Some properties on the surfaces of vector fields and its application to the Stokes and Navier–Stokes problems with mixed boundary conditions

International Journal of Differential Equations

2020

Self Cite

In this paper, we are concerned with the nonsteady Boussinesq system under mixed boundary conditions. The boundary conditions for fluid may include Tresca slip, leak and one-sided leak conditions, velocity, static (or total) pressure, rotation, and stress (or total stress) together, and the boundary 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 the boundary surface, we get variational formulation. The formulations consist of a variational inequality for velocity due to the boundary conditions of friction type and a variational equation for temperature. For the case of boundary conditions including the static pressure and stress, we prove that if the data of the problem are small enough and compatibility conditions at the initial instance are satisfied, then there exists a unique solution on the given interval. For the case of boundary conditions including the total pressure and total stress, we prove the existence of a solution without restriction on the data and parameters of the problem.

“…where S is the shape operator of the boundary surface (cf. (A.1) in [19]), 􏽥 v, 􏽥 u are expressions of v, u in a local coordinate system on Γ 3 , and…”

Section: Variational Formulations: E Case Of Static Pressurementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The Nonsteady Boussinesq System with Mixed Boundary Conditions including Conditions of Friction Type

International Journal of Differential Equations

2020

Self Cite

“…And also without discussing whether static pressure or total pressure (correspondingly stress or total stress) is suitable for real phenomena which is over our knowledge, we consider the problems with total pressure and total stress instead of static pressure and stress. Relying on the result in [32], we reflect all these boundary conditions into variational formulations of problems. Overcoming difficulty from one-sided leak boundary conditions, we get variational inequalities equivalent to the variational formulation for the problems.…”

Section: )mentioning

confidence: 99%

“…Theorem 2.1 (Theorem 2.1 in [32]) Suppose that v·n| Γ = 0. Then, on the surface Γ the following holds.…”

Section: Preliminary and Problemsmentioning

confidence: 99%

See 1 more Smart Citation

The steady Navier–Stokes and Stokes systems with mixed boundary conditions including one-sided leaks and pressure

Methods and Applications of Analysis

Cao

2016

Self Cite

In this paper we are concerned with the steady Navier-Stokes and Stokes problems with mixed boundary conditions involving Tresca slip, leak condition, one-sided leak conditions, velocity, pressure, rotation, stress and normal derivative of velocity together. Relying on the relations among strain, rotation, normal derivative of velocity and shape of boundary surface, we have variational formulations for the problems, which consist of five formulae with five unknowns. We get the variational inequalities equivalent to the formulated variational problems, which have one unknown. Then, we study the corresponding variational inequalities and relying the results for variational inequalities, we get existence, uniqueness and estimates of solutions to the Navier-Stokes and Stokes problems with the boundary conditions. Our estimates for solutions do not depend on the thresholds for slip and leaks.

Existence of a solution to the steady Magnetohydrodynamics‐Boussinesq system with mixed boundary conditions

Math Methods in App Sciences

2022

Self Cite

In this paper, we are concerned with the steady magnetohydrodynamics (MHD)‐Boussinesq system with mixed boundary conditions. The boundary conditions for fluid may include the stick, Tresca slip, leak, one‐sided leaks, pressure, vorticity, and stress conditions together, and the conditions for temperature may include Dirichlet, Neumann, and Robin conditions together. For the electromagnetic field, mixed boundary conditions are given. Also, the viscosity, magnetic permeability, electrical conductivity, thermal conductivity, and specific heat of the fluid depend on the temperature. Special attention is paid to the meaning of non‐homogeneous mixed boundary conditions for magnetic field. For the problem involving the static pressure and stress boundary conditions for fluid, it is proven that if the data of problem are small enough, then there exists a solution and the solution with small norm is unique when the viscosity, magnetic permeability, electrical conductivity, thermal conductivity, and specific heat are independent of the temperature. For the problem involving the total pressure and total stress boundary conditions for fluid, the existence of a solution is proven when a datum concerned with non‐homogeneous boundary conditions for magnetic field is small, but without smallness of other data.