Nucleus Accumbens and Its Role in Reward and Emotional Circuitry: A Potential Hot Mess in Substance Use and Emotional Disorders

1.   Introduction

Fluid-particle interaction model arises in many practical applications, and is of primary importance in the sedimentation analysis of disperse suspensions of particles in fluids. This model is one of the commonly used models nowadays in biotechnology, medicine, mineral processing and chemical engineering [27]-[25]. Usually, the fluid flow is governed by the Navier-Stokes equations for a compressible fluid while the evolution of the particle densities is given by the Smoluchowski equation [4], the system has the form:

where $\rho,u,\eta$, $P(\rho) = a\rho^\gamma$, $\Phi(x)$ denote the fluid density, velocity, the density of particles in the mixture, pressure, and the external potential respectively, $a>0,\gamma>1$. $\mu>0$ is the viscosity coefficient, and $3\lambda+2\mu\geq 0$ are non-negative constants satisfied the physical requirements.

There are many kinds of literatures on the study of the existence and behavior of solutions to Navier-Stokes equations (See [1]-[17]). Taking system (1) as an example, Carrillo $et\ al$ [4] discussed the the global existence and asymptotic behavior of the weak solutions providing a rigorous mathematical theory based on the principle of balance laws, following the framework of Lions [18] and Feireisl $et\ al$ [11,12]. Motivated by the stability arguments in [5], the authors also investigated the numerical analysis in [6]. Ballew and Trivisa [1] constructed suitable weak solutions and low stratification singular limit for a fluid particle interaction model. In addition, Mellet and Vasseur [20] proved the global existence of weak solutions of equations by using the entropy method on the asymptotic regime corresponding to a strong drag force and strong brownian motion. Zhang $et\ al$ [31] establish the existence and uniqueness of classical solution to the system (1).

Despite the important progress, there are few results of non-Newtonian fluid-particle interaction model. As we know, the Navier Stokes equations are generally accepted as a right governing equations for the compressible or incompressible motion of viscous fluids, which is usually described as

where $\Gamma$ denotes the viscous stress tensor, which depends on $E_{ij}(\nabla u)$, and

is the rate of strain. If the relation between the stress and rate of strain is linear, namely, $\Gamma = \mu E_{ij}(\nabla u)$, where $\mu$ is the viscosity coefficient, then the fluid is called Newtonian. If the relation is not linear, the fluid is called non-Newtonian. The simplest model of the stress-strain relation for such fluids given by the power laws, which states that

for $0<q<1$ (see[3]). In [16], Ladyzhenskaya proposed a special form for $\Gamma$ on the incompressible model:

For $\mu_0 = 0$, if $p<2$ it is a pseudo-plastic fluid. In the view of physics, the model captures the shear thinning fluid for the case of $1<p<2$ (see[19]).

Non-Newtonian fluid flows are frequently encountered in many physical and industrial processes [8,9], such as porous flows of oils and gases [7], biological fluid flows of blood [30], saliva and mucus, penetration grouting of cement mortar and mixing of massive particles and fluids in drug production [13]. The possible appearance of the vacuum is one of the major difficulties when trying to prove the existence and strong regularity results. On the other hand, the constitutive behavior of non-Newtonian fluid flow is usually more complex and highly non-linear, which may bring more difficulties to study such flows.

In recent years, there has been many research in the field of non-Newtonian flows, both theoretically and experimentally (see [14]-[26]). For example, in [14], Guo and Zhu studied the partial regularity of the generalized solutions to an incompressible monopolar non-Newtonian fluids. In [32], the trajectory attractor and global attractor for an autonomous non-Newtonian fluid in dimension two was studied. The existence and uniqueness of solutions for non-Newtonian fluids were established in [29] by applying Ladyzhenskaya's viscous stress tensor model.

In this paper, followed by Ladyzhenskaya's model of non-Newtonian fluid, we consider the following system

with the initial and boundary conditions

and the no-flux condition for the density of particles

where $\rho,u,\eta$, $P(\rho) = a\rho^\gamma$, $\Phi(x)$ denote the fluid density, velocity, the density of particles in the mixture, pressure, and the external potential respectively, $a>0,\gamma>1,\frac{4}{3}<p,q<2$. $\lambda>0$ is the viscosity coefficient, $\Omega$ is a one-dimensional bounded interval, for simplicity we only consider $\Omega = (0,1)$, $\Omega_T = \Omega\times[0,T]$.

The system describes a compressible shear thinning fluid-particle interaction system for the evolution of particles dispersed in a viscous non-Newtonian fluid and the particle is driven by non-Newtonian gravitational potential. To our knowledge, there still no existence results for (2)-(4) when $1<p,q<2$. The aim of this paper is to study the existence and uniqueness of strong solutions to this system. Throughout the paper we assume that $a = \lambda = 1$ for simplicity. In the following sections, we will use simplified notations for standard Sobolev spaces and Bochner spaces, such as $L^p = L^p(\Omega), H_0^1 = H_0^1(\Omega), C([0,T];H^1) = C([0,T];H^1(\Omega))$.

We state the definition of strong solution as follows:

Definition 1.1. The $(\rho,u,\Psi,\eta)$ is called a strong solution to the initial boundary value problem(2)-(4), if the following conditions are satisfied:

(ⅰ)

(ⅱ) For all $\phi \in L^\infty(0,T_*;H^1(\Omega))$, $\phi_t \in L^\infty(0,T_*;L^2(\Omega))$, for a.e. $t \in (0,T)$, we have

(ⅲ) For all $\varphi \in L^\infty(0,T_*;W_0^{1,p}(\Omega)\cap H^2(\Omega))$, $\varphi_t \in L^2(0,T_*;H_0^1(\Omega))$, for a.e. $t \in (0,T)$, we have

(ⅳ) For all $\psi \in L^\infty(0,T_*;H^2(\Omega))$, $\psi_t \in L^\infty(0,T_*; H^1(\Omega))$, for a.e. $t\in (0,T)$, we have

(ⅴ) For all $\vartheta \in L^\infty(0,T_*;H^2(\Omega))$, $\vartheta_t \in L^\infty(0,T_*;L^2(\Omega))$, for a.e. $t\in (0,T)$, we have

The main result of this paper is stated in the following theorem.

1.1.   Main theorem

Theorem 1.2. Let $\Phi\in C^2(\Omega)$, $\frac{4}{3}<p,q<2$ and assume that the initial data $(\rho_0,u_0,\eta_0)$ satisfy the following conditions

and the compatibility condition

for some $g\in L^2(\Omega)$. Then there exist a $T_*\in(0,+\infty)$ and a unique strong solution $(\rho,u,\Psi,\eta)$ to (2)-(4) such that

Remark 1. By using exactly the similar argument, we can prove the result also hold for the case $1 <p,q\leq\frac{4}{3}$. We omit the details here.

2.   A priori estimates for smooth solutions

In this section, we will prove the local existence of strong solutions. From the continuity equation $(2)_1$, we can deduce the conservation of mass

Because equation $(2)_2$ possesses always with singularity, we overcome this difficulty by introduce a regularized process, then by taking the limiting process back to the original problem. Namely, we consider the following system

with the initial and boundary conditions.

and $u_0\in H_0^1(\Omega)\cap H^2(\Omega)$ is the smooth solution of the boundary value problem

Provided that $(\rho,u,\eta)$ is a smooth solution of (10)-(15) and $\rho_0\geq \delta$, where $0<\delta\ll 1$ is a positive number. We denote by $M_0 = 1+ \mu_0+ \mu^{-1}_0+|\rho_0|_{H^1}+|g|_{L^2}$.

We first get the estimate of $|u_{0xx}|_{L^2}$. From (16), we have

Then

Applying Young's inequality, we have

thus

where $C$ is a positive constant, depending only on $M_0$.

Next, we introduce an auxiliary function

We will derive some useful estimate to each term of $Z(t)$ in terms of some integrals of $Z(t)$, then apply arguments of Gronwall's inequality to prove $Z(t)$ is locally bounded.

2.1.   Preliminaries

In order to prove the main Theorem, we first give some useful lemmas for later use.

Lemma 2.1. Let $u_{0}\in H^{1}_{0}(\Omega)\cap H^{2}(\Omega)$, $\rho_0\in H^{1}(\Omega)$, $\eta_0\in H^{2}(\Omega)$, $\Phi\in C^2(\Omega)$, $g\in L^2(\Omega)$, $u_0^\varepsilon$ is a solution of the boundary value problem

Then there are a subsequence $\{u^{\varepsilon_j}_0\}$, $j = 1,2,3,...,$ of $\{u^{\varepsilon }_0\}$ and $u_{0}\in H^{1}_{0}(\Omega)\cap H^{2}(\Omega)$ such that as $\varepsilon_j\rightarrow 0$,

Proof. According to (18), we have

Taking it by the $L^2$ norm, we have

then

Therefore, by the above inequality, as $\varepsilon_j\rightarrow 0$,

Thus, we can obtain $\{u^{\varepsilon_j}_{0x}\}$ is a Cauchy subsequence of $C^{\frac{3}{2}}(\Omega)$, for all $\alpha_1>0$, we find $N$, as $i, j > N$, and

Now, we prove that $\{u^{\varepsilon}_{0xx}\}$ has a Cauchy sequence in $L_2$ norm.

Let

For all $\alpha>0$, there exists $N$, as $i, j > N$, we can deduce that

With the assumption, we can obtain

where $C$ is a positive constant, depending only on $|\rho_0|_{H^{1}(\Omega)}$, $|g|_{L^{2}(\Omega)}$ and $|\eta_0|_{H^{2}(\Omega)}$. Using the following inequality,

where $0<\theta<1$.

By the simple calculation, we can get

where $C$ depending only on $p$, then

Substituting this into (18), we have

then there is a subsequence $\{u^{\varepsilon_j}_{0xx}\}$ and $\{u^{\varepsilon}_{0xx}\}$, such that

By the uniqueness of the weak convergence, we have

Since $(P(\rho_0)+\eta)_x+\eta_0 \Phi_x-\rho_0^{1\over 2}(g+\Phi_x)$ are independent of $\varepsilon$, the same that we obtain, as $\varepsilon_j\rightarrow 0$,

This completes the proof of Lemma 2.1.

Lemma 2.2.

where $C$ is a positive constant, depending only on $M_0$.

Proof. We estimates for $u$ and $\eta$ for later use. It follows from (11) that

We note that

Taking it by the $L^2$ norm and using Young's inequality, we have

On the other hand, by $(12)$, we have

Taking it by $L^2$-norm, using Young's inequality, which gives

This implies that

By (13), taking it by the $L^2$ norm, we have

Multiplying (10) by $\rho$, integrating over $\Omega$, we deduce that

Integrating it by parts, using Sobolev inequality, we obtain

Differentiating $(10)$ with respect to $x$, and multiplying it by $\rho_x$, integrating over $\Omega$, and using Sobolev inequality, we have

From (26) and (27) and the Gronwall's inequality, then lemma 2.2 holds.

Lemma 2.3.

where $C$ is a positive constant, depending only on $M_0$.

Proof. Multiplying $(13)$ by $\eta$, integrating the resulting equation over $\Omega_T$, using the boundary conditions (4) and Young's inequality, we have

Multiplying $(13)$ by $\eta_t$, integrating (by parts) over $\Omega_T$, using the boundary conditions (4) and Young's inequality, we have

Differentiating $(13)$ with respect to $t$, multiplying the resulting equation by $\eta_t$, integrating (by parts) over $\Omega_T$, we get

Combining (29)-(31), we obtain the desired estimate of Lemma 2.3.

Lemma 2.4.

where $C$ is a positive constant, depending only on $M_0$.

Proof. Using (10), we rewritten the (11) as

Multiplying (33) by $u_t$, integrating (by parts) over $\Omega_T$, we have

We deal with each term as follows:

By virtue of $(10)$, we have

Substituting the above into (34), using Sobolev inequality and Young's inequality, we have

To estimate (36), combining (35) we have the following estimates

In exactly the same way, we also have

which, together with (36) and (37), implies (32) holds.

Lemma 2.5.

where $C$ is a positive constant, depending only on $M_0$.

Proof. Differentiating equation $(11)$ with respect to $t$, multiplying the result equation by $u_t$, and integrating it over $\Omega$, we have

Note that

Let

from (24), it follows that

Combining (35), (40) can be rewritten into

Using Sobolev inequality, Young's inequality, $(11)$, (24) and (25), we obtain

In order to estimate $I_{11}$, we need to deal with the estimate of $|\Psi_{xt}|_{L^2}$. Differentiating (12) with respect to $t$, multiplying it by $\Psi_t$ and integrating over $\Omega$, we have

and

Let

then

Then (43) can be rewritten into

Using Young's inequality, combining the above estimates we deduce that

Substituting $I_j(j = 1,2,\ldots,11)$ into (42), and integrating over $(\tau,t)\subset (0,T)$ on the time variable, we have

To obtain the estimate of $|\sqrt{\rho}u_t(t)|_{L^2}^2$, we need to estimate $\lim\limits_{\tau\rightarrow 0}|\sqrt{\rho}u_t(\tau)|_{L^2}^2$. Multiplying (33) by $u_t$ and integrating over $\Omega$, we get

According to the smoothness of $(\rho,u,\eta)$, we have

Then, taking a limit on $\tau$ in (45), as $\tau\rightarrow 0$, we can easily obtain

This complete the proof of Lemma 2.5.

With the help of Lemma 2.2 to Lemma 2.5, and the definition of $Z(t)$, we conclude that

where $C,\tilde{C}$ are positive constants, depending only on $M_0$. This means that there exist a time $T_1>0$ and a constant $C$, such that

where $C$ is a positive constant, depending only on $M_0$.

3.   Proof of the main theorem

In this section, the existence of strong solutions can be established by a standard argument. We construct the approximate solutions by using the iterative scheme, derive uniform bounds and thus obtain solutions of the original problem by passing to the limit. Our proof will be based on the usual iteration argument and some ideas developed in [10]. Precisely, we first define $u^0 = 0$ and assuming that $u^{k-1}$ was defined for $k\geq 1$, let $\rho^k,u^k,\eta^k$ be the unique smooth solution to the following system

with the initial and boundary conditions

where

With the process, the nonlinear coupled system has been deduced into a sequence of decoupled problems and each problem admits a smooth solution. And the following estimates hold

where $C$ is a generic constant depending only on $M_0$, but independent of $k$.

In addition, we first find $\rho^k$ from the initial problem

with smooth function $u^{k-1}$, obviously, there is a unique solution $\rho^k$ on the above problem and also we could obtain that

Next, we will prove the approximate solution $(\rho^k,u^k,\eta^k)$ converges to a limit $(\rho^\varepsilon,u^\varepsilon, \eta^\varepsilon)$ in a strong sense. To this end, let us define

By a direct calculation, we can verify that the functions $\bar\rho^{k+1},\bar u^{k+1},\bar\eta^{k+1}$ satisfy the system of equations

Multiplying (56) by $\bar\rho^{k+1}$, integrating over $\Omega$ and using Young's inequality, we obtain

where $C_\zeta$ is a positive constant, depending on $M_0$ and $\zeta$ for all $t<T_1$ and $k\geq 1$.

Multiplying (57) by $\bar u^{k+1}$, integrating over $\Omega$ and using Young's inequality, we obtain

Let

then

To estimate the second term of (61), we have

On the other hand, multiplying (58) by $\bar\Psi^{k+1}$, integrating over $\Omega$, we obtain

Since

and

then

which implies

From (55), (62) and (64), (61) can be re-written as

where $B_\xi(t) = C(1+|u_{xt}^k(t)|_{L^2}^2$, for all $t\leq T_1$ and $k\geq 1$. Using (55) we derive

Multiplying (59) by $\bar \eta^{k+1}$, integrating over $\Omega$, using (55) and Young's inequality, we have

Combining (60), (65) and (66), we have

where $E_\zeta(t)$ is depending only on $B_\zeta(t)$ and $C_\xi$, for all $t\leq T_1$ and $k\geq 1$. Using (55), we obtain

Integrating (67) over $(0,t)\subset (0,T_1)$ with respect to t, using Gronwall's inequality, we have

From the above recursive relation, choose $\xi>0$ and $0<T_*<T_1$ such that $C\exp(C_\xi T_*)<\frac{1}{2}$, using Gronwall's inequality, we deduce that

where $C$ is a positive constant, depending only on $M_0$.

Therefore, as $k\rightarrow +\infty$, the sequence $(\rho^k,u^k,\eta^k)$ converges to a limit $(\rho^\varepsilon,u^\varepsilon,\eta^\varepsilon)$ in the following strong sense

By virtue of the lower semi-continuity of various norms, we deduce from the uniform estimate (55) that $(\rho^\varepsilon,u^\varepsilon,\eta^\varepsilon)$ satisfies the following uniform estimate

Since all of the constants are independent of $\varepsilon$, there exists a subsequence $(\rho^{\varepsilon_j},u^{\varepsilon_j}, \eta^{\varepsilon_j})$ of $(\rho^{\varepsilon},u^{\varepsilon},\eta^{\varepsilon})$, without loss of generality, we denote to $(\rho^{\varepsilon},u^{\varepsilon},\eta^{\varepsilon})$. Let $\varepsilon \rightarrow 0$, we can get the following convergence

and there also holds

For each small $\delta>0$, let $\rho_0^\delta = J_\delta * \rho_0+\delta$, where $J_\delta$ is a mollifier on $\Omega$, and $u_0^\delta\in H_0^1(\Omega)\cap H^2(\Omega)$ is a smooth solution of the boundary value problem

where $g^\delta\in C_0^\infty$ and satisfies $|g^\delta|_{L^2}\leq |g|_{L^2}$, $\lim\limits_{\delta\rightarrow 0^+}|g^\delta-g|_{L^2} = 0$.

We deduce that $(\rho^\delta,u^\delta,\eta^\delta)$ is a solution of the following initial boundary value problem

where $\rho_0^\delta\geq\delta, \frac{4}{3}<p,q<2$.

By the proof of Lemma 2.1, there exists a subsequence $\{u_0^{\delta_j}\}$ of $\{u_0^{\delta}\}$, as $\delta_j\rightarrow 0^+$, $u_0^{\delta}\rightarrow u_0$ in $H_0^1(\Omega)\cap H^2(\Omega)$, $-(|u_{0x}^{\delta_j}|^{p-2}u_{0x}^{\delta_j})_x\rightarrow -(|u_{0x}|^{p-2}u_{0x})_x$ in $L^2(\Omega)$, Hence, $u_0$ satisfies the compatibility condition (9) of Theorem 1.2. By virtue of the lower semi-continuity of various norms, we deduce that $(\rho,u,\eta)$ satisfies the following uniform estimate

where $C$ is a positive constant, depending only on $M_0$. The uniqueness of solution can also be obtained by the same method as the above proof of convergence, we omit the details here. This completes the proof.

Acknowledgments

The authors would like to thank the anonymous referees for their valuable suggestions.