# 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] Daoyuan Fang, Ting Zhang. Compressible Navier-Stokes equations with vacuum state in one dimension. Communications on Pure & Applied Analysis, 2004, 3 (4) : 675-694. doi: 10.3934/cpaa.2004.3.675 [2] Thomas Y. Hou, Ruo Li. Nonexistence of locally self-similar blow-up for the 3D incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 637-642. doi: 10.3934/dcds.2007.18.637 [3] Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213 [4] Hirofumi Notsu, Masato Kimura. Symmetry and positive definiteness of the tensor-valued spring constant derived from P1-FEM for the equations of linear elasticity. Networks & Heterogeneous Media, 2014, 9 (4) : 617-634. doi: 10.3934/nhm.2014.9.617 [5] Hakan Özadam, Ferruh Özbudak. A note on negacyclic and cyclic codes of length $p^s$ over a finite field of characteristic $p$. Advances in Mathematics of Communications, 2009, 3 (3) : 265-271. doi: 10.3934/amc.2009.3.265 [6] Bernold Fiedler, Carlos Rocha, Matthias Wolfrum. Sturm global attractors for $S^1$-equivariant parabolic equations. Networks & Heterogeneous Media, 2012, 7 (4) : 617-659. doi: 10.3934/nhm.2012.7.617 [7] Lei Liu, Li Wu. Multiplicity of closed characteristics on $P$-symmetric compact convex hypersurfaces in $\mathbb{R}^{2n}$. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020378 [8] Carlos Fresneda-Portillo, Sergey E. Mikhailov. Analysis of Boundary-Domain Integral Equations to the mixed BVP for a compressible stokes system with variable viscosity. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3059-3088. doi: 10.3934/cpaa.2019137 [9] Irena PawŃow, Wojciech M. Zajączkowski. Global regular solutions to three-dimensional thermo-visco-elasticity with nonlinear temperature-dependent specific heat. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1331-1372. doi: 10.3934/cpaa.2017065 [10] Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277 [11] Yimin Zhang, Youjun Wang, Yaotian Shen. Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1037-1054. doi: 10.3934/cpaa.2011.10.1037 [12] Lucas C. F. Ferreira, Jhean E. Pérez-López, Élder J. Villamizar-Roa. On the product in Besov-Lorentz-Morrey spaces and existence of solutions for the stationary Boussinesq equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2423-2439. doi: 10.3934/cpaa.2018115 [13] Xiaoming Wang. Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 543-556. doi: 10.3934/dcdss.2017027 [14] Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261 [15] Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271 [16] Charles Fulton, David Pearson, Steven Pruess. Characterization of the spectral density function for a one-sided tridiagonal Jacobi matrix operator. Conference Publications, 2013, 2013 (special) : 247-257. doi: 10.3934/proc.2013.2013.247 [17] Hong Seng Sim, Wah June Leong, Chuei Yee Chen, Siti Nur Iqmal Ibrahim. Multi-step spectral gradient methods with modified weak secant relation for large scale unconstrained optimization. Numerical Algebra, Control & Optimization, 2018, 8 (3) : 377-387. doi: 10.3934/naco.2018024 [18] Rafael Luís, Sandra Mendonça. A note on global stability in the periodic logistic map. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4211-4220. doi: 10.3934/dcdsb.2020094 [19] Lakmi Niwanthi Wadippuli, Ivan Gudoshnikov, Oleg Makarenkov. Global asymptotic stability of nonconvex sweeping processes. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1129-1139. doi: 10.3934/dcdsb.2019212 [20] Carlos Gutierrez, Nguyen Van Chau. A remark on an eigenvalue condition for the global injectivity of differentiable maps of $R^2$. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 397-402. doi: 10.3934/dcds.2007.17.397

2019 Impact Factor: 1.27

Article outline