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Existence of global weak solutions of $ p $-Navier-Stokes equations
| 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 |
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.
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D. Breit, Existence Theory for Generalized Newtonian Fluids, Academic Press, 2017.
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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. |
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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. |
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F. E. Browder,
Non-linear equations of evolution, Ann. of Math., 80 (1964), 485-523.
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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. |
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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.
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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.
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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.
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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. |
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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. |
| [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. |
| [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. |
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D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 2001. |
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P. Hartman and G. Stampacchia,
On some non-linear elliptic differential-functional equations, Acta Math., 115 (1966), 271-310.
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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.
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| [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. |
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E. H. Lieb and M. Loss, Analysis, American Mathematical Society, 2001.
doi: 10.1090/gsm/014. |
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P. Lindqvist, Notes on the Stationary $p$-Laplace Equation, Springer Briefs in Mathematics, Springer, 2019.
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| [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. |
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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. |
| [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. |
| [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. |
| [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. |
| [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. |
show all references
References:
| [1] |
D. Breit, Existence Theory for Generalized Newtonian Fluids, Academic Press, 2017.
|
| [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. |
| [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. |
| [4] |
F. E. Browder,
Non-linear equations of evolution, Ann. of Math., 80 (1964), 485-523.
doi: 10.2307/1970660. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [13] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 2001. |
| [14] |
P. Hartman and G. Stampacchia,
On some non-linear elliptic differential-functional equations, Acta Math., 115 (1966), 271-310.
doi: 10.1007/BF02392210. |
| [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.
|
| [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. |
| [17] |
E. H. Lieb and M. Loss, Analysis, American Mathematical Society, 2001.
doi: 10.1090/gsm/014. |
| [18] |
P. Lindqvist, Notes on the Stationary $p$-Laplace Equation, Springer Briefs in Mathematics, Springer, 2019.
doi: 10.1007/978-3-030-14501-9. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
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