# American Institute of Mathematical Sciences

## 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

* Corresponding author: Zhaoyun Zhang

Received  November 2020 Revised  January 2021 Published  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.

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
 [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 $1 show all references ##### References:  [1] D. Breit, Existence Theory for Generalized Newtonian Fluids, Academic Press, 2017. Google Scholar [2] D. Breit, Existence theory for stochastic power law fluids, J. Math. Fluid. Mech., 17 (2015), 295-326. doi: 10.1007/s00021-015-0203-z. Google Scholar [3] F. E. Browder, Nonlinear elliptic boundary value problems, Bull. Amer. Math. Soc., 69 (1963), 862-874. doi: 10.1090/S0002-9904-1963-11068-X. Google Scholar [4] F. E. Browder, Non-linear equations of evolution, Ann. of Math., 80 (1964), 485-523. doi: 10.2307/1970660. Google Scholar [5] X. Chen, A. Jüngel and J. -G Liu, A note on Aubin-Lions-Dubinskii lemmas, Acta Appl. Math., 133 (2014), 33-43. doi: 10.1007/s10440-013-9858-8. Google Scholar [6] X. Chen and J. -G Liu, Global weak entropy solution to Doi-Saintillan-Shelley model for active and passive rod-like and ellipsoidal particle suspensions, J. Differential Equations., 254 (2013), 2764-2802. doi: 10.1016/j.jde.2013.01.005. Google Scholar [7] X. Chen and J.-G Liu, Analysis of polymeric flow models and related compactness theorems in weighted spaces, SIAM J. Math. Anal., 45 (2013), 1179-1215. doi: 10.1137/120887850. Google Scholar [8] K. Cheng, C. Wang and S. M. Wise, An energy stable BDF2 Fourier pseudo-spectral numerical scheme for the square phase field crystal equation, Commun. Comput. Phys., 26 (2019), 1335-1364. doi: 10.4208/cicp.2019.js60.10. Google Scholar [9] W. Cong and J.-G. Liu, A degenerate$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
 [1] Bernd Kawohl, Jiří Horák. On the geometry of the p-Laplacian operator. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 799-813. doi: 10.3934/dcdss.2017040 [2] Sophia Th. Kyritsi, Nikolaos S. Papageorgiou. Positive solutions for p-Laplacian equations with concave terms. Conference Publications, 2011, 2011 (Special) : 922-930. doi: 10.3934/proc.2011.2011.922 [3] Leyun Wu, Pengcheng Niu. Symmetry and nonexistence of positive solutions to fractional p-Laplacian equations. Discrete & Continuous Dynamical Systems, 2019, 39 (3) : 1573-1583. doi: 10.3934/dcds.2019069 [4] Adam Lipowski, Bogdan Przeradzki, Katarzyna Szymańska-Dębowska. Periodic solutions to differential equations with a generalized p-Laplacian. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2593-2601. doi: 10.3934/dcdsb.2014.19.2593 [5] Shuang Wang, Dingbian Qian. Periodic solutions of p-Laplacian equations via rotation numbers. Communications on Pure & Applied Analysis, 2021, 20 (5) : 2117-2138. doi: 10.3934/cpaa.2021060 [6] Dimitri Mugnai. Bounce on a p-Laplacian. Communications on Pure & Applied Analysis, 2003, 2 (3) : 371-379. doi: 10.3934/cpaa.2003.2.371 [7] Ronghua Jiang, Jun Zhou. Blow-up and global existence of solutions to a parabolic equation associated with the fraction p-Laplacian. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1205-1226. doi: 10.3934/cpaa.2019058 [8] Carlo Mercuri, Michel Willem. A global compactness result for the p-Laplacian involving critical nonlinearities. Discrete & Continuous Dynamical Systems, 2010, 28 (2) : 469-493. doi: 10.3934/dcds.2010.28.469 [9] Tomás Caraballo, Marta Herrera-Cobos, Pedro Marín-Rubio. Global attractor for a nonlocal p-Laplacian equation without uniqueness of solution. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1801-1816. doi: 10.3934/dcdsb.2017107 [10] Robert Stegliński. On homoclinic solutions for a second order difference equation with p-Laplacian. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 487-492. doi: 10.3934/dcdsb.2018033 [11] Shanming Ji, Yutian Li, Rui Huang, Xuejing Yin. Singular periodic solutions for the p-laplacian ina punctured domain. Communications on Pure & Applied Analysis, 2017, 16 (2) : 373-392. doi: 10.3934/cpaa.2017019 [12] Meiqiang Feng, Yichen Zhang. Positive solutions of singular multiparameter p-Laplacian elliptic systems. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021083 [13] Maya Chhetri, D. D. Hai, R. Shivaji. On positive solutions for classes of p-Laplacian semipositone systems. Discrete & Continuous Dynamical Systems, 2003, 9 (4) : 1063-1071. doi: 10.3934/dcds.2003.9.1063 [14] Everaldo S. de Medeiros, Jianfu Yang. Asymptotic behavior of solutions to a perturbed p-Laplacian problem with Neumann condition. Discrete & Continuous Dynamical Systems, 2005, 12 (4) : 595-606. doi: 10.3934/dcds.2005.12.595 [15] Elisa Calzolari, Roberta Filippucci, Patrizia Pucci. Dead cores and bursts for p-Laplacian elliptic equations with weights. Conference Publications, 2007, 2007 (Special) : 191-200. doi: 10.3934/proc.2007.2007.191 [16] Yutaka Tsuzuki. Solvability of $p$-Laplacian parabolic logistic equations with constraints coupled with Navier-Stokes equations in 2D domains. Evolution Equations & Control Theory, 2014, 3 (1) : 191-206. doi: 10.3934/eect.2014.3.191 [17] Lucas C. F. Ferreira, Elder J. Villamizar-Roa. On the existence of solutions for the Navier-Stokes system in a sum of weak-$L^{p}$ spaces. Discrete & Continuous Dynamical Systems, 2010, 27 (1) : 171-183. doi: 10.3934/dcds.2010.27.171 [18] Yansheng Zhong, Yongqing Li. On a p-Laplacian eigenvalue problem with supercritical exponent. Communications on Pure & Applied Analysis, 2019, 18 (1) : 227-236. doi: 10.3934/cpaa.2019012 [19] Genni Fragnelli, Dimitri Mugnai, Nikolaos S. Papageorgiou. Robin problems for the p-Laplacian with gradient dependence. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 287-295. doi: 10.3934/dcdss.2019020 [20] Francesca Colasuonno, Benedetta Noris. A p-Laplacian supercritical Neumann problem. Discrete & Continuous Dynamical Systems, 2017, 37 (6) : 3025-3057. doi: 10.3934/dcds.2017130

2019 Impact Factor: 1.27