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Global existence of strong solutions of Navier–Stokes equations with non-Newtonian potential for one-dimensional isentropic compressible fluids

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Abstract

We consider strong solutions to the initial boundary value problems for the isentropic compressible Navier–Stokes equations in one dimension:

$$\rho\left\{\begin{array}{lll} t+(\rho u)_x=0\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\, {\rm in}\,(0,T)\times(0,1)\\ (\rho u )_t+(\rho u^2)_x+\rho \Phi_x-(\mu( \rho )u_x)_x+P_x=0\quad\quad {\rm in}\,(0,T)\times(0,1) \\\left(\left(\frac{\delta(\Phi_x)^2\,+\,1}{(\Phi_x)^2\,+\,\delta}\right)^{\frac{2-p}{2}}\Phi_x\right)_x=4\pi g(\rho-\frac{1}{|\Omega|}\int\nolimits_\Omega \rho dx\,\,\,\, )\quad\,\, {\rm in}\,(0,T)\times(0,1)\end{array}\right.$$

Here, the Φ is a non-Newtonian potential and strong solutions of the problem and obtains the uniqueness under the compatibility condition.

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Authors and Affiliations

  1. Institute of Mathematics, Jilin University, Changchun, 130012, Jilin, China

    Hongzhi Liu & Hongjun Yuan

  2. Inner Mongolia Finance and Economics College, Huhhot, 010051, China

    Hongzhi Liu, Jiezeng Qiao & Fanpei Li

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  1. Hongzhi Liu
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  2. Hongjun Yuan
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  3. Jiezeng Qiao
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Liu, H., Yuan, H., Qiao, J. et al. Global existence of strong solutions of Navier–Stokes equations with non-Newtonian potential for one-dimensional isentropic compressible fluids. Z. Angew. Math. Phys. 63, 865–878 (2012). https://doi.org/10.1007/s00033-012-0202-3

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  • DOI: https://doi.org/10.1007/s00033-012-0202-3

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