Space-time decay rate of high-order spatial derivative of solution for 3D compressible Euler equations with damping

1.   Introduction and main results

In this paper, we investigate the space-time decay rate of the $(\ell-1)$th-order and $\ell$th-order spatial derivatives of the strong solution for the 3D compressible Euler equations with damping, which takes the following form:

where $\left(x, t\right) \in \mathbb{R}^{3}\times \mathbb{R_+}$ is the spatial coordinate and time. $\rho = \rho(x, t)$ and $u = u(x, t)$ represent the density and velocity respectively. The pressure $p = p(\rho)$ satisfies the $\gamma$-law with the adiabatic exponent $\gamma > 1$. The constant $\lambda > 0$ models the damping effect.

1.1.   History of the problem

We review some closely related results to this topic as follows. For the one-dimensional Cauchy problem, ^[2,3,4,5,6] presented a series of results on convergence rates for the nonlinear diffusion waves, ^[7,8,9,10] considered the asymptotic behavior of weak entropy solutions in vacuum and ^[11] focused on the well-posedness for compressible Euler equations with physical vacuum singularity. We also refer the readers to ^[12,13] for the initial-boundary value problem. For multidimensional cases, Jang and Masmoudi ^[14] showed the well-posedness for the compressible Euler equations with physical vacuum singularity. Wang and Yang ^[15] gave the global existence and the pointwise estimates of the solution with small data, Sideris et al. ^[16] used an equivalent reformulation of the system (1.1) to obtain the effective energy estimates. Liao et al. ^[17] studied the $L^p$ convergence rate of the planar diffusion waves by using approximate Green functions and the energy method. Fang et al. ^[18] obtained the existence and asymptotic behavior of $C^1$ solutions in some Besov space by using the spectral localization method. Tan and Wang ^[19,20] improved the decay result of the velocity in the $L^2$-norm that is $(1+t)^{-\frac{5}{4}}$ by using different methods. As for the initial boundary value problem, we refer the interested readers to ^{[15,21,22,23]}. For the decay rate of the strong solution to the Cauchy problem (1.1), under the assumption that $\left\|\left(\rho_0-\bar{\rho}, u_0\right)\right\|_{H^\ell}$ with the integer $\ell \geq 3$ was sufficiently small and $\left\|\rho_0-\bar{\rho}\right\|_{L^1}+\left\|u_0\right\|_{L^{\frac{3}{2}}}$ was finite, Chen and Tan ^[24] showed the temporal decay rate for the solution of (1.1) to be as follows:

Based on (1.2), for any integer $\ell\geq3$, Kim ^[1] obtained that the space-time decay rate of the solution for the system (1.1) is $O\left(t^{-\frac{3}{4}-\frac{k}{2}+\frac{\sigma}{2}}\right), 0\leq k \leq \ell-2$.

Noticing that there is no space-time decay rate of the $(\ell-1)$th-order or $\ell$th-order spatial derivative of solution to (1.1), the main motivation behind this paper is to significantly contribute to the resolution of this issue. More precisely, we establish the space-time decay rates of the $(\ell-1)$th-order and $\ell$th-order spatial derivatives of the strong solution for (1.1).

1.2.   Reformulation

In this paper, as in ^[16,25], we reformulate the Cauchy problem (1.1) as follows. Introduce the sound speed

and set $\bar{\mu} = \mu(\bar{\rho})$ to correspond to the sound speed at a background density $\bar{\rho} > 0$. Define

Then, (1.1) can be rewritten as

where

1.3.   Notations

$C$ and $C_i$ are time independent constants, which may vary in different places. $L^{p}$ and $H^{\ell}$ denote the usual Lebesgue space $L^{p}\left(\mathbb{R}^{3}\right)$ and Sobolev spaces $H^{\ell}\left(\mathbb{R}^{3}\right) = W^{\ell, 2}\left(\mathbb{R}^{3}\right)$ with norms $\|\cdot\|_{L^{p}}$ and $\|\cdot\|_{H^{\ell}}$ respectively. We denote $\|(f, g)\|_{X}: = \|f\|_{X}+\|g\|_{X}$ for simplicity. The notation $f \lesssim g$ means that $f \leq C g$. We often drop the $x$-dependence of the differential operators, that is $\nabla f = \nabla_{x} f = \left(\partial_{x_{1}} f, \partial_{x_{2}} f, \partial_{x_{3}} f\right)$ and $\nabla^{k}$ denotes any partial derivative $\partial^{\alpha}$ with the multi-index $\alpha, |\alpha| = k$. For any $\sigma \in \mathbb{R}$, denote the weighted Lebesgue space by $L_{\sigma}^{p}\left(\mathbb{R}^{3}\right)(2 \leq p < +\infty)$, where

Then, we can define the weighted Sobolev space as follows:

1.4.   Main results

We extend the work of Kim ^[1], which showed the space-time decay rate of the $k(0\leq k\leq\ell-2, \ell \geq 3)$th-order derivative of the strong solution for the system (1.3) as follows in Lemma 1.1. Based on (1.2) and the result of Kim ^[1], we can prove the space-time decay rate of the $k(0\leq k \leq \ell)$th-order derivative of the strong solution. It covers the results of Kim ^[1]. For the convenience of the readers, we outline the space-time decay rates of the $(\ell-1)$th-order and $\ell$th-order derivatives of the solution in the following Theorem 1.2.

Lemma 1.1. (Refer to Theorem 1.2 in ^[1]) For any integer $\ell\geq3$, the initial data $\left(n_{0}, u_{0}\right) \in H^{\ell}\left(\mathbb{R}^{3}\right)$, where $\left\|\left(n_0, u_0\right)\right\|_{H^\ell}$ is sufficiently small and $\left\|n_0\right\|_{L^1}+\left\|u_0\right\|_{L^{\frac{3}{2}}}$ is finite; Then, the strong solution $(n, u)$ of the system (1.3) such that

for all $0 \leq k \leq \ell-2$, $\sigma\geq 0$ and $t > T$, where $T$ is large enough.

Theorem 1.2. For any integer $\ell\geq3$, the initial data $\left(n_{0}, u_{0}\right) \in H^{\ell}\left(\mathbb{R}^{3}\right)\bigcap H_{\sigma}^{\ell}\left(\mathbb{R}^{3}\right)$, where $\left\|\left(n_0, u_0\right)\right\|_{H^\ell}$ is sufficiently small and $\left\|n_0\right\|_{L^1}+\left\|u_0\right\|_{L^{\frac{3}{2}}}$ is finite; Then, the strong solution $(n, u)$ of the system (1.3) such that

for all $\sigma\geq 0$ and $t > T$, where $T$ is large enough.

Remark 1.3. Kim ^[1] did not give the $(\ell-1)$th-order and $\ell$th-order spatial derivatives of the strong solution for the system (1.3) since the term $\left\|\nabla^k(n, u)(\cdot, t)\right\|_{L^{\infty}}(k = 0, \ldots, \ell-2)$ is involved in the following energy inequality

where $\hat{\mathcal{K}}(t) = \sum_{\beta = 0}^k \left\|\nabla^\beta(n, u)\right\|_{L_{\sigma}^{2}}^{2}(0\leq k \leq \ell-2)$. Noticing that the estimate in (1.6) is the sum of the estimates of $I_j(1\leq j\leq 5)$ given in (3.3). To prove (1.5), we need to develop some new thoughts, and our strategy can be outlined as follows. First, we make full use of the structure of $(1.3)$ to reduce the order of the spatial derivative of the solution (see (3.6) and (3.8)). Second, we make delicate energy estimates for $I_j$ respectively, and we employ different weighted energy methods for $0\leq k \leq \ell-2, k = \ell-1, k = \ell$ in the process. We can refer the readers to the proofs of (3.9) and (3.17) for more details.

Now, let us outline the strategies of proving Theorem 1.2 and explain the main difficulties in the process. We use the strategy of induction, delicate weighted energy estimates and the interpolation trick to prove Theorem 1.2. According to (1.4), Theorem 1.2 holds for $k = 0$ and $k = 1$. Using the strategy of induction and delicate weighted energy estimates, one has

where $\widehat{ \mathrm{E}}(t): = \left\|\nabla^k(n, u)\right\|_{L_{\sigma}^{2}}^{2}$. Combining the interpolation trick, we can prove that Theorem 1.2 holds for $0 \leq k \leq \ell$. The main difficulty is that Lemma 2.2 does not work in weighted Lebesgue space $L_\sigma^2$. To overcome the difficulty, we fully use the structure of $(1.3)$ to reduce the order of the spatial derivative of the solution, and we make delicate weighted energy estimates. For the sake of simplicity, we only take the trouble term $\langle |x|^{2 \sigma} \nabla^{\ell} n, \nabla^{\ell}(n\operatorname{div}u)\rangle$ in (3.6) as an example. First, by fully using the structure of $(1.3)_1$ to obtain an equation of $\operatorname{div}u$ given in (3.4), and by substituting (3.4) into this term, one has

Next, we will focus on the main trouble terms $K_{j}(j = 1, 2, 3)$. For the term $K_1$, by employing integration by parts, it holds that

For the term $K_2$, we have

where $\left\|\nabla^{\ell}n\right\|_{L_{\sigma}^{2}}^{2}-\langle |x|^{2 \sigma}|\nabla^{\ell} n|^2, \left(\frac{n}{\lambda+\nu n}\right)\rangle$ is equivalent to $\left\|\nabla^{\ell}n\right\|_{L_{\sigma}^{2}}^{2}$, since $\left\|n_0\right\|_{H^{\ell}}$ is sufficiently small, and the fact that

Noticing that Lemma 2.2 does not work in $L_\sigma^2$, for the term $K_3$, we need to employ some new ideas. The key observation here is to use Hölder's inequality skillfully to get

where

With (1.8)–(1.10) in hand, we can bound the trouble terms $K_1$, $K_2$ and $K_3$ properly.

The paper is organized as follows. In Section 2, we present some lemmas, which are used frequently throughout this paper. In Section 3, using the strategy of induction, delicate weighted energy estimates and the interpolation trick, we prove Theorem 1.2.

2.   Preliminaries

Lemma 2.1. (Gagliardo-Nirenberg inequality) Let $0 \leq i, j \leq k$; Then,

where a satisfies

Especially, when $p = 3, q = r = 2, i = j = 0$ and $k = 1$, combining Cauchy's inequality, we have

when $p = \infty, q = r = 2, i = 0, j = 1$ and $k = 2$, combining Cauchy's inequality, we have

and

while $i = j = 0, k = 1, a = 1$ and $p = q = r = 2$, combining Minkowski's inequality, we have

Proof. This is a special case of ^[26] and some simple inferences.

Lemma 2.2. Assume that the function $f(\varrho)$ satisfies

then for any integer $k \geq 0$ and $p \geq 2$, we have

Proof. Refer to Lemma A.4 of ^[24] for $p = 2$ and the Lemma 2.2 of ^[27] for $p \geq 2$.

Lemma 2.3. The vector function $f\in C_{0}^{\infty}\left(\mathbb{R}^{\mathrm{3}}\right)$ and bounded scalar function $g$ such that

Proof. The left side of the above inequality can be rewritten as

Using Hölder's inequality, we have

Lemma 2.4. (Interpolation inequality with weights) If $p, r \geqslant 1, s+n / r, \alpha+n /p, \beta+n / q > 0$ and $0 \leqslant\theta \leqslant 1$ then

for $f \in C_{0}^{\infty}\left(\mathbb{R}^{\mathrm{n}}\right)$ provided that

and

Especially, while $s = p = q = 2, \theta = \frac{\sigma-1}{\sigma}, s = \sigma-1, \alpha = \sigma$ and $\beta = 0$, we have

Proof. We compute

Thus, we complete the proof of Lemma 2.4.

Lemma 2.5. (Gronwall-type Lemma) Let $\alpha_{0} > 1, \alpha_{1} < 1, \alpha_{2} < 1$, and $\beta_{1} < 1, \beta_{2} < 1$. Assume that a continuously differential function $F:[1, \infty) \rightarrow [0, \infty)$ satisfies

where $C_{0}, C_{1}, C_{2}, C_{3}, K_{0}\geq0$ and $\sigma_{i} = \frac{1-\alpha_{i}}{1-\beta_{i}} > 0$ for $i = 1, 2$. Assume that $\sigma_{1} \geq \sigma_{2}$, then, there exists a constant $C^{*}$ depending on $\alpha_{0}, \alpha_{1}, \beta_{1}, \alpha_{2}, \beta_{2}, K_{0}$ and $C_{i}, i = 1, 2, 3$, for all $t \geq 1$, such that $F(t) \leq C^{*}t^{\sigma_{1}}.$

Proof. We can refer to Lemma 2.1 of ^[28].

3.   The proof of Theorem 1.2

According to (1.2), for all $0\leq k \leq \ell$ and $t > T$, where $T$ is large enough, we have

We will take the strategy of induction to prove Theorem 1.2 as follows. According to (1.4), Theorem 1.2 holds for $k = 0$ and $k = 1$. By the general step of induction, assume that the estimate (1.5) holds for $0\leq m\leq k-1(2\leq k\leq \ell)$, i.e.,

for $0\leq m\leq k-1$. Then, we need to verify that (3.2) holds for $m = k$. Applying $\nabla^{k}$ to each equation of $(1.3)_{1}$ and $(1.3)_{2}$, multiplying the $(1.3)_{1}$ and $(1.3)_{2}$ by $\left|x\right|^{2\sigma} \nabla^{k} n$ and $\left|x\right|^{2\sigma} \nabla^{k} u$ respectively, summing them up and then integrating over $\mathbb{R}^{3}$, we have

For $0\leq k \leq \ell$, there exist the terms involving $\nabla^{k+1}(n, u)$ in (3.3). To reduce the order of $\nabla^{k+1}(n, u)$, we have to use the equations $\operatorname{div} u$ and $\nabla n$ as derived from $(1.3)$ as follows:

Substituting (3.4) and (3.5) into (3.3), we have

Applying Lemma 2.3 and Cauchy's inequality, one has

For $J_2$, using integration by parts, we obtain

Using Minkowski's inequality, Hölder's inequality, Lemmas 2.3 and 2.1 (Gagliardo-Nirenberg inequality), (3.1) and (3.2), and Cauchy's inequality, we have

For $J_3$, we have

Using integration by parts, we obtain

Applying Minkowski's inequality, Hölder's inequality, Lemmas 2.3 and 2.1 (Gagliardo-Nirenberg inequality), (3.1), (3.2) and Cauchy's inequality, we have

Applying Minkowski's inequality, Hölder's inequality, Lemmas 2.2 and 2.1 (Gagliardo-Nirenberg inequality), (3.1), (3.2) and Cauchy's inequality, we obtain

Substituting (3.10)–(3.14) into (3.9), we have

For $J_4$, we have to use the equation $n_t$ derived from $(1.3)_1$ as follows:

Applying (3.16), $J_4$ can be rewritten as

Applying Minkowski's inequality, Hölder's inequality, Lemmas 2.2 and 2.1 (Gagliardo-Nirenberg inequality), (3.1), (3.2) and Cauchy's inequality, we obtain

Substituting (3.18)–(3.22) into (3.17), we have

Using the same arguments as $J_2$, $J_3$ and $J_ 4$ for $J_5$, $J_6$ and $J_7$ respectively, we have

Substituting (3.7), (3.8), (3.15), (3.23) and (3.24) into (3.6), and noticing that $\epsilon$ is small enough, then there exists a large enough $T$ such that

for all $t > T$. Defining

it is obvious that there exist two positive constants $\overline{C }$ and $\underline{C}$ such that $\underline{C }\left\|\nabla^k(p, u)\right\|_{L_{\sigma}^{2}}^{2}\leq H(t) \leq \overline{C }\left\|\nabla^k(p, u)\right\|_{L_{\sigma}^{2}}^{2}$. Thus, $H(t)$ is equivalent to $\left\|\nabla^k(p, u)\right\|_{L_{\sigma}^{2}}^{2}$, and (3.25) can be rewritten as

Substituting (2.3) and (3.1) into (3.26), we have

Denoting $\widehat{ \mathrm{E}}(t): = \left\|\nabla^k(n, u)\right\|_{L_{\sigma}^{2}}^{2}$, we can obtain

If $\sigma > \frac{3}{2}+k$, then we can apply Lemma 2.5 with $\alpha_{0} = \frac{5}{4} > 1, \alpha_{1} = \left(\frac{3}{4}+\frac{k}{2}\right) \frac{2}{\sigma} < 1$ and $\beta_{1} = \frac{\sigma-1}{\sigma} < 1$, $\sigma_{1} = \frac{1-\alpha_{1}}{1-\beta_{1}} = -\frac{3}{2}-k+\sigma$. Thus,

for all $t > T$. The Theorem 1.2 is proved for all $\sigma > \frac{3}{2}+k$ and the conclusion for the case of $[0, \frac{3}{2}+k]$ is proved by Lemma 2.4 (Interpolation inequality with weights). More precisely, by combining (3.1) and (3.28), we have

for all $t > T$ and $\sigma_{0} \in[0, \sigma]$, where $[0, \frac{3}{2}+k] \subset [0, \sigma](\sigma > \frac{3}{2}+k)$. Thus, we have covered the proof of Theorem 1.2.

Acknowledgments

This work was partially supported by the National Natural Science Foundation of China ($\#$12271114), Guangxi Natural Science Foundation ($\#$2019JJG110003, $\#$2019AC20214), Innovation Project of Guangxi Graduate Education ($\#$YCSW2023133) and Key Laboratory of Mathematical and Statistical Model (Guangxi Normal University), Education Department of Guangxi Zhuang Autonomous Region. The author is grateful to Prof. Yinghui Zhang for the helpful discussions.

Conflict of interest

The author declares that there is no conflicts of interest.