Mixed radial-angular bounds for Hardy-type operators on Heisenberg group

1.   Introduction

In this paper, we consider the Cauchy problem for the following two-dimensional inhomogeneous incompressible viscoelastic rate-type fluids with stress-diffusion:

where the unknowns $\rho = \rho(x, t)$, $u = (u^1(x, t), u^2(x, t))$ and $b = b(x, t)$ stand for the density, velocity of the fluid and the spherical part of the elastic strain, respectively. $P$ is a scalar pressure function, which guarantees the divergence-free condition of the velocity field. The coefficients $\nu$ and $\sigma$ are two positive constants. In addition, we suppose that $e(\cdot)$ is a smooth convex function about $b$ and $e(0)\leq0$, $e'(0) = 0$, $e''(b)\leq C_0$, where $C_0$ is a positive constant depending on the initial data. The class of fluids is the elastic response described by a spherical strain ^[3]. Compared with ^[3], we have added the divergence-free condition to investigate the effect of density on viscoelastic rate-type fluids, while the divergence-free condition is for computational convenience.

It is easy to observe that for $\sigma = 0$, the system (1.1) degenerates two distinct systems involving the inhomogeneous Navier-Stokes equation for the fluid and a transport equation with damped $e'(b)$. Numerous researchers have extensively studied the well-posedness concern regarding the inhomogeneous Navier-Stokes equations; see ^{[1,7,8,9,11,14]} and elsewhere. However, the transport equation has a greater effect on the regularity of density than on that of velocity. Additionally, due to the presence of the damped term $e'(b)$, the initial elasticity in system (1.1) exhibits higher regularity compared to the initial velocity.

In the case where $\sigma > 0$, system (1.1) resembles the inhomogeneous magnetohydrodynamic (MHD) equations, with $b$ as a scalar function in (1.1) that does not satisfy the divergence condition found in MHD equations. It is essential to highlight that the system (1.1) represents a simplified model, deviating from standard viscoelastic rate-type fluid models with stress-diffusion to facilitate mathematical calculations. Related studies on system (1.1) can be found in ^[3,4,15]. In particular, Bulíček, Málek, and Rodriguez in ^[5] established the well-posedness of a 2D homogeneous system (1.1) in Sobolev space. Our contribution lies in incorporating the density equation into this established framework.

Inspired by ^[11,18], we initially establish a priori estimates for the system (1.1). Subsequently, by using a Friedrich's method and the compactness argument, we obtain the existence and uniqueness of the solutions. Our main result is as follows:

Theorem 1.1. Let the initial data $(\rho_0, u_0, b_0)$ satisfy

where $m, M$ are two given positive constants with $m < M$. Then system (1.1) has a global solution $(\rho, u, b)$ such that, for any given $T > 0$, $(t, x)\in[0, T)\times\mathbb{R}^2$,

Moreover, if $\nabla\rho_0\in L^4(\mathbb{R}^2)$, then the solution is unique.

Remark 1.1. Compared to the non-homogeneous MHD equations, handling the damping term $e'(b)$ poses a challenge, so that we cannot obtain the time-weighted energy of the velocity field. To explore the uniqueness of the solution, it is necessary to improve the regularity of the initial density data.

The key issue to prove the global existence part of Theorem 1.1 is establishing the a priori $L^\infty(0, T; H^1(\mathbb{R}^2))$ estimate on $(u, \nabla b)$ for any positive time $T$. We cannot directly estimate the $L^2$ estimate of $(u, b)$, which mainly occurs in the velocity term $\mathrm{div}\, (\nabla b\otimes\nabla b-\frac{1}{2}\left|\nabla b\right|^2\mathbb{I})$. Therefore, we need to estimate the $L^2$ of the $\nabla b$ equation. Afterwards, the $L^2$ estimation of equation $b$ was affected by a damping term $e'(b)$, so we made an $L^2$ estimation of equation $e'(b)$. Finally, to show the $L^\infty(0, T; H^1(\mathbb{R}^2))$ of $u$, we also need an estimate of the second derivative of $b$. In summary, we found that the initial value of the $b$ equation needs to be one derivative higher than the initial value of the $u$ equation.

Concerning the uniqueness of the strong solutions, a common approach is to consider the difference equations between two solutions and subsequently derive some energy estimates for the resulting system differences based on the fundamental natural energy of the system. However, for system (1.1), the presence of a damping term $e'(b)$ of the equation $b$ and density equation prevents the calculation of the time-weighted energy of the velocity field. To research the solution's uniqueness, we need to enhance the regularity of the initial density data.

The paper is structured as follows: Section 2 presents prior estimates for system (1.1). In Section 3, we establish the existence and uniqueness of Theorem 1.1.

2.   A priori estimates

Proposition 2.1. Assume that $m, M$ are two given positive constants and $0 < m \leq M < \infty$, the initial data $\rho_{0}$ satisfies $0 < m\le \rho_{0}\le M < +\infty$, and the initial data $(\sqrt{\rho_0}u_0, \nabla b_0)\in L^{2}(\mathbb{R}^{2})\times L^{2}(\mathbb{R}^{2})$. Let $(\rho, u, b)$ be a smooth solution of system (1.1), then there holds for any $t > 0$,

where C is a constant depending only on $\sigma$, $\nu$.

Proof. First, any Lebesgue norm of $\rho_0$ is preserved through the evolution, and $0 < m\le \rho(t)\le M < +\infty$.

To prove (2.2), taking the $L^{2}$ inner product of the second equation of (1.1) with $u$ and integrating by parts, then we obtain

where we used the fact that

Multiplying the third equation of (1.1) by $-\sigma\Delta b$ and integrating by parts, we obtain

Thanks to the convexity of $e(b)$, we know

By inserting (2.5) into (2.4), combining the result with (2.3), one yields

Integrating it with respect to time, we have

On the other hand, applying $0 < m\le \rho \le M < + \infty$, which together with (2.6) implies

which, along with inequality (2.6), yields (2.2). □

Proposition 2.2. Under the assumptions of Proposition 2.1, the corresponding solution ($\rho$, u, b) of the system (1.1) admits the following bound for any $t > 0$:

where $C$ is a positive constant depending on $m$, $M$, $u_0$, $\rho_0$, and $\nabla b_0$.

Proof. First, we obtain by taking $L^2$ inner product of $(1.1)_3$ with $e'(b)$ that

Integrating with respect to time, we obtain

Similarly, multiplying $(1.1)_3$ by $b$, we have

after using (2.8) and Grönwall's inequality, we obtain

In the following, applying Laplace operator $\Delta$ to $(1.1) _3$ and multiplying the resulting equation by $\Delta b$; additionally, multiplying $(1.1)_2$ by $u_t$ and $(1.1)_3$ by $b_t$, respectively, then integrating them on $\mathbb{R}^2$ and adding up all these results together, we obtain

Utilizing Gagliardo-Nirenberg's, Hölder's, Young's inequalities (2.2), we estimate the first term as follows:

Similarly, by direct calculations, the other terms can be bounded as

Next, according to the regularity theory of the Stokes system in Eq $(1.1)_2$, it follows that

after multiplying by $\frac{1}{8}$, we arrive at

Substituting the estimates $I_1-I_6$ into (2.10) and combining inequality (2.11), we have

which, along with Grönwall's inequality (2.2), (2.8), and (2.9), leads to

which completes the proof of Proposition 2.2. □

Proposition 2.3. Under the assumptions of Proposition 2.2, there holds

where $C$ is a positive constant depending on $m$, $M$, $u_0$, $\rho_0$ and $b_0$.

Proof. Taking the derivative of Eq $(1.1)_2$ with respect to time $t$, then multiplying $u_t$ on both sides of the resulting equation and integrating by parts gives

Next, we compute each term on the right-hand side of the equation above one by one using estimates (2.2) and (2.7). The bound of the first term has been estimated as

By using Gagliardo-Nirenberg's, Hölder's, and Young's inequalities and (2.2), we have

Similarly,

and

Inserting these estimates into (2.14), we have

Now we turn to the $b$ equation of (1.1). Differentiating $(1.1)_3$ with respect to $t$, we obtain

Multiplying it by $b_t$ and $-\Delta b_t$, integrating the resulting equation, and summing up these results, due to the divergence-free condition $\mathrm{div}\, u = 0$, we obtain

Summing up (2.15) and (2.16) yields that

Applying (2.7) and Grönwall's inequality to the above inequality, we obtain

What's more, by the same argument of $\|u\|_{L^\infty (L^2)}$ in Proposition 2.1, we have

which completes the proof of Proposition 2.3. □

Proposition 2.4. Under the assumption of Proposition 2.3, it holds that for any $t > 0$:

and

Proof. Again, it follows from the regularity of the Stokes system

By Propositions 2.1–2.3, we obtain

and

which leads to (2.17). Finally, we recall that the density $\rho$ satisfies

Applying the operator $\nabla$ to both sides of the above equation yields

By applying the $L^p$ estimate to the above equation, combined with the divergence free condition implies

The Grönwall's inequality implies

We thus complete the proof of Proposition 2.4. □

3.   Proof of Theorem 1.1

The section is to prove Theorem 1.1. For any given $\rho_0$ and $(u_0, b_0)\in H^s(\mathbb{R}^2)\times H^{s+1}(\mathbb{R}^2)$, we define the initial data

where $\eta_\epsilon$ is the standard Friedrich's mollifier with $\epsilon > 0$. With the initial data $(\rho_0^{\epsilon}, u_0^{\epsilon}, b_0^{\epsilon})$, the system (1.1) has a unique global smooth solution $(\rho^{\epsilon}, u^{\epsilon}, b^{\epsilon})$. From Propositions 2.1 and 2.2, we obtain

By standard compactness arguments and Lions-Aubin's Lemma, we can obtain a subsequence denoted again by $(u^{\epsilon}, b^{\epsilon})$, that $(u^{\epsilon}, b^{\epsilon})$ converges strongly to $(u, b)$ in $L^2(\mathbb{R}^+; H^{s_1})\times L^2(\mathbb{R}^+; H^{s_2})$, as $\epsilon\rightarrow 0$, for $s_1 < 2$ and $s_2 < 3$. By the definition of $\rho_0^{\epsilon}$ and let $\epsilon\rightarrow 0$, we find that the limit $\rho$ of $\rho^{\epsilon}$ satisfies $m\leq\rho\leq M.$

Next, we shall prove the uniqueness of the solutions. Assume that $(\rho_i, u_i, b_i)\, (i = 1, 2)$ be two solutions of system (1.1), which satisfy the regularity propositions listed in Theorem 1.1. We denote

Then the system for $(\tilde{\rho}, \tilde{u}, \tilde{b}, \tilde{P})$ reads

where

Setting $\nu = \sigma = 1$ in what follows.

Step 1: Taking $L^2$ inner product to the second equation of (3.1) with $\tilde{u}$, we have

By Hölder's and interpolation inequalities, we have

Similarly,

where

Hölder's inequality implies

By substituting above estimates (3.3)–(3.5) into (3.2), we have

where

Step 2: Taking $L^2$ inner product to the third equation of (3.1) with $\tilde{b}-\Delta\tilde{b}$, we obtain

We shall estimate each term on the right-hand side of (3.7). For the first term of (3.7), using Hölder's inequality, we have

Meanwhile, we have

where $\xi$ is a function between $b_2$ and $b_1$.

Moreover,

and

By inserting (3.8)–(3.11) into (3.7), one yields

Step 3: We will derive the estimate of $\|\tilde{\rho}\|_{L^2}$ as follows:

Step 4: Summing up the above estimates, that is, (3.6), (3.12), and (3.13), we obtain

where

Noticing the fact that $\int_0^t \big(1+\mathcal{F}_3(\tau)+\mathcal{F}_4(\tau)+\mathcal{F}_5(\tau)\big)\mathrm{d}\tau\le Ct+C$ and that $\|f\|_{L^\infty}^2\leq\|f\|_{L^2}\|\nabla^2f\|_{L^2}$, we can obtain that there exists a small $\epsilon_0$ such that

for $t\in [0, \epsilon_0]$. Therefore, we obtain $\tilde{\rho}(t) = \tilde{u}(t) = \tilde{b}(t)\equiv 0$ for any $t\in [0, \epsilon_0]$. The uniqueness of such strong solutions on the whole time interval $[0, +\infty)$ then follows by a bootstrap argument.

Moreover, the continuity with respect to the initial data may also be obtained by the same argument in the proof of the uniqueness, which ends the proof of Theorem 1.1.

4.   Conclusions

This paper focuses on two-dimensional inhomogeneous incompressible viscoelastic rate-type fluids with stress-diffusion. We have established its global solution, and the uniqueness of the solution in specific situations is also proved in this paper.

Author contributions

Xi Wang and Xueli Ke: Conceptualization, methodology, validation, writing-original draft, writing-review & editing. All authors have read and approved the final version of the manuscript for publication.

Acknowledgments

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

Conflict of interest

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