Existence and uniqueness of solution for a class of non-Newtonian fluids with non-Newtonian potential and damping

1.   Introduction and main result

We consider the following class of 1D non-Newtonian fluids

where $u$, $\rho$, $M = \rho u$ and $P = A \rho^{\gamma}(A > 0, \gamma > 1)$ denote velocity, the unknown density, momentum and pressure, respectively, $g > 0$ is the acceleration of gravity and $\Phi$ is the gravitational potential. The constant $\alpha > 0$ models friction. Without losing generality, throughout the paper we take $\alpha = 1$. The initial and boundary value conditions of Eq (1.1) are as follows

$\Omega$ is considered as a one-dimensional bounded interval here. Furthermore, for simplicity, we only assume the $\Omega = I = (0, 1)$, $\Omega_{T} = I\times(0, T)$. The initial density $\rho_{0} \geq 0$, $p$ and $q$ are given constants, and they are both studied in the case of less than 2, where since the method of study is similar for $1 < p < \frac{4}{3}$ and $\frac{4}{3} < p < 2$, we next study only the case of $\frac{4}{3} < p < 2$.

According to classical Newtonian fluid mechanics, in parallel fluids, the shear force is proportional to shear velocity, and its proportion is the viscosity coefficient, i.e.,

Generally, we call a fluid with the above properties a Newtonian fluid. Accordingly, a fluid does not have this property is called a non-Newtonian fluid. For non-Newton fluids, $\Gamma\left(\rho, \nabla u \right)$ has a reasonable choice (see Ladyzhenskaya^[1])

and

In chemistry, biomechanics, glaciology, geology, and blood rheology, there are many problems in non-Newtonian fluids, which lead to an interest in studying non-Newtonian fluids ^[2,3,4]. There are many theoretical and experimental studies in non-Newtonian fluid flow.

In this paper, the vacuum condition i.e., the initial density, is zero. Strictly speaking, a vacuum, a gas state below atmospheric pressure in a given space, is a physical phenomenon. In real life, vacuum distillation, vacuum drying, and vacuum concentration are typical vacuum cases. The role of vacuum in vacuum distillation is mainly to reduce the boiling point temperature of substances and reduce the influence of temperature factors on substances. Vacuum drying and concentration both use a vacuum environment to accelerate the volatilization and evaporation of specific substances or enable the whole process to be completed at lower conditions. In particular, non-Newtonian fluids can expand in a vacuum. According to the experiment, as the air in the vacuum bottle decreases, tiny bubbles gradually appear on the surface of the non-Newtonian fluid, and the bubbles expand until they spill the container out. In any case, the theoretical knowledge of non-Newtonian fluids under vacuum must be continuously refined. To this end, this article allows for an initial vacuum.

In 1996, J. Málek, J. Ne$\check{c}$as, M. Rokyta, M. R$\dot{u}\check{z}$i$\check{c}$ka divided non-Newnewton flows with regard to $p$ in the monograph ^[5] : when $1 < p < 2$, we call such a fluid a shear thinning fluid, when $p > 2$, we call such a fluid a shear thickening fluid. For non-Newtonian fluids, Yuan Hongjun and Xu Xiaojing ^[6] studied existence of a solution and whether it is unique of a class of non-Newtonian fluid solution with singularity and vacuum. Takashi Suzuki and Takayuki Kobayashi ^[7] proved the existence of weak solution to the Navier-Stokes-Poisson equation. Meng Qiu and Yuan Hongjun ^[8] proved the existence and uniqueness of a class of local solution under conditions where compressible non-Newtonian fluids with a non-Newtonian bit potential in a one-dimensional bounded interval. Song Yukun, Yuan Hongjun, and Yang Chen ^[9] investigated the existence and uniqueness of a class of local solution in the presence of isentropic compressible non-Newtonian fluids in a one-dimensional bounded area. Liu Hongzhi, Yuan Hongjun, Qiao Jiezeng and Li Fanpei ^[10] constructed the global existence of robust solution of Navier-Stokes equations with non-Newtonian potential. Li Huapeng and Yuan Hongjun ^[11] demonstrated the local existence and uniqueness of 1D non-Newtonian fluid solution with damping.

The damping item comes from resistance to fluid motion. Model (1.1) describes more natural phenomena. For example, porous media flow. We can refer to ^{[12,13,14,15]} contents of the damp item. However, for non-Newtonian fluid, there is no degradation result with non-Newtonian potential and damp item. We construct a system (1.1)–(1.2) with local existence and uniqueness of the strong solution of non-Newtonian fluids with the non-Newtonian potentials and friction damping. The result is the following theorem:

Theorem 1. Assume that

and that there is a function $g \in L^{2}(I)$, makes the following equation true almost everywhere on I:

then there exists a small time $T_{*} \in(0, +\infty)$ and a unique strong solution $(\rho, u, \Phi)$ to the initial boundary value problem (1.1)–(1.2) such that:

For the above theorem, we will be divided it into four parts to prove. In the first part, we use the iterative method to get the approximate solution system of problems (1.1)–(1.2) and then make a consistent estimate of its approximate solution. In the second part, the convergence of the approximate solution is proved by the weak convergence method. In the third and fourth parts, we demonstrated that a locally strong solution to problems (1.1)–(1.2) exists uniquely.

2.   Uniform estimates

Lemma 1. (Embedding inequality). Assume that $f = 0$ on $\partial \Omega$, here $\Omega \in R^1$ is bounded and open, $f \in C^{2+\alpha}(\bar{\Omega})$. Then

where $d(\Omega)$ denotes the length of $\Omega$.

See the literature ^[4] for proof.

The system of Eqs (1.1) we studied contains more unknowns, and $(1.1)_{2}$ and $(1.1)_{3}$ are non-linear, so we cannot get a direct solution (from Abelian theorem, the fifth and higher order algebraic equations have no analytical solution). Therefore, we are inspired to seek approximate solutions to the system of equations.

Therefore, we apply an iterative approach to problems (1.1)–(1.2), which yields its approximate solution system.

we take the initial mass $m_0 = \int_0^1 \rho_0(x) \mathrm{d} x > 0$, $P^k = \mathrm{P}\left(\rho^k\right) = A\left(\rho^k\right)^\gamma$, $A > 0$, $\gamma > 1$, $\rho_0^\delta = \delta+\rho_0*J_\delta$, $\delta > 0$,

For problem (2.5), $u_{0}^{\varepsilon} \in H^{2}(I)\cap H_{0}^{1}(I)$ is a smooth solution to it

Then, we will conduct a consistent estimation of the approximate solution and prove that the limit of the approximate solution is just the solution of the Eqs (1.1)–(1.2).

In order to do this, we will first get the uniform estimate on $u_{0} ^{\varepsilon}$. The $u_{0}^{\varepsilon}$ is known from the smooth solution of the boundary value problem

then

Using Young's inequality, we have

with the help of the Lemma 1, we get

where $C > 0$ is a constant that depends only on $M_{0}$, which may not necessarily be fixed. Next, we denote

For any fixed integer $K$, define

then we will prove that $J_{K}(t)$ is locally bounded for $\frac{4}{3} < p < 2$. We estimate each term in $J_{K}(t)$ in the following sections.

2.1.   Estimate for $\left|u^{k}(t)\right|_{w_{0}^{1, p}(I)}$

Multiplying (2.2) by $u_{t}^{k}$, Integrating over $(0, 1)$ concerning $x$ and integrating over $(0, t)$ to $s$ gives, we can get

We firstly compute the second term of (2.11), we obtain

and

Substituting (2.12), (2.13) into (2.11), by $(2.9)$ and Young's inequality, we have

By (2.1), we get

Then the above inequality can be expressed as

where $0 < \eta \ll 1$. To estimate the right part of the (2.14), we have the following estimates

Using (2.1), we have

By virtue of $(2.2)$, we have

then we have

taking the above inequality by $L^{2}$ norm, using Young's inequality, we obtain

We deal with $\left|\Phi_{x x}^k\right|_{L^2(I)}$, by (2.3) we have

taking it by $L^2$-norm, using Young's inequality and Lemma 1, we get

then

Using (2.14) and the above inequality, we get

for all $k, 1 \leq k \leq K$.

2.2.   Estimate for $\left|\sqrt{\rho^{k} u_{t}^{k}}(t)\right|_{L^{2}(I)}$

We differentiate $(2.2)$ with respect to $t$, and multiply it by $u_{t}^{k}$, and integrating it over $(0, 1)$ with respect to $x$, we obtain

Since

let

by $(2.20)$, we have

Then (2.22) can be rewritten as

Using Sobolev embedding theorem and Young's inequality, we obtain

In order to estimate $I_{12}$, we need to deal with $\Phi_{x t}^k$. Differentiating (2.3) with respect to $t$, multiplying it by $\Phi_t^k$ and integrating over $(0, 1)$, we have

By (2.23), we have

Let

then

we have

Using Young's inequality, combining the above estimate we obtain

Substituting ${I_j}(j = 1, 2, \ldots, 12)$ into (2.24), integrating over $(\tau, t)$ on time variable, we have

then, from the above recursive relation, for $1 \leq k \leq K$, we obtain

Thus, we deduce from (2.25) that

where $C$ is a positive constant, depending only on $M_{0}$.

To obtain the estimate of $\left|\sqrt{\rho^{k}} u_{t}^{k}(t)\right|_{L^{2}(I)}^{2}$, we need to estimate

Using (2.2), we get

Since $\left({{\rho ^k}, {u^k}, \Phi^{k}} \right)$ is a smooth solution, we obtain

Thus, using (2.9), we deduce

Taking a limit on $\tau$ for inequality $(2.26)$, we obtain, as $\tau\rightarrow 0,$

2.3.   Estimate $|\rho^{k}(s)|_{H^{1}(I)}$

We differentiate (2.1) concerning $x$, multiply it by $\rho_{x}^{k}$, integrating it over (0, 1) for $x$, and using Sobolev embedding theorem, we obtain

applying Gronwall's inequality, it follows that

Substituting (2.20) into the above inequality, we get

Using (2.29) and (2.1), we have

By virtue of (2.20), (2.27), (2.29) and (2.30), we conclude that

where $C_{1}, C_{2}$ are two positive constants, depending only on $M_{0}$. By the definition of $J _{K}(t)$, we obtain

If

then we take $T_{1} = T$. On the other hand, if

we can find $t_{0} \in(0, T)$, such that

So we have

and

so

then we have

Given this inequality, we can acquire a short-time $T_1 > 0$ such that:

3.   Convergence of approximate solution

It is demonstrated that the approximate solution $\left(\rho^{k}, u^{k}, \Phi^{k}\right)$ strongly converge to the solution of the Eqs (1.1)–(1.2) with positive density. We give the following definition

then we verify that $\left(\bar{\rho}^{k+1}, \bar{u}^{k+1}, \bar{\Phi}^{k+1}\right)$ satisfy the system of equations

the initial boundary value conditions are given as follows

Multiplying (3.1) by $\bar{\rho}^{k+1}$, integrating over $I$ with respect to $x$, we deduce that

where $B_{\eta}^{k} = C\left|u_{x x}^{k}\right|_{L^{2}(I)}+C_{\eta}|\rho^{k}|^{2}_{H^{1}(I)}$, for all $t \leq T_{1}$ and $k \geq 1$.

Multiplying (3.2) by $\bar{u}^{k+1}$, integrating over $I$ with respect to $x$, using (3.1), Hölder inequality and Lemma 1, we obtain

Let

so

using (3.4) and (3.5), we have

Using (2.34), (3.7) and Young's inequality, (3.5) could be rewritten as

where $E_{\eta}^{k}(t) = C\left(1+\left|u_{x t}^{k}(t)\right|_{L^{2}(I)}^{2}\right)$, for all $t < T_{1}$ and $k \geq 1$. Using (2.31), we derive

According to Eq (2.3)

let's multiply both sides of this equation by $\bar{\Phi}^{k+1}$, about $x$ in $(0, 1)$ integral, available

then

By combining the above formula, H$\ddot{o}$lder inequality and Lemma 1 are obtained

Collecting (3.4), (3.8) and (3.9), we deduce that

Using Gronwall's inequality, we have

Then, we choose $\eta > 0$ and then $T_{*} > 0$ so small that $T_{*} < T_{1}$ and $C \exp \left(C_{\eta} T_{*}\right) < 1 / 2$, we get

Hence, we combine the above inequalities, in view of Gronwall's inequality, we deduce that

Therefore, we conclude that the full sequence $\left(\rho^{k}, u^{k}, \Phi^{k}\right)$ converges to a limit $(\rho, u, \Phi)$ in the following strong sense:

Combining (3.3) and the convergence of (3.12), we can get

From the lower semi-continuity of the norm, we get:

4.   Existence

The proof of existence should be completed in three steps, namely, taking limits on $k\rightarrow \infty$, $\varepsilon\rightarrow0^{+}$ and $\delta\rightarrow0^{+}$. Since the method is similar, we will only describe the process of taking limits on $\delta\rightarrow0$ below. The first two steps can be found in the literature ^[4].

We take $\delta$ to be a very small positive number, let $\rho_{0}^{\delta} = J_{\delta} * \rho_{0}+\delta$, $J_{\delta}$ is a mollifier on $I$, $u_{0}^{\delta} \in H_{0}^{1}(I) \cap H^{2}(I)$ is the unique smooth solution of the boundary value problem:

there exists $g^{\delta} \in C_{0}^{\infty}(I)$ satisfies

For $\rho_{0}^{\delta} = J_{\delta}\ast\rho_{0}+\delta,$ there is a subsequence $\left\{ \left({\rho _{\rm{0}}^{{\delta _{\rm{j}}}}, u_{0}^{\delta_{j}}}\right) \right\} of \left\{\left({\rho _{\rm{0}}^\delta, u_{0}^{\delta} }\right) \right\}$, as $\delta_{j} \rightarrow0^{+}$ satisfing

Therefore, $\left(\rho_{0}, u_{0}\right)$ satisfies the following problem

There exists a $T_{*} \in(0, +\infty)$, the initial-boundary value problem

admits a unique solution $\left(\rho^{\delta}, u^{\delta}, \Phi^{\delta}\right)$. Moreover, $\left(\rho^{\delta}, u^{\delta}, \Phi^{\delta}\right)$ satisfies the uniform estimate

According to the above uniform estimate, by the lower semi-continuity of norm, as $\delta_{j} \rightarrow0,$ we deduce the following uniform estimate:

5.   Uniqueness

Suppose $\left(\rho_{1}, u_{1}, \Phi_{1}\right)$ is a strong solution to the problem (1.1)–(1.2), $\left(\rho_{2}, u_{2}, \Phi_{2}\right)$ is also a strong solution to the problem (1.1)–(1.2), then we have

where $h = -u_{2}-u_{2 t}-u_{2} u_{2 x} \in L^{2}\left(0, T_{*}; L^{2}(I)\right)$.

Then using Hölder inequality together with Lemma 1, we have

By (3.7), we have

where $\mu_{0} = \frac{1}{\left(\left|u_x(t)\right|_{L^{\infty}\left(0, t; L^{\infty}(I)\right)}+\left|\bar{u}_x(t)\right|_{L^{\infty}\left(0, t; L^{\infty}(I)\right)}\right)^{2-p}}$.

Then, following from (5.1), by choosing $\varepsilon = \left(\mu_{0}\right)^{\frac{p-2}{2}} / 8$, we derive

where $0 < A(t) = C\left(1+|h-\Phi_{2x}|_{L^{2}(I)}^{2}+\left|u_{2 x}\right|_{L^{\infty}(I)}^{2}\right) \in L^{1}\left(0, T_{*}\right)$.

As is known from the definition of a strong solution, we take $\varphi = \rho_{1}-\rho_{2}$, then

where $0 < B(t) = C\left(\left|\rho_{1}\right|_{H^{1}(I)}+\left|u_{2 x}\right|_{L^{\infty}(I)}\right) \in L^{1}\left(0, T_{*}\right)$. Similarly, we have

where $0 < B(t) = C\left(\left|\rho_{1}\right|_{H^{1}(I)}+\left|u_{2 x}\right|_{L^{\infty}(I)}\right) \in L^{1}\left(0, T_{*}\right)$. Similarly, we have

Combining $(5.2)$, $(5.3)$ and $(5.4)$, we obtain

where $H(t) = A(t)+B(t)+D(t) \in L^{1}\left(0, T_{*}\right)$. Using Gronwall's inequality, we can get

then

we can get

Therefore

Acknowledgments

The authors thank Beihua University for funding and supporting this work through The Science and Technology Research Project of the Jilin Provincial Education Department (Grant No.JJKH20220040KJ).

Conflict of interest

The authors declare there is no conflicts of interest.