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

comes from a host of applied fields such as the theory of non-Newtonian fluid, the study of water infiltration through porous media, and combustion theory; one can refer to [1–4] and the references therein. Here p > 1,Di = ∂/∂xi, a ∈ C(Ω), Ω ⊂ R is a bounded domain with the appropriately smooth boundary ∂Ω. If a(x) > c > 0, the equations with the type of (1) have been extensively studied; one can refer to [5–7] and the references therein. If a(x) ≥ 0, one wants to obtain the well-posedness of the equation; the initial value


Introduction and the Main Results
The quasilinear parabolic equation comes from a host of applied fields such as the theory of non-Newtonian fluid, the study of water infiltration through porous media, and combustion theory; one can refer to [1][2][3][4] and the references therein.Here  > 1,   = /  ,  ∈ (Ω), Ω ⊂ R  is a bounded domain with the appropriately smooth boundary Ω.If () >  > 0, the equations with the type of (1) have been extensively studied; one can refer to [5][6][7] and the references therein.If () ≥ 0, one wants to obtain the well-posedness of the equation; the initial value  (, 0) =  0 () ,  ∈ Ω is invariably imposed.But the boundary value condition  (, ) = 0, (, ) ∈ Ω × (0, ) may be overdetermined.Yin and Wang [8] made a more important devoting to the problem; they classified the boundary into three parts: the nondegenerate boundary, the weakly degenerate boundary, and the strongly degenerate boundary, by means of a reasonable integral description.The boundary value condition should be supplemented definitely on the nondegenerate boundary and the weakly degenerate boundary.On the strongly degenerate boundary, they formulated a new approach to prescribe the boundary value condition rather than defining the Fichera function as treating the linear case.Moreover, they formulated the boundary value condition on this strongly degenerate boundary in a much weak sense since the regularity of the solution is much weaker near this boundary.In a word, instead of the whole boundary condition (3), only a partial boundary condition is imposed in [8], where Σ  ⊆ Ω.
In our paper, for simplism, we assume that (),   (), and (, ) are  1 functions, and  () > 0,  ∈ Ω,  () = 0,  ∈ Ω; (5) the equation is degenerate on the boundary.In our previous works [9,10], we have shown that such degeneracy may result in the fact that the weak solution of the equation lacks the regularity to define the trace on the boundary.Accordingly, how to construct a suitable function, which is independent of the boundary value condition, to obtain the stability of the weak solutions, becomes formidable.The main aim of the paper is to solve the corresponding problem by introducing a new kind of the weak solutions.

2
Journal of Function Spaces Definition 1. Function (, ) is said to be a weak solution of (1) with the initial value (2), if and for any function The initial value is satisfied in the sense of that lim The existence of the solution can be proved in a similar way as that in [8]; we omit the details here.In our paper, we mainly are concerned about the stability of the weak solutions without any boundary value condition.
Theorem 3. Let , V be two nonnegative solutions of (1) with the initial values  0 , V 0 , respectively.If 1 <  ≤ 2 and then the stability of the weak solutions is true in the sense of (11).
Let us give a comparison between Theorems 2 and 3. To see that, we specially assume that Then it is easy to know that if  < 1/( − 1), then condition (10) is satisfied; while  <  − 1, then condition (12) is true.
Theorem 5. Let , V be two solutions of ( 1) with the differential initial values  0 (), V 0 (), respectively.Then there exists a positive constant  ≥ 1 such that In particular, for any small enough constant  > 0, If  0 = V 0 , by the arbitrariness of , one can see that (, ) = V(, ), ..(, ) ∈   ; the uniqueness of the solution is true.
We have used some techniques in [9].But there are many essential improvements in our paper.The main results of my previous work [9] were established on the assumption of that where () is the distance function from the boundary.Condition ( 19) is much stronger than the usual homogeneous boundary value condition (3), so the conclusions in [9] are not perfect.But in my new paper, we have introduced the new kind of the weak solutions (Definition 1); also we can establish the stability of the weak solutions without any boundary value condition.
Corollary 6.Let , V be two weak solutions of (1) with the initial values  0 (), V 0 (), respectively.If ( 9) is true and it is supposed that then the stability is true without any boundary value condition.
Proof.If (35) is true, then (30) is true by (21).Thus the corollary can be proved in a similar way as that of Theorem 2

The Local Stability
Proof of Theorem 5. Let , V be two solutions of (1) with the initial values  0 (), V 0 (), respectively.From the definition of the weak solution, if () = , for any  1 ∈  1 0 (Ω),  2 ∈  ∞ (0, ; In particular, we choose where  [,] is the characteristic function on [, ] and the constant  ≥ 1. Clearly, For the second term on the right-hand side of (54), since  The proof is complete.