On a non-Newtonian fluid type equation with variable diffusion coefficient

1.   Introduction

The mathematical modelling of various physical processes, where spatial heterogeneity has a primary role, usually results in the derivation of nonlinear evolution equations with variable diffusion, or dispersion. As pointed out by Karachalios-Zographopoulos in ^[18], to name but a few, equations of such a type have been successfully applied to the heat propagation in heterogeneous materials ^[5,12,16,17], the study of transport of electron temperature in a confined plasma ^[7], the propagation of varying amplitude waves in a nonlinear medium ^[24], the study of electromagnetic phenomena in nonhomogeneous superconductors ^[3,13,14,15] and the dynamics of Josephson junctions ^[8,9], the epidemiology and the growth and control of brain tumors ^[21]. It is not possible that all the characteristics of these applications explained in one equation due to its adaptability. In this paper, we focus on the non-Newtonian fluids equations. Applications of non-Newtonian fluids in wide range in many fields like fiber coating and crude oil extraction and many more have fascinated many researchers, one can refer to ^[22,25] etc. In Newtonian fluids, the viscosity does not change, while in non-Newtonian fluids, the viscosity changes when under force/stress to either more liquid or more solid, and so the non-Newtonian fluids are fluids that describe the relationship between deformation rates and stress none linearly, and they do not follow Newton's law of viscosity.

Let us give three explicit equations derived from Newtonian flow or non-Newtonian flow. The first one is in the study of water infiltration through porous media, Darcy's linear relation

satisfactorily describes the flow conduction provided that the velocities are small. Here $V$ represents the seepage velocity of water, $\theta$ is the volumetric moisture content, $K(\theta)$ is the hydraulic conductivity and $\phi$ is the total potential, which can be expressed as the sum of a hydrostatic potential $\psi(\theta)$ and a gravitational potential $z$

If it is assumed that infiltration takes place in a horizontal column of the medium, then the continuity equation has the form

Then we have

with $D(\theta) = K(\theta)\psi'(\theta)$. Certainly, water is the most usual Newtonian fluid. Also, Eq (1.5) is called as a porous medium equation.

The second one is to consider a compressible fluid flow in a homogeneous isotropic rigid porous medium. Then the volumetric moisture content $\theta$, the seepage velocity $\vec{V}$ and the density of the fluid are governed by the continuity equation

For non-Newtonian fluid, according to Chapter 2 of ^[27], the linear Darcy's law is not longer valid, because the influence of many factors such as the molecular and ion effects needs to be concerned. Instead, one has the following nonlinear relation

where $\rho\vec{V}$ and $P$ denote the momentum velocity and pressure respectively, $\lambda > 0$ and $\alpha > 0$ are some physical constants. Combing (1.3) with (1.4), one has

The third one is to consider the flows in fractured media ^[20]. Let $\varepsilon$ be the size ratio of the matrix blocks to the whole medium and let the width of the fracture planes and the porous block diameter be in the same order. If the permeability ratio of matrix blocks to fracture planes is of order $\varepsilon^{p_{\varepsilon}}$, where $p_{\varepsilon}$ is a positive oscillating constant, then the nonlinear Darcy law combined with the continuity equation leads to the following equation

where $u^{\varepsilon}$ is the density of the fluid, $\omega^{\varepsilon}, k^{\varepsilon}$ are the porosity and the permeability of the medium.

Equations (1.4)–(1.6) can be abstracted as

and the quantity $p > 1$ 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. While $a(x)$ is only a nonnegative function, it reflects the characteristic of the medium which may not be homogenous. The most achievements before are focused on the case $a(x) = 1$,

which is simply called as the Non-Newtonian fluid equation usually. The existence, the uniqueness, the regularity and the long-time behaviors of weak solutions to this equation have been studied in ^{[1,2,4,19,27,34]} etc.

In this paper, we consider the following non-Newtonian fluid type equation with a variable, nonnegative diffusion coefficient $a(x)$:

where $D_i = \frac{\partial}{\partial x_i}$, $0\leq a(x) \in C(\overline{\Omega})$, $b_i(x, t) \in C^1(\overline{Q_T})$, and $f(u, x, t)\in C^1(\mathbb{R}\times\overline{Q_T})$, $\Omega\subset \mathbb{R}^{N}$ is a bounded domain with a smooth boundary $\partial \Omega$. The most difference between Eq (1.9) and (1.8) is that the diffusion $a(x)$ in Eq (1.9) may degenerate at some points of $\overline{\Omega}$.

A special case of (1.9) is that $b_i(x, t) = b_i(x)$ and $a(x)\in C^1(\overline{\Omega})$ satisfies

In this case, the corresponding well-posed problem has been considered in ^[28,31,32] recently. In general, the initial value condition

is always needed, but instead of the usual boundary value condition

only a partial boundary value condition

is imposed, where $\Sigma_p = \Sigma_1\times (0, T)$ and $\Sigma_1$ is a relatively open subset of $\partial \Omega$ in ^[28], even it can be an empty set sometime in ^[29,30,31]. In this paper, different ^{[6,10,23,28,29,30,31]}, since $b_i(x, t)$ depends on the time variable $t$, we find that $\Sigma_p$ can not be expressed as a cylindrical space as $\Sigma_1\times (0, T)$. Instead, the partial boundary value condition is only imposed on a submanifold of $\Sigma_p\subseteq\partial\Omega\times(0, T)$ (the details are given in (1.16) or (3.2) below).

Now, let us give the definition of weak solution and supply some other related backgrounds.

Definition 1.1. A function $u(x, t)$ is said to be a weak solution of the initial-boundary value problem of Eq (1.9), if

and for any function $g(s)\in C^1(\mathbb{R})$ with $g(0) = 0$, $\varphi_1 \in {C^1_0(\Omega)}$ and $\varphi_2\in L^\infty(0, T; W_{loc}^{1, p}(\Omega))$, there holds

The initial value condition is satisfied in the sense of

Moreover, the partial boundary value condition is imposed as

If $b_i(x, t) = b_i(x)$, $f(u, x, t) = f(x, t)-c(x, t)u$, the existence of weak solution has been proved in ^[28]. In addition, if there is

then for a weak solution of Eq (1.9), we have

and the trace of $u(x, t)$ on the boundary $\partial\Omega$ can be defined in the classical sense ^[28,29]. If the inequality (1.17) is true and $b_i(x, t) = b_i(x)$, then a similar partial boundary value condition as (1.16) has been imposed in ^[28]. In this paper, we mainly consider the case of that the inequality (1.17) is not true, i.e.,

then we can not define the trace of $u(x, t)$ in the classical sense. So, the first dedication of this paper lies in that we give a generalization of the trace of $u(x, t)$ on the boundary $\partial\Omega$ in a special way. In details, inspired by ^[28,32], we can define the trace of $u(x, t)$ on the boundary $\partial\Omega$ as

where

and

is the super limit. In what follows, we only simply denote ${\rm{ess}}\sup \lim\limits_{\lambda\rightarrow 0}f(\lambda)$ as $\lim\limits_{\lambda\rightarrow 0}f(\lambda)$. The rationality of such a generalization of the classical trace will be specified later in this paper.

If $f(u, x, t)$ is a continuous function and is Lipchitz continuous about the variable $u$, then the existence of weak solutions of the initial-boundary value problem of Eq (1.9) can be proved in a similar way as those ^[28,32], we don't prepare to prove the existence of weak solutions again. We mainly pay attention to the stability or the uniqueness of weak solutions by a generalized Gronwall inequality.

Theorem 1.2. Let $u(x, t)$ and $v(x, t)$ be two weak solutions of the initial-boundary value problem of Eq (1.9), and with the same homogeneous the partial boundary condition

Here $\Sigma$ has the form (1.16). If $2 > p > 1$, $a(x)$ satisfies (1.10), and

then

Moreover, we can obtain a local stability of weak solutions as follows.

Theorem 1.3. Let $u(x, t)$ and $v(x, t)$ be two solutions of Eq (1.9) with the differential initial values $u_0(x)$ and $v_0(x)$ respectively, but no any boundary value condition is required. if $p > 1$, $a(x)$ satisfies (1.10), whether (1.17) or (1.19) is true, and

then

Different from Theorem 1.2, in this theorem, there is not any boundary value condition imposed. Actually, the uniqueness of weak solution to Eq (1.9) can be obtained without conditions (1.21) and (1.23).

Theorem 1.4. Let $a(x)\geq 0$ satisfy (1.10), $p > 1$, $u(x, t)$ and $v(x, t)$ be two weak solutions of Eq (1.9) with the initial values $u_0(x) = v_0(x)$. If $a(x)$ satisfies (1.10) and one of the following assumptions is true.

ⅰ) $a(x)$ satisfies (1.17), $u(x, t)$ and $v(x, t)$ are with the same partial boundary value condition

in the sense of the classical trace. Here $\vec{b} = \{b_i\}$, ${\rm{div}}(\vec{b}(x, t)) = \sum_{i = 1}^N\frac{\partial b_i(x, t)}{\partial x_i}$.

ⅱ) Whether $a(x)$ satisfies (1.17) or (1.19), but

Then

One can see that all theorems above admit the case of (1.19), so the generalization of classical trace to the general form (1.20) is the most novelty of this paper. One can refer to the appendix for more details. Certainly, some other restrictions on the convective coefficient $b_i(x, t)$, i.e., the inequalities (1.21), (1.23) and (3.1), are imposed. How to relieve these restrictions to obtain the same conclusions? This is a question worth discussing thoroughly.

2.   The generalized Gronwall inequality

Let us review the classical Gronwall inequality.

Gronwall inequality: Let $x(t)$ and $c(t)$ be two nonnegative integral functions and $a(t)$ be a bounded function on $[0, T]$. If

then

It is well-known that there are many applications of the Gronwall inequality in PDE, one can refer to ^[11,27] etc. In this paper, we find a generalization of the Gronwall inequality, and use it to prove the stability theorems of the degenerate parabolic equation (1.9).

Lemma 2.1. Let $x(t)$ and $c(t)$ be two nonnegative integral functions on $t\in [0, T]$, $a(t)$ be a bounded function. If there is a constant $0 < l\leq1$ such that

then

where $c$ is a constant depending on $\int_0^{T}x(\tau)d\tau$.

Proof. If $l = 1$, there is nothing to be proved. When $l < 1$, by (2.3), using the Young inequality, we have

By (2.1) and (2.2), we have (2.4).

3.   The proofs of theorems

For small $\eta > 0$, let

Obviously, $h_{\eta}(s)\in C(\mathbb{R})$, and

Proof of Theorem 1.2. From the definition of weak solution, if $g(s) = s$, for any $\varphi_1 \in {C^1_0(\Omega)}$ and $\varphi_2\in L^\infty\left(0, T; W_{loc}^{1, p}(\Omega)\right)$ we have

Denote $\Omega_{\varepsilon} = \{x\in\Omega: a(x) > \varepsilon\}$. Let $\xi$ be

By a process of limit, we can choose

in (3.2), where, $\chi_{[\tau, s]}$ is the characteristic function on $[\tau, s]\subset(0, T)$. Then

where $Q_{\tau s} = \Omega\times[\tau, s]$.

A straightforward calculation leads to

Since $1 < p < 2$, we have

By this inequality, one can see that the right-hand side of (3.4) tends to $0$ as $\varepsilon\rightarrow 0$.

Noticing that

if denoting

by the partial boundary value condition (1.16), we have

Meanwhile, since $1 < p < 2$, we have $\frac{p}{p-1} > 2$. Due to that $u(x, t), v(x, t)\in L^{\infty}(Q_T)$, by (1.17), we have

Naturally, we have

Since $u(x, t), v(x, t)\in L^{\infty}(Q_T)$ and $f(u, x, t)\in C^1(\mathbb{R}\times\overline{Q_T})$, by (3.4), (3.6)–(3.8), from (3.3), we easily obtain

where $l\leq1$.

By virtue of the generalized Gronwall Lemma 2.1, choosing $x(s) = \int_{\Omega}[u(x, s)-v(x, s)]^2dx$, we have

and letting $\tau\rightarrow 0$, we arrive at the desire.

Proof of Theorem 1.3. Let $u(x, t)$ and $v(x, t)$ be two solutions of Eq (1.9) with the initial values $u_0(x)$ and $v_0(x)$ respectively, but without any boundary value condition. From the definition of the weak solution, we can choose $\chi_{[\tau, s]}a(x)(u-v)$ as a test function, where $\chi_{[\tau, s]}$ is the characteristic function on $[\tau, s]\subset [0, T)$. Denoting $Q_{\tau s} = \Omega\times[\tau, s]$, then we have

In particular,

Clearly,

For the second term on the right hand side of (3.10), we have

If $p\geq 2$,

If $1 < p < 2$, by then the Hölder inequality

Meanwhile, by (1.23),

since $u(x, t), v(x, t)\in L^{\infty}(Q_T)$, we easily get that

Here, $p' = \frac{p}{p-1}$ as usual.

At the same time, we have

and since $|f(u, x, t)-f(v, x, t)|\leq c$, by that $u(x, t), v(x, t)\in L^{\infty}(Q_T)$, using the Hölder inequality, we have

Also,

At last, by (3.10)–(3.18), we let $\lambda\rightarrow0$ in (3.9). Then

where $q < 1$. By (3.19), by the generalized Gronwall Lemma 2.1, choosing

we have

Thus, by the arbitrary of $\tau$, we have

The proof is complete.

In what follows, we study the uniqueness of weak solution to Eq (1.9).

Proof of Theorem 1.4. Let $u(x, t)$ and $v(x, t)$ be two weak solutions of Eq (2.1) with the same initial value $u_0(x) = v_0(x)$.

For any given small $\delta > 0$, we denote $D_{\delta} = \{x\in\Omega: |w| = |u-v| > \delta\}$. Without loss of the generality, we may assume that there is a $\delta > 0$ such that the measure $\mu(D_{\delta}) > 0$. Let $\varphi_{\lambda}(\xi)$ be a even function. When $\xi\geq 0$, it its defined as

where $\lambda$ is small enough satisfying $\delta > 2\lambda > 0$, $1 > \beta > 0$, and we define that

By multiplying Eq (1.9) by $\varphi_{\lambda}(w) = \varphi_{\lambda}(u-v)$, integrating over $Q_t = \Omega\times (0, t)$, since $a(x) = 0$ when $x\in \partial \Omega$, using the condition (3.2) in ⅰ) or the condition (3.1) in ⅱ), we have

Let us analyse the three terms in the righthand side of this equality. First, the monotone inequality of the operator $\Delta$ yields

Secondly, since $u(x, t), v(x, t)\in L^{\infty}(Q_T)$,

Thirdly, let $t_0 = \inf\{\tau\in (0, t]: w > \lambda\}$. Then

From (3.23)–(3.25), we have

Letting $\lambda\rightarrow 0$, we get the contradiction.

At the end of this section, we would like to point that, $\varphi_{\lambda}(w) = \varphi_{\lambda}(u-v)$ does not satisfy the request of the test function $g(\varphi_1\varphi_2)$, so in the proof of Theorem 1.4, we do not use the equality (1.14) in Definition 1.1. However, since the condition (3.2) in ⅰ) or the condition (3.1) in ⅱ), all the boundary integrals disappears in (3.22), and the proof of Theorem 1.4 is true.

4.   Conclusions

The non-Newtonian fluid equation with a variable diffusion is more applicable than the usual fluid equation. In the PDE theory, since the variable diffusion $a(x)$ may be degenerate on some points of $\overline{\Omega}$, one can not deduce the inequality

and the weak solution matching up with the equation does not belong to $L^p(0, T; W_0^{1, p}(x))$. Such a fact makes the boundary value condition can not be imposed in the classical way. In this paper, a new kind of weak solution introduced, the trace is generalized in accordance with the weak solution defined, and it is found that a partial boundary value condition on a submanifold of $\partial\Omega\times (0, T)$ is enough to ensure the weak solution well-posedness. In fact, the definitions, the theorems and the methods used in this paper are able to be generalized to study the well-posed problem of the other degenerate parabolic equations such as

and

where $\alpha > 0$ is a constant, $p(x, t) > 1$ is a continuous function.

Acknowledgments

This work is partially supported by Natural Science Foundation of Fujian Province (no: 2021J011193) and supported by NSFC-52171308, China.

Conflict of interest

The authors declare no conflict of interest.

A.   Appendix: The generalizations of the trace

In this appendix, we give a simple discussion of the trace. Without loss the generality, we assume that $f(x)\geq 0$. It is well-known that, $C_0^{\infty}(\Omega)$ is dense in $W_0^{1, p}(\Omega)$, and so the trace of $f(x)\in W^{1, p}(\Omega)$ on the boundary $\partial \Omega$

is defined as the limit of a sequence $f_{\varepsilon}(x)\in C_0^{\infty}(\Omega)$ as

As above in this paper, we denote $\Omega_{\lambda} = \{x\in \Omega: d(x) > \lambda\}$ for a small enough positive number $\lambda$, and denote by $\bf{B}$ the closure of the set $C_0^{\infty}(Q_T)$ with respect to the norm

Yin-Wang ^[28] defined the trace of $u\in \bf{B}$, $u(x, t) = 0$ on $\Sigma_{2}$ as

One can see that if $a(x) = d(x)$, then $d_{x_i} = n_i$ with that $\vec{n} = \{n_i\}$ is the inner normal. Then the Definition (A.2) is just the same as that of (1.20). The reminder is to show that when $a(x)$ satisfies (1.10), $a(x)\neq d(x)$, the Definition (A.2) is equivalent to that of (1.20).

At first, since $a(x)$ satisfies (1.10) and $\int_{\Omega}a(x)|\nabla u|^pdx < \infty$, we know

and $u\in BV(\Omega_{\lambda})$, the traces of

and

can be well defined in the sense of (A.1). Thus, the definition of (1.20) or the definition of (A.2) itself has a explicit meaning. In fact, recalling the BV function space $BV(\Omega)$, i.e. $\left|\frac{\partial f}{\partial x_i}\right|$ is a regular measure, and

then $BV(\Omega)$ is a Banach space under the norm

By the trace $f^+(y)$ of $f(x)\in BV(\Omega)$ on the boundary $\partial \Omega$ is defined as the limit of $f(x)$ along the normal (Lemma 1 of ^[26]). Thus, if we denote that

then for a $f(x)\in BV(\Omega)\bigcap L^{\infty}(\Omega)$, we have

Secondly, for a smooth bounded domain $\Omega$, there is a finite open cover such that

where $\Sigma_{\alpha} = B_{\alpha}\bigcap\partial\Omega$. If $\Sigma_{\alpha}\neq\emptyset$, after using the flattening technique, we may assume that

and

and

Let $a(x)\in C^1(\overline{\Omega})$ satisfy

Since $a(x) > 0$ when $x\in B_{1}^{+}$, and there is a small enough $\delta_0$ such that when $0 < x_2 < \delta$

For small enough $\lambda < \delta_0$, we set

and denote the local coordinate representation $B_{\alpha}$ and $\Sigma_{\alpha}$ as above. By the coordinate transformation

the domain $E_{x\lambda}$ is transformed to $E_{y\lambda} = \Sigma_{\alpha}\times (0, \lambda)$.

In $E_{y\lambda}$, it is clear of that $d(y, \Sigma_{\alpha}) = y_{N}$. Since $f\in BV(E_{x\lambda})$ or $f\in BV(E_{y\lambda})$, as (A.3), we have

When $f$ is nonnegative and $f|_{\partial\Omega} = 0$, then the equality (A.6) can be re-calculated as

where $M\left(\frac{\partial a}{\partial x_N}\right) = \sup_{x\in \Sigma_{\alpha}}\frac{\partial a}{\partial x_N}$. In other words, when $f\in BV(\Omega)$ is nonnegative and $f|_{\partial\Omega} = 0$, (A.7) implies that

Thirdly, by (A.4), the coordinate transformation (A.5) has a inverse transformation

Recalling the definition of (1.20), i.e.,

without loss the generality, under the inverse transformation (A.9), we may assume that the domain $(\Omega_{\varepsilon}\setminus \Omega_{2\varepsilon})\cap \Omega_{1t}$ is transformed to a domain $D_{\varepsilon y}\cap \Omega_{1t}$

Then, similar as the discussion of (A.7) and (A.8), the definition of (1.20) is equivalent to that

From these point, one can see that the definition of (1.20) is just a reasonable version the definition of (A.2), which is the generalized definition of the trace of $u\in \bf{B}$ with degeneracy on the boundary. Actually, such a local analysis of the definition of the trace in $BV(\Omega)$ has appeared in our previous work ^[33]. Since ^[33] is unpublished till now, we re-give the details here.