# Methods and Applications of Analysis

## Volume 27 (2020)

### Mixed boundary value problems of the system for steady flow of heat-conducting incompressible viscous fluids with dissipative heating

Pages: 87 – 124

DOI: https://dx.doi.org/10.4310/MAA.2020.v27.n2.a1

#### Authors

Tujin Kim (Institute of Mathematics, State Academy of Sciences, Pyongyang, North Korea)

Daomin Cao (School of Mathematics and Information Science, Guangzhou University, Guangzhou, Guangdong, China; and Institute of Applied Mathematics, AMSS, Chinese Academy of Sciences, Beijing China)

#### Abstract

In this paper we are concerned with the equation for steady flow of heat-conducting incompressible viscous Newtonian fluids with dissipative heating under mixed boundary conditions. The boundary conditions for fluid may include Tresca slip, leak condition, one-sided leak conditions, velocity, pressure, rotation, stress together and the conditions for temperature may include Dirichlet, Neumann and Robin conditions together. Relying on the relations among strain, rotation, normal derivative of velocity and shape of boundary surface, we get variational formulations consisted of a variational inequality for velocity and a variational equation for temperature, which are equivalent to the original PDE problems for smooth solutions. Then, we study the existence of solutions to the variational problems. To this end, first we study the existence of solutions to auxiliary problems including a parameter for approximation and two or three parameters concerned with the norms of velocity and temperature. Then we determine the parameters concerned with the norms of velocity and temperature in accordance with the data of problems, and we get the existence of solutions by passing to limits as the parameter for approximation goes to zero.

#### Keywords

heat-conducting fluids, dissipative heating, variational inequality, mixed boundary conditions, Tresca slip, leak boundary conditions, one-sided leaks, pressure boundary condition, existence

#### 2010 Mathematics Subject Classification

35J87, 35Q35, 49J40, 76D03, 76D05, 80A20

The research of T. Kim was partly supported by Guangzhou University, P. R. China.

The work of D. Cao was partially supported by NSFC (No. 11771469 and No. 11831009). Cao was also supported by the Key Laboratory of Random Complex Structures and Data Science, Chinese Academy of Sciences (2008DP173182).