EnGui Fan,Zhijun Qiaoand ManWai Yuen

1 School of Mathematical Sciences,Shanghai Center for Mathematical Sciences and Key Laboratory of Mathematics for Nonlinear Science,Fudan University,Shanghai 200433,China

2 School of Mathematical and Statistical Science Univerisity of Texas—Rio Geande Valley,Edinburg,TX 78539,United States of America

3 Department of Mathematics and Information Technology,The Education University of Hong Kong,10 Lo Ping Road,Tai Po,New Territories,Hong Kong,China

4 Author to whom any correspondence should be addressed.

Abstract In this paper,we prove the existence of general Cartesian vector solutions u=b(t)+A(t)x for the Ndimensional compressible Navier–Stokes equations with density-dependent viscosity,based on the matrix and curve integration theory.Two exact solutions are obtained by solving the reduced systems.

Keywords:compressible Navier–Stokes equations with density-dependent viscosity,Cartesian solutions,symmetric and anti-symmetric matrix,quadratic form,curve integration

1.Introduction

The Navier–Stokes equations play a very important role in fluids,oceanography and atmospheric dynamics.The system has been investigated extensively and intensively.There are much progress made on the local strong solutions and the global weak solutions,for example[1–8].There are interesting papers on analytical solutions of the Navier–Stokes equations for the special functions h(ρ)and g(ρ)[9,10,16,19].However,these known analytical solutions are not explicit.Based on the new matrix theory and decomposition technique,An,Fan and Yuen proved the existence of the Cartesian solutions for the compressible Euler equations(1.6)[22].Then Chow,Fan and Yuen further generalized to the damped Euler equations[33].

In section 2,we show that the compressible Navier–Stokes equations with density-dependent viscosity have the Cartesian solutions if A fulfills appropriate matrix equations.By solving the reduced systems,two solvable cases are provided in sections 3 and 4.

2.Existence of the Cartesian solutions

3.First reduction:constant matrix A

4.Second reduction:time-dependent matrix A

Acknowledgments

This research is partially supported by the National Science Foundation of China(Grant No.11 271 079;10 671 095)and RG 11/2015-2016R from the Education University of Hong Kong.

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