Global Smooth Radially Symmetric Solutions to a Multidimensional Radiation Hydrodynamics Model

In the modeling of astrophysical flows, reentry problems, or high temperature combustion phenomena, we have to deal with high-temperature fluids and when the temperatures of the fluids are more than 10000K, radiative effect should be taken into consideration ([1], [2]).

In fact, when fluid interacts with radiation through energy exchange, since the momentum caused by the radiation can be neglected, while the radiative flux must be added into the energy equation since the transport of energy carried by the radiation process is more important, we can then use the following radiative Euler equations, which is a compressible Euler system coupled with an elliptic equation for radiation flux, as an approximate system to describe the motion of radiation hydrodynamics (For more Physical background, we refer to Chapter \@slowromancapxi@ and \@slowromancapxii@ in [23], and for derivation of the equation for radiation flux, one can also refer to [9] and [26]):

$$\left\{\begin{aligned} &\rho_{t}+\nabla_{x}\cdot(\rho\bm{u})=0,\\ &(\rho{\bf u})_{t}+\nabla_{x}\cdot(\rho({\bf u}\otimes{\bf u})+\nabla_{x}P=0,\\ &\left(\rho\left(e+\frac{|{\bf u}|^{2}}{2}\right)\right)_{t}+\nabla_{x}\cdot\left(\left(\rho\left(e+\frac{|{\bf u}|^{2}}{2}\right)+P\right){\bf u}\right)+\nabla_{x}\cdot{\bf q}=0,\\ &-\nabla_{x}\left(\nabla_{x}\cdot{\bf q}\right)+a_{1}{\bf q}+b_{1}\nabla_{x}\left(\theta^{4}\right)=0.\end{aligned}\right.$$

(1.1)

Here $a_{1}>0,b_{1}>0$ are positive constants. The primary dependent variables are the fluid density $\rho$, the fluid velocity ${\bf u}\in\mathbb{R}^{3}$, the absolute temperature $\theta$, and the radiative heat flux ${\bf q}\in\mathbb{R}^{3}$. The pressure $P$, the internal energy $e$, and the other three thermodynamic variables the density $\rho$, the absolute temperature $\theta$, and the specific entropy $s$ are related through Gibbs’ equation $de=\theta ds-Pd\rho^{-1}$.

Throughout this paper, we consider only ideal, polytropic gases:

	$\displaystyle P$	$\displaystyle=$	$\displaystyle R\rho\theta=A\rho^{\frac{1+C_{v}}{C_{v}}}{\rm exp}\left(\frac{s}{C_{v}}\right),$
	$\displaystyle\theta$	$\displaystyle=$	$\displaystyle\frac{A}{R}\rho^{\frac{1}{C_{v}}}{\rm exp}\left(\frac{s}{C_{v}}\right),$		(1.2)
	$\displaystyle\rho$	$\displaystyle=$	$\displaystyle\left(\frac{A}{R}\right)^{\frac{C_{v}}{C_{v}+1}}P^{\frac{C_{v}}{C_{v}+1}}{\rm exp}\left(-\frac{s}{C_{v}+1}\right),$

where $R>0,A>0$ (specific gas constant) and $C_{v}>0$ (specific heat at constant volume) are positive constants. Without loss of generality, we assume in the rest of this paper that $a_{1}=b_{1}=R=1$.

Due to the high nonlinearities of the rariative Euler equations (1.1) and the complex physical phenomena described by it, its mathematical theory is one of the hottest topics in the field of nonlinear partial differential equations. The main observation is that, although solutions of the compressible Euler system will generally develop singularity no matter how smooth and how small the initial data are (one can refer to John [10], Lax [12], Sideris [22] and the book [3]), it is believed that the radiation effect does imply some dissipative mechanism, which can guarantee the global regularity of the solutions to radiative Euler equations at least for small initial data. One of the main mathematical problems concerning the radiative Euler equations (1.1) is to justify the above expectation rigorously.

For such a problem, the results for one-dimensional case are quite complete, one can refer to Kawashima et al. [11] for the global existence of classical solutions and Deng and Yang [4] for pointwise structure. About the stability of elementary waves, one can refer to Lin et al. [15] and [16] respectively for the existence and stability of shock wave, Lin [14] for rarefaction wave and Wang and Xie [27] for viscous contact wave. As for the composite of several elementary waves, there are Fan et al. [6] about two viscous shock waves, Xie [28] about viscous contact wave and rarefaction waves. There are results with respect to some asymptotic limit of the radiation hydrodynamics towards elementary waves, we refer to Huang and Li [8], Wang and Xie [25] and Rohde et al. [20].

While for the multidimensional case, to the best of our knowledge, no result was available up to now. The main purpose of this paper is to show that the initial-boundary value problem of such a radiative Euler equations in a three-dimensional bounded concentric annular domain does admit a unique global smooth radially symmetric solution provided that the initial data is sufficiently small.

Now we transfer the system in our case to a desired form. Since all thermodynamics variables $\rho,\ \theta,\ e,\ P$ as well as the entropy $s$ can be written as functions of any two of them, if we take $P$ and $s$ as independent variables, the system (1.1) can be rewritten in terms of $P$, $\bm{u}$, $s$ and $\bm{q}$ as

$$\left\{\begin{aligned} &P_{t}+\left(\bm{u}\cdot\nabla\right)P+\frac{C_{v}}{C_{v}+1}P{\rm div}\bm{u}+\frac{1}{C_{v}}{\rm div}\bm{q}=0,\\ &\bm{u}_{t}+(\bm{u}\cdot\nabla)\bm{u}+\frac{\nabla P}{\rho}=0,\\ &s_{t}+(\bm{u}\cdot\nabla)s+\frac{{\rm div}\bm{q}}{P}=0,\\ &-\nabla{\rm div}\bm{q}+\bm{q}+\frac{4\theta^{3}}{C_{v}+1}\left(\frac{\nabla P}{\rho}+\theta\nabla s\right)=0.\end{aligned}\right.$$

(1.3)

Here to deduce the last equation in (1.3), we have used the fact that

	$\displaystyle-\nabla{\rm div}\bm{q}+\bm{q}+4\theta^{3}\nabla\theta$	$\displaystyle=-\nabla{\rm div}\bm{q}+\bm{q}+4\theta^{3}\left(\frac{1}{C_{v}+1}\nabla\theta+\frac{C_{v}}{C_{v}+1}\nabla\theta\right)$
		$\displaystyle=-\nabla{\rm div}\bm{q}+\bm{q}+4\theta^{3}\left\{\frac{1}{C_{v}+1}\nabla\theta+\frac{C_{v}}{C_{v}+1}\nabla\left[A\rho^{\frac{1}{C_{v}}}{\rm exp}\left(\frac{s}{C_{v}}\right)\right]\right\}$
		$\displaystyle=-\nabla{\rm div}\bm{q}+\bm{q}+4\theta^{3}\left(\frac{1}{C_{v}+1}\nabla\theta+\frac{1}{C_{v}+1}\frac{\theta}{\rho}\nabla\rho+\theta\nabla s\right)$
		$\displaystyle=-\nabla{\rm div}\bm{q}+\bm{q}+\frac{4\theta^{3}}{C_{v}+1}\left(\frac{\nabla P}{\rho}+\theta\nabla s\right)=0.$

We consider the system (1.3) in a bounded concentric annular domain $\Omega=\{\bm{x}\ \big{|}\ \bm{x}\in\mathbb{R}^{3},\ |\bm{x}|=r,\ 0<a<r<b<+\infty\}$ with prescribed initial data

$\displaystyle\big{(}P(0,\bm{x}),\bm{u}(0,\bm{x}),s(0,\bm{x})\big{)}=(P_{0}(\bm{x}),\bm{u}_{0}(\bm{x}),s_{0}(\bm{x}))\quad{\rm for}\ \ \bm{x}\in\Omega$

(1.4)

and boundary conditions

$$\bm{u}(t,\bm{x})=0,\quad\bm{q}(t,\bm{x})=0\quad{\rm on}\ (t,\bm{x})\in[0,+\infty)\times\partial\Omega.$$

(1.5)

Now we study the spherically symmetric classical solution to the initial boundary problem (1.3)–(1.5) with the form that

	$\displaystyle P(t,\bm{x})=\rho(t,\bm{x})\theta(t,\bm{x})=\tilde{\rho}(t,r)\tilde{\theta}(t,r)=\tilde{P}(t,r),$
	$\displaystyle\bm{u}(t,\bm{x})=\frac{\tilde{u}(t,r)}{r}\bm{x},\ s(t,\bm{x})=\tilde{s}(t,r),\ \bm{q}(t,\bm{x})=\frac{\tilde{q}(t,r)}{r}\bm{x}.$

The corresponding spherical form of the initial boundary value problem (1.3)–(1.5) reads

$$\left\{\begin{aligned} &\tilde{P}_{t}+\tilde{u}\tilde{P}_{r}+\frac{C_{v}+1}{C_{v}}\tilde{P}\frac{(r^{2}\tilde{u})_{r}}{r^{2}}+\frac{1}{C_{v}}\frac{(r^{2}\tilde{q})_{r}}{r^{2}}=0,\\ &\tilde{u}_{t}+\tilde{u}\tilde{u}_{r}+\frac{\nabla\tilde{P}}{\tilde{\rho}}=0,\\ &\tilde{s}_{t}+\tilde{u}\tilde{s}_{r}+\frac{(r^{2}\tilde{q})_{r}}{r^{2}\tilde{P}}=0,\\ &-\left[\frac{(r^{2}\tilde{q})_{r}}{r^{2}}\right]_{r}+\tilde{q}+\frac{4\tilde{\theta}^{3}}{C_{v}+1}\left(\frac{\tilde{P}_{r}}{\tilde{\rho}}+\tilde{\theta}\tilde{s}_{r}\right)=0\end{aligned}\right.$$

(1.6)

with the corresponding initial and boundary conditions

	$\displaystyle\big{(}\tilde{P}(0,r),\tilde{u}(0,r),\tilde{s}(0,r)\big{)}$	$\displaystyle=(\tilde{P}_{0},\tilde{u}_{0},\tilde{s}_{0})(r),\qquad$	$\displaystyle a\leq r\leq b,$		(1.7)
	$\displaystyle(\tilde{u}(t,a),\tilde{q}(t,a))$	$\displaystyle=(\tilde{u}(t,b),\tilde{q}(t,b))=0,\qquad$	$\displaystyle t\geq 0.$		(1.8)

Now we interpret the initial boundary value problem (1.3)–(1.8) into Lagrangian coordinates. We now use $\eqref{cns1}_{1}$ to find another set of variables $(t^{\prime},x)$ such that

$$\left\{\begin{aligned} r&=r(t^{\prime},x)\\ t&=t^{\prime}\end{aligned}\right.$$

and $x$ is not dependent on $t$, which is called Lagrangian coordinate. Define coordinate transformation that

$$\left\{\begin{aligned} &r(t^{\prime},x)=r_{0}(x)+\int_{0}^{t^{\prime}}\tilde{u}(\tau,r(\tau,x))\mathrm{d}\tau,\\ &t=t^{\prime},\end{aligned}\right.$$

(1.9)

where $r_{0}(x)$ is defined by following way: First, define the function $h(z)$ as

$\displaystyle h(z):=\int_{a}^{z}y^{2}\tilde{\rho}_{0}(t,y)\ \mathrm{d}y.$

Since that $\tilde{\rho}_{0}(y)>0$ (which is assumed in Theorem 1.1), function $h(z)$ is invertible and $h^{-1}(\cdot)$ is well defined. Then, $r_{0}(x)$ is defined as

$\displaystyle r_{0}(x)=h^{-1}(x).$

(1.10)

We claim that the coordinate $(t^{\prime},x)$ is Lagrangian coordinate. The first thing we want to show is that $x$ is truly not dependent on $t$. So we now prove that $\partial_{t}x=0$ and

$\displaystyle x=\int_{a}^{r(t^{\prime},x)}y^{2}\tilde{\rho}(t,y)\ \mathrm{d}y=\int_{a}^{r(t,x)}y^{2}\tilde{\rho}(t,y)\ \mathrm{d}y.$

(1.11)

From $\eqref{cns1}_{1}$ and (1.9), we can see that

	$\displaystyle\partial_{t}\int_{a}^{r(t,x)}y^{2}\tilde{\rho}(t,y)\ \mathrm{d}y$	$\displaystyle=r^{2}\tilde{\rho}(t,r)\partial_{t}r-\int_{a}^{r(t,x)}y^{2}\tilde{\rho}_{t}(t,y)\ \mathrm{d}y$
		$\displaystyle=r^{2}\tilde{\rho}(t,x)\partial_{t}r-\int_{a}^{r(t,x)}\mathrm{d}\left[y^{2}\tilde{\rho}\tilde{u}(t,y)\right]$
		$\displaystyle=r^{2}\tilde{\rho}\left(\partial_{t}r-\tilde{u}\right)=0.$		(1.12)

The second equality holds because after spherically symmetric transform, the equation of $\tilde{\rho}(t,r)$ takes the form that

$\displaystyle\tilde{\rho}_{t}+\frac{(r^{2}\tilde{\rho}\tilde{u})_{r}}{r^{2}}=0.$

(1.13)

We integrate (1.12) over $(0,t)$ to get that

$\displaystyle\int_{a}^{r(t,x)}y^{2}\tilde{\rho}(t,y)\ \mathrm{d}y=\int_{a}^{r_{0}(x)}y^{2}\tilde{\rho}_{0}(y)\ \mathrm{d}y=h(r_{0})=x.$

Without loss of generality, we set $h(b)=1$, so that $x\in[0,1]$. Since $t^{\prime}=t$, without confusion, we still use $t$ to denote the time variable and we have

$\displaystyle\tilde{P}(t,r)=\tilde{\rho}(t,r)\tilde{\theta}(t,r)=\rho(t,x)\theta(t,x)=P(t,x),\ \tilde{u}(t,r)=u(t,x),\ \tilde{s}(t,r)=s(t,x),\ \tilde{q}(t,r)=q(t,x).$

The next thing we need to know is that whether this coordinate transformation is well-posed. From the a priori estimates which we are going to derive later , it can be seen that $\rho>0$, so the coordinate transformation is well-posed and identities (1.11), (1.9) imply

$\displaystyle r_{t}(t,x)=u(t,x),\quad r_{x}(t,x)=\frac{1}{r^{2}\rho(t,x)}.$

(1.14)

By virtue of (1.14), the system (1.6) is reformulated to that of $(P,u,s,q)(t,x)$, which takes the form

$$\left\{\begin{aligned} &P_{t}+\frac{C_{v}+1}{C_{v}}P\rho(r^{2}u)_{x}+\frac{1}{C_{v}}\rho(r^{2}q)_{x}=0,\\ &u_{t}+r^{2}P_{x}=0,\\ &s_{t}+\frac{(r^{2}q)_{x}}{\theta}=0,\\ &-r^{2}\rho\left[\rho(r^{2}q)_{x}\right]_{x}+q+\frac{4r^{2}\theta^{3}}{C_{v}+1}\left(P_{x}+Ps_{x}\right)=0,\end{aligned}\right.$$

(1.15)

where $t>0$, $x\in\mathbb{I}:=(0,1)$. The corresponding initial and boundary conditions are

	$\displaystyle(P(0,x),u(0,x),s(0,x))$	$\displaystyle=(P_{0}(x),u_{0}(x),s_{0}(x)),\qquad$	$\displaystyle x\in\mathbb{I},$		(1.16)
	$\displaystyle(u(t,0),q(t,0))$	$\displaystyle=(u(t,1),q(t,1))=0,\qquad$	$\displaystyle t\geq 0,$		(1.17)

where $(P_{0},u_{0},s_{0}):=(\tilde{P}_{0},\tilde{u}_{0},\tilde{s}_{0})\circ r_{0},$ the symbol $\circ$ denotes composition, and $r_{0}$ is defined by (1.10).

To state our main result, we first introduce some notations.

Firstly, the initial data $\left(\partial_{t}^{k}P(0,x)\right.$,$\partial_{t}^{k}u(0,x)$,$\partial_{t}^{k}s(0,x)$,$\left.\partial_{t}^{k-1}q(0,x)\right)$ (for any integer $k\geq 1$) are defined inductively by following procedure: first apply $\partial_{t}^{k-1}$ ($k=1,2,\cdots$) on system (1.15); second obtain $\partial_{t}^{k-1}q(0,x)$ by letting $t=0$ in the last elliptic equation and solving it; Third obtain $\left(\partial_{t}^{k}P(0,x),\partial_{t}^{k}u(0,x),\partial_{t}^{k}s(0,x)\right)$ by solving for $\left(\partial_{t}^{k}P(t,x),\partial_{t}^{k}u(t,x),\partial_{t}^{k}s(t,x)\right)$ in the first three equations and evaluating at $t=0$.

In order to clarify the space of solutions, we introduce the space

$\displaystyle X_{m}\left([0,T];\mathbb{I}\right):=\bigcap_{k=0}^{m}C^{k}\left([0,T];H^{m-k}\left(\mathbb{I}\right)\right)\quad\left(\ \text{for some positive integer }m\ \right)$

with norm

$\displaystyle{\left|\kern-0.96873pt\left|\kern-0.96873pt\left|f\right|\kern-0.96873pt\right|\kern-0.96873pt\right|}_{m,T}=\sup_{0\leq t\leq T}{\left|\kern-0.96873pt\left|\kern-0.96873pt\left|f(t)\right|\kern-0.96873pt\right|\kern-0.96873pt\right|}_{m}=\sup_{0\leq t\leq T}\sum_{k=0}^{m}\left\|\partial_{t}^{k}f(t)\right\|_{H^{m-k}(\mathbb{I})}.$

According to the above definitions, the value ${\left|\kern-0.96873pt\left|\kern-0.96873pt\left|\left(P,u,s,q,q_{x}\right)(0)\right|\kern-0.96873pt\right|\kern-0.96873pt\right|}_{m}$ is well-defined. Furthermore, if we assume $\|(P_{0}-1,u_{0},s_{0}-1)\|_{H^{m}}$ is sufficiently small, it holds that

$\displaystyle{\left|\kern-0.96873pt\left|\kern-0.96873pt\left|\left(P,u,s,q,q_{x}\right)(0)\right|\kern-0.96873pt\right|\kern-0.96873pt\right|}^{2}_{m}\leq C_{1}\left\{\|(P_{0}-1,u_{0},s_{0}-1)\|^{2}_{H^{m}}\right\}\|(P_{0}-1,u_{0},s_{0}-1)\|^{2}_{H^{m}}.$

(1.18)

Here and in the rest of this paper, we will frequently use $C\{\cdot\}$, $C_{i}\{\cdot\}$, $G_{j}\{\cdot\}$ to denote some positive constants which are continuous nondecreasing functions of the quantities listed in braces.

Especially, $\interleave\cdot\interleave_{m,tan}$ is the norm when derivatives with respect to $x$ are not included. That is,

$\displaystyle{\left|\kern-0.96873pt\left|\kern-0.96873pt\left|f(t)\right|\kern-0.96873pt\right|\kern-0.96873pt\right|}_{m,tan}=\sum_{k=0}^{m}\left\|\partial_{t}^{k}f(t)\right\|_{L^{2}(\mathbb{I})}.$

Our ultimate goal is to prove the following theorem concerning the global existence of classical solution to the initial boundary problem (1.15)–(1.17) when the initial data is a small perturbation of a constant state which, for simplicity, is taken as (1,0,1). (The corresponding equilibrium state for $(\rho,u,\theta)$ is $(c_{\rho},0,c_{\theta})$. We can see from (1) that $c_{\rho}$ and $c_{\theta}$ are some positive constants.)

We can see it from the a priori estimates in Section 2 that the main difficulty to obtain the global solvability results from the high-dimensional space variable $r$, because the spatial derivative of $r$ (referring to $\eqref{r_eq}_{2}$) is not small even with small initial data. Furthermore, we can see $r$ comes from the Lagrangian coordinate transformation, which gives us a beautiful structure of system (1.15). However, $r$ as a function of $\rho$ and $u$, its definition closely related to the nonlinear structure of mass-conservation equation (1.13). This dependency will cause troubles when we try to obtain the local solution of linearized system. (One can refer to the comments under (3.6), (3.8) and (3.36).)

As usual, the main proof of Theorem 1.1 can be divided into two parts, the a priori estimates and the local existence result, which are established respectively in Section 2 and 3. If these two results stand, then with the method of continuity (see for an example that [24, p. 5972-5973]), Theorem 1.1 holds.

To simplify the a priori estimates, we reformulate our system around the equilibrium state by taking change of variables as that $(P,u,s,q)\rightarrow(P+1,u,s+1,q)$ and the system (1.15) is reformulated as

$$\left\{\begin{aligned} &P_{t}+\frac{C_{v}+1}{C_{v}}(P+1)\rho(r^{2}u)_{x}+\frac{1}{C_{v}}\rho(r^{2}q)_{x}=0,\\[5.69054pt] &(r^{2}u)_{t}+r^{4}P_{x}=2ru^{2},\\[5.69054pt] &s_{t}+\frac{(r^{2}q)_{x}}{\theta}=0,\\ &q-r^{2}\rho^{2}(r^{2}q)_{xx}+\frac{4r^{2}\theta^{3}}{C_{v}+1}\left(P_{x}+s_{x}\right)=\frac{1}{2}r^{2}(\rho^{2})_{x}(r^{2}q)_{x}-\frac{4r^{2}\theta^{3}}{C_{v}+1}Ps_{x},\end{aligned}\right.$$

(2.1)

as well as the linearized form that

$$\left\{\begin{aligned} &P_{t}+\frac{C_{v}+1}{C_{v}}c_{\rho}(r^{2}u)_{x}+\frac{1}{C_{v}}(r^{2}q)_{x}=\mathbb{S}_{1},\\ &u_{t}+r^{2}P_{x}=0,\\[5.69054pt] &s_{t}+\frac{1}{c_{\theta}}(r^{2}q)_{x}=\mathbb{S}_{3},\\ &q-r^{2}c_{\rho}^{2}(r^{2}q)_{xx}+\frac{4r^{2}c_{\theta}^{3}}{C_{v}+1}\left(P_{x}+s_{x}\right)=\mathbb{S}_{4},\end{aligned}\right.$$

(2.2)

where

	$\displaystyle\mathbb{S}_{1}=\frac{C_{v}+1}{C_{v}}(c_{\rho}-\rho)(r^{2}u)_{x}+\frac{C_{v}+1}{C_{v}}\rho P(r^{2}u)_{x}+\frac{1}{C_{v}}(1-\rho)(r^{2}q)_{x},$
	$\displaystyle\mathbb{S}_{3}=\left(\frac{1}{c_{\theta}}-\frac{1}{\theta}\right)(r^{2}q)_{x},$
	$\displaystyle\mathbb{S}_{4}=r^{2}(\rho^{2}-c_{\rho}^{2})(r^{2}q)_{xx}+\frac{4r^{2}}{C_{v}+1}(c_{\theta}^{3}-\theta^{3})\left(P_{x}+s_{x}\right)+\frac{1}{2}r^{2}(\rho^{2})_{x}(r^{2}q)_{x}-\frac{4r^{2}\theta^{3}}{C_{v}+1}Ps_{x}.$

The corresponding initial and boundary conditions are given by that

	$\displaystyle(P(0,x),u(0,x),s(0,x))$	$\displaystyle=(P_{0}-1,u_{0},s_{0}-1)(x),\qquad$	$\displaystyle x\in\mathbb{I},$		(2.3)
	$\displaystyle(u(t,0),q(t,0))$	$\displaystyle=(u(t,1),q(t,1))=0,\qquad$	$\displaystyle t\geq 0,$		(2.4)

For later use, we list some Sobolev inequalities refined with respect to norm $\interleave\cdot\interleave_{m}$ as follows

We define the solution spaces with parameters $T>0$ and $N>0$ that

	$\displaystyle A_{m}\left([0,T];\mathbb{I};N\right)=\left\{f(t,x)\ \Big{|}\ (t,x)\in[0,T]\times\mathbb{I},\ {\left|\kern-0.96873pt\left|\kern-0.96873pt\left|f\right|\kern-0.96873pt\right|\kern-0.96873pt\right|}^{2}_{m,tan}+{\left|\kern-0.96873pt\left|\kern-0.96873pt\left|f\right|\kern-0.96873pt\right|\kern-0.96873pt\right|}^{2}_{m-1}\leq N\right\},$
	$\displaystyle B_{m}\left([0,T];\mathbb{I};N\right)=\left\{f\in X_{m}\left([0,T];\mathbb{I}\right)\Big{|}\ {\left|\kern-0.96873pt\left|\kern-0.96873pt\left|f\right|\kern-0.96873pt\right|\kern-0.96873pt\right|}_{m}\leq N\right\}.$

The a priori estimates $(P,u,s,q)$ in a certain solution space are derived in this subsection as follows.

The proof of Proposition 2.2 is divided into the following three lemmas.