler equations and $p$-Navier-Stokes equations, J. Differential Equations., 264 (2018), 4707-4748.  doi: 10.1016/j.jde.2017.12.023.  Google Scholar

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P. Lindqvist, Notes on the Stationary $p$-Laplace Equation, Springer Briefs in Mathematics, Springer, 2019. doi: 10.1007/978-3-030-14501-9.  Google Scholar

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A. Matas and J. Merker, Existence of weak solutions to doubly degenerate diffusion equations, Appl. Math., 57 (2012), 43-69.  doi: 10.1007/s10492-012-0004-0.  Google Scholar

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G. J. Minty, On a monotonicity method for the solution of nonlinear equations in Banach spaces, Proc. Nat. Acad. Sci. U.S.A., 50 (1963), 1038-1041.  doi: 10.1073/pnas.50.6.1038.  Google Scholar

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G. J. Minty, Monotone (nonlinear) operators in Hilbert space, Duke Math. J., 29 (1962), 341-346.  doi: 10.1215/S0012-7094-62-02933-2.  Google Scholar

[22]

J. Shen, C. Wang, X. Wang and S. M. Wise, Second-order convex splitting schemes for gradient flows with Ehrlich-Schwoebel type energy: Application to thin film epitaxy, SIAM J. Numer. Anal., 50 (2012), 105-125.  doi: 10.1137/110822839.  Google Scholar

[23]

J. Simon, Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar

[24]

C. Wang, X. Wang and S. M. Wise, Unconditionally stable schemes for equations of thin film epitaxy, Discrete Contin. Dyn. Syst., 28 (2010), 405-423.  doi: 10.3934/dcds.2010.28.405.  Google Scholar

show all references

References:
[1] D. Breit, Existence Theory for Generalized Newtonian Fluids, Academic Press, 2017.   Google Scholar
[2]

D. Breit, Existence theory for stochastic power law fluids, J. Math. Fluid. Mech., 17 (2015), 295-326.  doi: 10.1007/s00021-015-0203-z.  Google Scholar

[3]

F. E. Browder, Nonlinear elliptic boundary value problems, Bull. Amer. Math. Soc., 69 (1963), 862-874.  doi: 10.1090/S0002-9904-1963-11068-X.  Google Scholar

[4]

F. E. Browder, Non-linear equations of evolution, Ann. of Math., 80 (1964), 485-523.  doi: 10.2307/1970660.  Google Scholar

[5]

X. Chen, A. Jüngel and J. -G Liu, A note on Aubin-Lions-Dubinskii lemmas, Acta Appl. Math., 133 (2014), 33-43.  doi: 10.1007/s10440-013-9858-8.  Google Scholar

[6]

X. Chen and J. -G Liu, Global weak entropy solution to Doi-Saintillan-Shelley model for active and passive rod-like and ellipsoidal particle suspensions, J. Differential Equations., 254 (2013), 2764-2802.  doi: 10.1016/j.jde.2013.01.005.  Google Scholar

[7]

X. Chen and J.-G Liu, Analysis of polymeric flow models and related compactness theorems in weighted spaces, SIAM J. Math. Anal., 45 (2013), 1179-1215.  doi: 10.1137/120887850.  Google Scholar

[8]

K. Cheng, C. Wang and S. M. Wise, An energy stable BDF2 Fourier pseudo-spectral numerical scheme for the square phase field crystal equation, Commun. Comput. Phys., 26 (2019), 1335-1364.  doi: 10.4208/cicp.2019.js60.10.  Google Scholar

[9]

W. Cong and J.-G. Liu, A degenerate $p$-Laplacian Keller-Segel model, Kinet. Relat. Models., 9 (2016), 687-714.  doi: 10.3934/krm.2016012.  Google Scholar

[10]

E. DiBenedetto and M. A. Herrero, Non-negative solutions of the evolution p-Laplacian equation. Initial traces and Cauchy problem when $1 <p<2$, Arch. Rational Mech. Anal., 111 (1990), 225-290.  doi: 10.1007/BF00400111.  Google Scholar

[11]

M. Dreher and A. Jüngel, Compact families of piecewise constant functions in $L^p(0, T;B)$, Nonlinear Anal., 75 (2012), 3072-3077.  doi: 10.1016/j.na.2011.12.004.  Google Scholar

[12]

W. Feng, A. J. Salgado, C. Wang and S. M. Wise, Preconditioned steepest descent methods for some nonlinear elliptic equations involving p-Laplacian terms, J. Comput. Phys., 334 (2017), 45-67.  doi: 10.1016/j.jcp.2016.12.046.  Google Scholar

[13]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 2001.  Google Scholar

[14]

P. Hartman and G. Stampacchia, On some non-linear elliptic differential-functional equations, Acta Math., 115 (1966), 271-310.  doi: 10.1007/BF02392210.  Google Scholar

[15]

J. Leray and J.-L. Lions, Quelques résultats de Vi$\check{s}$ik sur les problèmes elliptiques nonlinéaires par les méthodes de Minty-Browder, Bull. Soc. Math. France., 93 (1965), 97-107.   Google Scholar

[16]

L. Li and J.-G. Liu, $p$-Euler equations and $p$-Navier-Stokes equations, J. Differential Equations., 264 (2018), 4707-4748.  doi: 10.1016/j.jde.2017.12.023.  Google Scholar

[17]

E. H. Lieb and M. Loss, Analysis, American Mathematical Society, 2001. doi: 10.1090/gsm/014.  Google Scholar

[18]

P. Lindqvist, Notes on the Stationary $p$-Laplace Equation, Springer Briefs in Mathematics, Springer, 2019. doi: 10.1007/978-3-030-14501-9.  Google Scholar

[19]

A. Matas and J. Merker, Existence of weak solutions to doubly degenerate diffusion equations, Appl. Math., 57 (2012), 43-69.  doi: 10.1007/s10492-012-0004-0.  Google Scholar

[20]

G. J. Minty, On a monotonicity method for the solution of nonlinear equations in Banach spaces, Proc. Nat. Acad. Sci. U.S.A., 50 (1963), 1038-1041.  doi: 10.1073/pnas.50.6.1038.  Google Scholar

[21]

G. J. Minty, Monotone (nonlinear) operators in Hilbert space, Duke Math. J., 29 (1962), 341-346.  doi: 10.1215/S0012-7094-62-02933-2.  Google Scholar

[22]

J. Shen, C. Wang, X. Wang and S. M. Wise, Second-order convex splitting schemes for gradient flows with Ehrlich-Schwoebel type energy: Application to thin film epitaxy, SIAM J. Numer. Anal., 50 (2012), 105-125.  doi: 10.1137/110822839.  Google Scholar

[23]

J. Simon, Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar

[24]

C. Wang, X. Wang and S. M. Wise, Unconditionally stable schemes for equations of thin film epitaxy, Discrete Contin. Dyn. Syst., 28 (2010), 405-423.  doi: 10.3934/dcds.2010.28.405.  Google Scholar

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