A New Kind of Weak Solution of Non-Newtonian Fluid Equation

Abstract

If the non-Newtonian fluid equation with a diffusion coefficient is degenerate on the boundary, the weak solution lacks the regularity to define the trace on the boundary. By introducing a new kind of weak solutions, the stability of the solutions is established without any boundary condition.

The Evolutionary p(x)-Laplacian Equation with a Partial Boundary Value Condition

“…Last but not least, no matter whether < − − 1 or not, the uniqueness of the weak solutions is always true. Actually, similar as [19], we can prove the following theorem. However, for the simplicity of the paper, we will not give the details of the proof of Theorem 4 in what follows.…”

“…and the initial value (2) We first introduced this kind of the weak solutions in our previous paper [19], in which the following equation was studied:…”

On the well-posedness problem of the electrorheological fluid equations

“…By this token, if b(x) = 0, our previous papers [9][10][11] showed that the boundary value condition (1.3) may be redundant, the uniqueness of the weak solutions can be proved only depending on the initial value (1.2). Accordingly, in this paper, we will construct a suitable test function to obtain the stability of the weak solutions independent of the boundary value condition (1.3).…”

A New Method to Deal with the Stability of the Weak Solutions for a Nonlinear Parabolic Equation

“…has been studied in [7][8][9][10]. What catches our attention is since ( ) is degenerate on the boundary, to obtain the regularity of the weak solutions on the boundary becomes difficult, and the trace on the boundary can not be defined in the classical sense.…”