Hölder inequality applied on a non-Newtonian fluid equation with a nonlinear convection term and a source term

Consider a non-Newtonian fluid equation with a nonlinear convection term and a source term. The existence of the weak solution is proved by Simon’s compactness theorem. By the Hölder inequality, if both the diffusion coefficient and the convection term are degenerate on the boundary, then the stability of the weak solutions may be proved without the boundary value condition. If the diffusion coefficient is only degenerate on a part of the boundary value, then a partial boundary value condition is required. Based on this partial boundary, the stability of the weak solutions is proved. Moreover, the uniqueness of the weak solution is proved based on the optimal boundary value condition.


Introduction and the main results
The evolutionary equation related to the p-Laplacian u t = div a(x)|∇u| p-2 ∇u (1.1) arises in the fields of mechanics, physics and biology. For instance, in the theory of non-Newtonian fluids, the quantity p is a characteristic of the medium, the media with p > 2 are called dilatant fluids and those with p < 2 are called pseudoplastics; if p = 2 they are Newtonian fluids. If a(x) ≡ 1, there is a tremendous amount of work on the existence, the uniqueness and the regularity of the weak solutions of the equation, one can refer to Refs. [1][2][3][4][5][6][7] and the references therein. Zhao [8] had studied the equation u t = div |∇u| p-2 ∇u + f (∇u, u, x, t), (1.2) and revealed some essential differences coming from the term f (∇u, u, x, t). Yin-Wang [9] had studied the equation revealed how the degeneracy of the diffusion coefficient a(x) affects the boundary value condition, where D i = ∂ ∂x i , a ∈ C(Ω) and a(x) ≥ 0. In this paper, we consider where Ω is a bounded domain in R N with appropriately smooth boundary, p > 1, Q T = Ω × (0, T), a(x) ∈ C 1 (Ω), a(x) ≥ 0 and a(x) > 0, x ∈ Ω, (1.5) the nonlinear convection b i (s, x, t) ∈ C(R × Q T ), the source term f (s, x, t) ∈ C(R × Q T ). Comparing with [9], we must pay attention on how these two nonlinear terms affect the well-posedness problem of Eq. Drawing on the experience of the linear degenerate parabolic theory, to study the wellposedness of the solutions of Eq. (1.4), the initial value is always necessary. While, the usual Dirichlet boundary value condition may be overdetermined. So it is only a partial boundary condition imposed in [9], where Σ p ⊆ ∂Ω. In particular, if Σ p = ∅, then there is not any boundary value condition. But the partial boundary condition [9] is in a weaker sense than the trace. The methods used in what follows are different from those in [9], we still use the sense of the trace to define the boundary value condition (1.7) or (1.8). Roughly speaking, we will show that the condition can substitute the boundary value condition (1.7). But if (1.9) is not right, only the partial boundary value condition (1.8) is required, we need to find the explicit formulas of Σ p and judge which one is the best.
The definition of the weak solutions follows a Banach space which is defined as follows. For every fixed t ∈ [0, T], let and denote by V t (Ω) its dual space. By W(Q T ) we denote the Banach space is the dual space of W(Q T ) (the space of linear functionals over W(Q T )).
and for any function The initial value is satisfied in the sense of that   The first aim of this paper is to prove the following stability theorems without any boundary value condition.

Theorem 1.5 Let u(x, t), v(x, t) be two solutions of (1.4) with the initial values u
The stability (1.17) is true.
Here and the hereafter, for any positive small δ > 0, Ω δ = {x ∈ Ω : a(x) > δ}. An interesting corollary from Theorem 1.5 is that, if Ω a(x) -1 p-1 dx < ∞, then without the condition (1.16), only if the condition (1.17) holds, the stability (1.18) is true. Additionally, the second inequality of (1.16) implies that g i (x)| x∈∂Ω = 0. In fact, the condition (1.17) can be replaced by the other conditions. The following theorem is one of results expected. Moreover, by choosing suitable test function, using the Hölder inequality, we can prove another stability theorem without any boundary value condition.
then the stability (1.18) is true.
The second aim of this paper is to prove the stability theorems based on the partial boundary value condition (1.8).

Theorem 1.8 Let u(x, t)
, v(x, t) be two solutions of (1.4) with the initial values u 0 (x), v 0 (x), respectively, and with the same partial boundary value condition In particular, for any small enough constant δ > 0, , the same conclusion as of Theorem 1.9 had been proved by the author in his previous work [10]. So, essential progress of this paper is that we do not assume that a(x)| x∈∂Ω = 0, the best partial boundary value condition is (1.24). This fact also remains an open problem: whether the partial boundary value condition (1.21) can be replaced by (1.24).

The existence of the weak solutions
This section considers the weak solution of the initial-value problem for Eq. (1.4). It is supposed that u 0 satisfies (1.15) By the results of [10,Sect. 8], also referring to [11], we have the following important lemma.
We consider the following regularized problem: , it is well known that the above problem has a unique classical solution [12,13].
According to the maximum principle [2], there is a constant c only dependent on u 0 L ∞ (Ω) but independent on ε, such that Multiplying (2.1) by u ε and integrating it over Q T , we easily have For small enough δ > 0, since p > 1, by (1.5) and (2.5), Using the Young inequality, we can show that Similar to the evolutionary p-Laplacian equation, we can prove that for any function ϕ ∈ C 1 0 (Q T ). Then If we denote Ω ϕ = supp ϕ, then (2.14) Now, for any ϕ 1 ∈ C 1 0 (Q T ), ϕ 2 (x, t) ∈ L 1 (0, T; W 1,p loc (Ω)), it is clearly that By the fact that C ∞ 0 (Ω ϕ 1 ) is dense in W 1,p (Ω ϕ 1 ), by a limit process, we have which implies that Again by a limit process, ϕ 1 can be chosen as in Definition 1.1.
At last, we are able to prove (1.14) as in [14], then u is a solution of Eq. (1.4) with the initial value (1.6) in the sense of Definition 1.1. Thus we have Theorem 1.3. By a similar method to [15], one easily proves the following lemma, we omit the details here.

The stability without the boundary value condition
For any given positive integer n, let g n (s) be an odd function, and where c is independent of n.
This is Corollary 2.1 of [11]. By a similar analysis, one can generalize Lemma 3.1.
In the first place, By Lemma 3.2, using the Lebesgue dominated convergence theorem, Since ∇φ n = n∇a(x) when x ∈ Ω \ Ω 1 n , in the other places, it is identical to zero, by the assumption of (1.19), we have Ω a(x) |∇u| p-2 ∇u -|∇v| p-2 ∇v · ∇φ n g n (uv) dx Since a(x) ∈ C 1 (Ω), by (3.5), In the second place, since b i (s, x, t) satisfies the condition (1.16) using the Lebesgue dominated convergence theorem, we have Last but not least, by condition (1.18) and g i (x)| x∈∂Ω = 0, we clearly have where l ≤ 1 By (3.10), we easily to get and by the arbitrariness of τ , we have

Proofs of Theorem 1.6 and Theorem 1.7
Proof of Theorem 1.6 Let u(x, t) and v(x, t) be two weak solutions of Eq. (1.4) with the initial values u 0 (x), v 0 (x), respectively.
For large enough m, let (4.1) By a limit process, we can choose χ [τ ,s] g n (φ m (uv)) as the test function, then Certainly, we still have By Lemma 3.2, using the Lebesgue dominated convergence theorem, As before, ∇φ m = m∇a(x) when x ∈ Ω \ Ω 1 m , in the other places, it is identical to zero, by the fact that using the Lebesgue dominated convergence theorem, we have In the second place, since b i (s, x, t) satisfies the condition (1.18) using the Lebesgue dominated convergence theorem, we have By a limit process, we can choose χ [τ ,s] g n (ϕ m (uv)) as the test function, then As before, ∇φ m = m∇a(x) when x ∈ Ω \ Ω 1 m , in the other places, it is identical to zero.
Since Ω a(x)|∇u| p dx < ∞, by that a(x) > 0 when x ∈ Ω, we have which yields (4.14) By the fact that lim n→∞ g n (s)s = 0, and the assumption of (1.20), using the Lebesgue dominated convergence theorem, we have Since b i (s, x, t) is a Lipschitz function, using Lebesgue dominated convergence theorem, we have obviously.
Last but not least, by (1.21) using the Lebesgue dominated convergence theorem, we have Now, after letting n → ∞, let m → ∞ in (4.10). Then we have the conclusion.

The usual boundary value condition
p-1 dx < ∞, then we can define the trace of u on the boundary ∂Ω. If one imposes the usual boundary value condition (1.7), the stability of the weak solutions is true. For the completeness of the paper, we also give this conclusion and its proof here. and with the initial values u 0 (x), v 0 (x), respectively, then Proof By a limit process, we can choose χ [τ ,s] g n (uv) as a test function. Then As usual, we have Moreover, similar to [15], we can prove that Now, let n → ∞ in (5.2). Then Let τ → 0. Then, by the Gronwall inequality, we have Theorem 5.1 is proved.
The interesting problem is that, since a(x) may be degenerate on the boundary, the usual boundary value condition (1.7) is overdetermined [9]. Obviously, Theorem 1.8 has solved this problem partially.
Proof of Theorem 1.8 If u(x,t), v(x, t) are two solutions of (1.4) with the initial values u 0 (x), v 0 (x), respectively, and with the same partial boundary value condition (1.21) Let φ n (x) be defined as in the proof of Theorem 1.5. By the assumption we can choose χ [τ ,s] φ n (x)g n (uv) as a test function. By the condition (1.23), similar to the proof Theorem 1.5, we are able to show the conclusion of Theorem 1.8, we omit the details here.

The uniqueness of the solution
In this section, we will prove Theorem 1.9. Then we may choose χ [τ ,s] (uv)a β as a test function, where β ≥ 1 is a constant, where Q τ ,s = Ω × (τ , s). By that |∇a| < c, and