Blow-up criterion for the 3-D inhomogeneous incompressible viscoelastic rate-type fluids with stress-diffusion

1.   Introduction

In this paper, we investigate the global well-posedness of the following system:

where $u = (u^1, u^2, u^3)$ and $b = b(x, t)$ represent the velocity field of the fluid and the spherical component of the elastic strain tensor, respectively. The scalar pressure function is denoted by $\Pi = \Pi(x, t)$. Generally, the viscosity coefficients $\nu$ and $\sigma$ are positive constants, and $e(\cdot)$ denotes a smooth convex function of $b$. Furthermore, it is assumed that the second derivative of $e$ with respect to $b$, which is denoted as $e''(b)$, is bounded by a constant $C$.

The system we study in this paper controls the motion of non-Newtonian fluids, which is described by a simplified viscoelastic rate-type model incorporating a stress-diffusion component. It retains many qualitative characteristics of more complex viscoelastic rate-type models, which makes it an important tool for studying the mathematical properties of non-Newtonian fluids. In this paper, we present a proof regarding the blow-up criterion applicable to this model. In the context of differential equations, when we say that the solution of an equation containing the "time" variable undergoes "blow-up", it typically means that the domain of definition of the solution is finite, and some kind of "undesirable" situation occurs at the endpoint of the time interval: The solution may tend towards infinity or it may lose smoothness (which can cause the differential equation to lose physical meaning), among other possibilities. This is an important and challenging phenomenon. As time progresses, whether it is displacement, velocity, heat, field intensity, or any other form of response, they will all tend towards infinity.

On the other hand, the presence of the stress-diffusion term in the governing equations is significant for several reasons. First, it enhances the qualitative mathematical properties of the governing equations to a certain degree. Second, the existence of the diffusion term has a profound impact on the dynamical behavior predicted by the system of governing equations, as exemplified by its utilization in modeling the shear banding phenomenon. There are two interpretations of the stress-diffusion term: One views it as a consequence of a nonlocal energy storage mechanism, while the other sees it as a result of a nonlocal entropy production mechanism. These different interpretations of the stress-diffusion mechanism lead to distinct evolution equations for temperature. Hence, understanding the thermodynamic foundation of the derived models is crucial when examining the nonlinear stability of equilibrium resting states. Moreover, in the current model, the presence of the additional stress tensor disrupts the original structure outlined in ^[1]. Growth strain induces stress, which, in turn, affects the diffusion process through chemical potential and diffusivity. Consequently, substantial modifications must be made to this part of the theory.

This model combines features of the Oldroyd-B and Giesekus models, and incorporates diffusion terms. Moreover, the fluid type studied in this investigation is characterized by its elastic response, which can be represented by spherical strain (a scalar multiplier of the identity tensor), specifically a scalar multiplication of the identity tensor. A detailed derivation process of the mathematical model can be found in the appendix of ^[2] (carrying on the developments outlined in ^[3,4,5]). Although mathematically complex, the model is capable of capturing the complex microstructure of fluids, which is essential for predicting and simulating the flow of non-Newtonian fluids. The rate-type fluid models with stress-diffusion are popular in the modeling of shear and vorticity banding phenomena; see, for example, the reviews in ^[6,7,8]. The results of this model provide profound insights into the behavior of viscoelastic fluids and offer mathematical tools and theoretical foundations for fluid dynamics problems in engineering and scientific applications. This task is particularly interesting if one considers fluid models that include terms that are associated with the elastic properties of the material.

When $b = 0$, the system reduces to the incompressible Navier-Stokes equations. The theoretical investigation of these equations was initiated by the foundational works of Leray ^[9] and Hopf ^[10] on the global existence of a weak solution. Regarding the local well-posedness of strong solutions, Ciga and Miyakawa ^[11], Kato ^[12], and others researched it using semigroup theory. Abidi, Gui, and Zhang ^[13] explored the global well-posedness of solutions in Besov spaces for systems with a bounded density. More recently, the authors of ^[14] also investigated the global existence and uniqueness of solution to two-dimensional (2-D) inhomogeneous incompressible Navier-Stokes equations in critical spaces.

Moreover, in the case where $\sigma = 0$, our Eq (1.1) can be conceptualized as the coupling of two systems: The inhomogeneous Navier-Stokes equation governing a fluid with constant density, and a transport equation with the damped term $e'(b)$. As we mentioned above, research on the three-dimensional (3-D) incompressible Navier-Stokes equation is indeed extensive, with numerous studies, such as ^[1,15,16] and others. On the other hand, for when $\sigma > 0$, Wang and Zhang ^[17] investigated the global well-posedness of the 2-D Boussinesq system incorporating the temperature-dependent viscosity and thermal diffusivity. Abidi and Zhang ^[18] explored the global well-posedness of a 3-D Boussinesq system with variable viscosity.

For the system we study in this paper, M. Bulíček, J. Málek, Vit Pr$\mathring{u}$sǎ, and E. Süli have conducted some related research in recent years. They ^[3] established the long-time and large-data existence of a weak solution to the evolutionary spatially periodic problem associated with the set of governing equations for compressible fluids with variable density in 3-D space. M. Bathory, M. Bulíček, and J. Málek ^[19] explored a global-in-time and large-data existence theory, within the context of weak solutions, to a class of homogeneous incompressible rate-type viscoelastic fluids confined in a closed 3-D container.

Compared with these reference papers, our primary focus is on the simplified model of viscoelastic fluids with stress-diffusion, which was derived in ^[2]. In this model, we specifically consider the scenario where the extra stress tensor, which describes the elastic response of the fluid, is purely spherical. Furthermore, we have incorporated the divergence-free condition to investigate the incompressible situation. In this paper, we have rigorously proven, for the first time, the existence and uniqueness of the local solution to the system using the energy method. Additionally, we have obtained global well-posedness under the assumption of small initial data through the application of a blow-up criterion.

Our main result reads as follows:

Theorem 1.1. Assume that when $s > \frac{3}{2}$ and the initial data $(u_0, b_0)$ satisfy $u_0\in H^{s}(\mathbb{R}^{3})$, $b_0\in H^{s+1}(\mathbb{R}^{3})$ and $\mathrm{div}\, u_0 = 0$, the following hold:

(1) (Local existence) There exists a small time $T^* > 0$ such that (1.1) has a unique strong solution $(u, b)$ on $(0, T^*)$ satisfying

(2) (Blow-up criterion) $T^*$ is the finite blow-up time of $(u, b)$ if and only if

(3) (Global existence) If there exists a small constant $\varepsilon > 0$ such that

then $T^* = \infty$.

The remainder of this paper is structured as follows. In Section 2, we introduce preliminary knowledge that will be subsequently used throughout the text. Next, the primary focus of the subsequent sections is on proving Theorem 1.1. In Section 3, we utilize Friedrich's method to construct approximate solutions for the system in (1.1). By employing a priori estimates to the approximate system, we establish the existence and uniqueness of the local strong solution for (1.1). In Section 4, we derive a blow-up criterion by a contradiction argument to determine whether the local solution exhibits blow-up behavior. Finally, we demonstrate the global existence of the solution under the assumption that the initial data are sufficiently small, thereby completing the proof of (1.1).

Let us complete this section with the notations we are going to use in the following.

(1) If $u$, $v$ are two functions, let us set $u: = v$ to define u as equal to v.

(2) For X being a Banach space and I being an interval of $\mathbb{R}$, $1\le q\le \infty$, we denote by $L^q(I; X)$ the set of measurable function on I with values in X. Sometimes, for convenience in this paper, we write it as $L^q(X)$.

(3) For $m\in \mathbb{N}$, $f_i\in L^p\, (i = 1, 2, ..., m)$, we write $\|(f_1, f_2, ..., f_m)\|_{L^p}: = \sum\limits_{i = 1}^{m}\|f_i\|_{L^p}$.

(4) For $a\lesssim b$, we mean that there exists a uniform constant C, which may be different in different lines, such that $a\le Cb$.

(5) For $m > 1$, $x\in \mathbb{R}^{n}$, and a vector-valued function $u(x) = (u^{1}, u^{2}, ..., u^{m})$, we define

2.   Preliminary

First, in order to facilitate the construction of an approximate equation for the system in (1.1), we introduce two projection operators $P_n$ and $P$ here.

Definition 2.1. (see ^[20]) Let us define the Helmholtz projection operator $P$ with a divergence-free condition by

where $R_j$ denotes the Riesz transform and satisfies $\mathcal {F}(R_jf)(\xi) = -\frac{\mathrm{i}\xi_j}{|\xi|}\mathcal{F}f(\xi)$.

Definition 2.2. (see ^[21]) The projection operator $P_n$ is defined as follows:

where the Fourier operator $\mathcal{F}f(\xi): = \int_{\mathbb{R}^3}{f(x)e^{-ix\xi}}dx$, and the characteristic function $1_{B_n}$ is defined on the surface of a sphere $B_n$ centered at the origin with a radius $n$.

Clearly, based on the definitions of $P_n$ and $P$ provided above, it is straightforward to deduce that $P_nP = PP_n$ and $P_n^2 = P_n$. The proofs of these statements are omitted here for brevity.

Next, we recall some basic definitions and properties related to the Littlewood-Paley theory (see ^[21] for more details). Assume that $\psi$ and $\theta$ are two functions in $C^\infty(\mathbb{R}^3)$ with

and $\hat{\theta}(\xi)+\sum\limits_{j\geq0}\hat{\psi}(2^{-j}\xi) = 1\, \, (\forall\, \xi \, \in \mathbb{R}^3)$.

Definition 2.3. (see ^[21]) We define the Littlewood-Paley operator $\Delta_j$ in $\mathbb{R}^3$ as follows:

In particular, utilizing the definition of the Littlewood-Paley operator provided above, we can re-describe the Sobolev space $H^s(\mathbb{R}^3)$.

Definition 2.4. (see ^[22]) Let $s\in \mathbb{R}$; the Sobolev space $H^s(\mathbb{R}^3)$ can then be defined as

We set the norm as

Definition 2.5. (see ^[22]) We define the space $\tilde{L}^{\infty}(0, T;{H}^s)$ by the norm:

Next, we introduce some lemmas that will be used in the following sections.

Lemma 2.1. (see ^[23]) Assume that $k\in \mathbb{N}$, $1\le p\le q\le \infty$, and $j\geq 0$, $\alpha$ is a multi-index (i.e., an element of $\mathbb{N}^3$), $\forall f\in L^p(\mathbb{R}^3)$, and there exists a positive constant $C$ which is independent of $j$ such that

Lemma 2.2. (see ^[24]) Let $s > 0$ and $f\in H^s(\mathbb{R}^3)\cap L^\infty(\mathbb{R}^3)$; $F(\cdot)$ is a smooth function on $\mathbb{R}$ and $F(0) = 0$. The following thus holds

Lemma 2.3. (see ^[25]) Let $s > 0$ and set $\varLambda^s: = \sqrt{-\Delta}$, then

and for $\forall f, g\in H^s(\mathbb{R}^3)\cap L^\infty(\mathbb{R}^3)$, we have

Lemma 2.4. (see ^[26]) (Lions-Aubin lemma) Assume that $X\hookrightarrow \hookrightarrow Y\hookrightarrow Z$, where X, Z are reflexible spaces, and X is dense in Z, then let $W: = \{u\in L^{p_0}(0, T;X), \, u_t\in L^{p_1}(0, T;X), \, and\, 1 < p_0, p_1\le \infty\}$. It then holds that $W\hookrightarrow\hookrightarrow L^{p_0}(0, T;Y)$.

3.   Local well-posedness

This section is devoted to the proof of the local well-posedness of Eq (1.1). To clearly illustrate the process, we will divide it into five steps.

Step 1. Construct approximate equations.

We use the Friedrich's method to construct the approximate solutions. Recall the projector operator $P_n$ and the Helmholtz projection operator P stated in Definitions 2.1 and 2.2, respectively, and then utilize them to introduce the following approximate system for (1.1):

where we used the fact that

Step 2. Energy estimates to approximate equations.

First, we take the scalar product of $(3.1)_1$ with $u_n$ and integrate the resulting expression over $\mathbb{R}^3$, which gives

Next, by multiplying $(3.1)_2$ by $-\sigma \Delta b_n$ and integrating in space, we have

Hence, by summing up (3.3) and (3.4), we get

By then integrating it over $[0, T_n]$, one eventually gets

In addition, taking the $L^2$ inner product of the $b_n$ equation with $P_n(e' (P_nb_n))$ gives

Next, doing integration by parts and integrating it over $[0, T_n]$, which, along with $(3.6)$ shows that

Therefore, from (3.6) and (3.7), we observe that each term in $(3.1)_1$ and $(3.1)_2$ is bounded in $L^2(\mathbb{R}^3)$. Consequently, it follows from the Cauchy-Lipschitz theorem that there exists a unique solution $(u_n, b_n)\in C(0, T_n; L^2)$ to the system in (3.1). Due to the uniqueness of the solution, one has $P_nu_n = u_n$ and $P_nb_n = b_n$. Hence the approximate system in (3.1) simplifies to the following form:

Moreover, the inequalities in (3.6) and (3.7) can also be reformulated as follows:

In what follows, we focus on estimating $\|u_n\|_{ H^s }$ and $\|b_n\|_{ H^{s +1}}$.

Applying the operator $\varLambda^s$ to $(3.8)_1$ and taking the $L^2$ inner product of the resulting equation with $\varLambda^su_n$, followed by integrating it over $\mathbb{R}^3$, one obtains

Similarly, we perform the same operation on the $b_n$ equation. Applying the $\varLambda^s$ operator to $(3.8)_2$ and multiplying the resulting expression by $\varLambda^sb_n$, $-\varLambda^s \Delta b_n$, by integrating them over $\mathbb{R}^3$, we then have

and

Summing Eqs (3.10)–(3.12) gives

According to Hölder's inequality, Young's inequality, and Lemmas 2.2 and 2.3, we can estimate each term on the right side of the above inequality:

where $J(\|b_n\|_{L^\infty}) = (1+\|b_n\|_{L^\infty})^{\lfloor s \rfloor+2}$.

Hence, plugging these estimates into the above inequality yields

which, together with Grönwall's inequality, gives rise to

where we defined

Subsequently, assume that $T_n$ is the maximal existence time of the approximate system in $(3.8)$. Furthermore, there exists a uniform lower bound for $T_n$, which ensures that the limit of time is in a certain interval. We omit the detailed proof here; the readers can refer to ^[27]. In addition, we set

Since $J(\cdot)$ is an increasing function and $s > \frac{3}{2}$, it follows from the Sobolev embedding theorem that when $t\in[0, \widetilde{T_n})$,

Clearly, $\exp\{J(4E_n(0))t+4E_n(0)t\}$ is an increasing and continuous function with respect to $t$. Therefore, we can choose $T^*\in[0, \widetilde{T_n})$ to be sufficiently small such that $\exp\{J(4E_n(0))t+4E_n(0)t\} < 2$. Consequently, for any $t\in[0, T^*]$, one has

which implies that

Finally, with the help of Lemma 2.3 and Sobolev's embedding theorem, it follows from $(3.8)_1$ that

which, combined with $(3.15)$, shows that

Similarly, performing the same operation on $(3.8)_2$ yields

Thus, we have completed the desired estimates for the approximate solutions.

Step 3. The existence of a local solution for the system in (1.1).

Based on the estimates of (3.15) to (3.17) derived in Step 2, the Lions-Aubin compactness theorem guarantees the existence of the subsequence $(u_{n_k}, b_{n_k})$ of $(u_n, b_n)$ and a function $(u, b)\in L^\infty(0, T^*; H^s)\cap L^2(0, T^*; H^{s+1})\times L^\infty(0, T^*; H^{s+1})\cap L^2(0, T^*; H^{s+2})$ such that for any $s' < s$, when $k\rightarrow \infty$,

Additionally, it is straightforward to observe that $(u, b)$ satisfies Eq (1.1) by taking the limit of the approximate equations; thus we have proved the uniqueness of the system in (1.1).

Step 4. Continuity of the solution.

To establish the continuity of the solution, we apply the operator $\Delta_j$ to both sides of (3.5) and then integrate it on $[0, T^*]$. Subsequently, multiplying the integrated expression by $2^{2js}$ and ultilizing Definition 2.5 gives rise to

Therefore, we easily infer the improved estimates for $(u, b)$ that

Meanwhile, Definition 2.4 indicates that there exists a positive integer $N$ such that

Hence, for any $t\in (0, T^*)$ and any $\delta$ such that $t+\delta\in (0, T^*)$, one obtains

where we have used Hölder's inequality and Lemma 2.1.

Taking $\delta$ which is small enough that $2N2^{2sN}|\delta|\|\partial_tu(t)\|_{L^2(0, T^*; L^2)}^2\le\frac{\varepsilon}{2}$, one can easily check that for any $\varepsilon > 0$, a $\delta > 0$ exists such that

which yields

On the other hand, by the same argument, the fact $b\in \tilde{L}^\infty(0, T^*; H^{s+1})$ ensures that $b\in C(0, T^*; H^{s+1})$. Therefore, the continuity of the solution $(u, b)$ to Eq (1.1) has been proved.

Step 5. Uniqueness of the solution.

Let $(u^{(1)}, b^{(1)}, \Pi^{(1)})$ and $(u^{(2)}, b^{(2)}, \Pi^{(2)})$ be two solutions of Eq (1.1) with the same initial data. We denote $\tilde{u}: = u^{(2)}-u^{(1)}$, $\tilde{b}: = b^{(2)}-b^{(1)}$, and $\tilde{\Pi}: = \Pi^{(2)}-\Pi^{(1)}$; $(\tilde{u}, \tilde{b}, \tilde{\Pi})$ then solves the following equation:

As we proved in Step 2, we can apply the standard $L^2$ energy estimate to $\tilde{u}$ and $\tilde{b}$, which gives

Combining integration by parts with Gagliardo-Nirenberg inequality, H$\ddot{o}$lder's inequality, and Young's inequality, we can estimate the right side of the inequality above as follows:

where $\xi$ is a function between $b^{(1)}$ and $b^{(2)}$. Putting all these estimates into the former inequality and utilizing the Sobolev embedding theorem, we can then rewrite the inequality as follows:

Note that

Grönwall's inequality then yields

Therefore, one can easily conclude that

Hence the uniqueness of the solution has been proved.

4.   Blow-up criteria

In this section, we prove the blow-up criterion of the system in (1.1) by a contradiction argument.

First of all, let $T^* < \infty$ be the maximal existence time of the solution and assume that

Taking the $L^2$ inner product of $(1.1)_1$ with $-\Delta u$, which, along with Hölder's and the Gagliardo-Nirenberg inequality, yields

Next, by applying the Laplace operator $\Delta$ to $(1.1)_2$ and multiplying the resulting expression by $\Delta b$, and then integrating the resulting expression over $\mathbb{R}^3$, we obtain

As we computed in the previous example, each term on the right-hand side of the inequality can also be estimated as follows:

Hence, it follows from (3.9) and a simple simplification that

Then, by virtue of Grönwall's inequality and (4.1), one deduces that

Furthermore, the Sobolev embedding theorem gives

Thus, with (4.2) and (4.3), by using the same method as employed in proving Eq (3.13), one eventually finds that when $s > \frac{3}{2}$,

The result implies that the solution can be extended even when $t = T^*$, which contradicts the definition of $T^*$. Therefore, we have

Finally, the converse is readily obvious from (4.4).

5.   Global existence

First, for simplicity, we define

Assuming that the solution $(u, b)$ blows up at $T^*$ and $T^* < \infty$, by utilizing the same proof of (3.14), one can find that when $t\in[0, T^*)$,

then

On the basis of the blow-up criterion we proved in Section 4, we conclude that $T^* = \infty$. Therefore, the proof of Theorem 1.1 is completed.

6.   Conclusions

In this paper, we studied a streamlined model that shares numerous similarities with the more complicated viscoelastic rate-type models commonly used in modeling fluids with complex microstructures. We have proven the blow-up result for this fluid, which implies the global well-posedness of the 3-D incompressible viscoelastic rate-type fluid system. In other words, this model has been mathematically proven to be thermodynamically consistent, and the blow-up criteria implies that these models are capable of describing the long-term behavior of fluids under small initial conditions and external forces. There are still many issues to explore concerning this equation, such as studying the stability of the solution or the existence of the smooth solution in the case of variable density, among others. These results will provide important preliminary insights into the mathematical properties of those more complex and practically significant models of non-Newtonian fluids used in applications.

Use of Generative-AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The author would like to thank the anonymous referees for their suggestions which made the paper more readable.

Conflict of interest

The author has no relevant financial or non-financial interests to disclose. The author has no competing interests to declare that are relevant to the content of this article.