Vanishing viscosity limit to the planar rarefaction wave for the two-dimensional radiative hydrodynamics

1.   Introduction

The two-dimensional radiative hydrodynamics are formulated as follows:

where $\rho \ge 0, {\boldsymbol{u}}: = (u_1, u_2)^{\mathsf{T}}$, $p$ and $\theta$ represent the fluid's density, velocity, pressure, and absolute temperature, ${\boldsymbol{q}}: = (q_1, q_2)^{\mathsf{T}}$ is the radiative heat flux. Moreover, $E: = e+\frac{1}{2}\left | {\boldsymbol{u}} \right | ^2$ is the specific total energy with the internal energy $e$. All variables mentioned above depend on $t$ and ${\boldsymbol{x}}$, where $t$ is time and ${\boldsymbol{x}}: = (x_1, x_2)\in \Omega$ is the spatial variable. Here, we are concerned with the viscous fluid flow in an infinitely long flat nozzle domain $\Omega : = \mathbb{R} \times \mathbb{T}$ with a real line $\mathbb{R}$ and a one-dimensional unit flat torus $\mathbb{T}: = \mathbb{R}/\mathbb{Z}$.

The viscous stress tensor $\mathcal{T}$ is given by

where $\mathbb{D}({\boldsymbol{u}}) = \frac{\nabla {\boldsymbol{u}}+(\nabla {\boldsymbol{u}})^{\mathsf{T}} } {2}$ stands for the deformation tensor, $\mathbb{I}$ represents the 2 × 2 identity matrix, parameters $\mu _1$ and $\lambda _1$ represent the shear and bulk viscosity coefficients of the fluid, respectively, and they both are constants satisfying

Moreover, the constant $\kappa_1 > 0$ denotes the heat-conductivity coefficient, and $a_1 > 0, b_1 > 0$ depend only on the fluid itself.

Setting

where $\epsilon > 0$ is the vanishing parameter, and constant $\mu, \lambda, \kappa, a, b$ are the prescribed uniformly in parameter $\epsilon$. Such a setting for $a_1$ and $b_1$ is borrowed from ^[1]. Specifically, we investigate the ideal polytropic fluids such that the pressure $p$ and the internal energy $e$ are given by the following state equations:

where $S$ is the entropy, $\gamma > 1$ is the adiabatic exponent, and both $A$ and $R$ are positive constants.

Under the Assumption (1.3), letting $\epsilon$ tend to zero, we formally derive that the solutions to the two-dimensional compressible Navier-Stokes equations with radiation term (1.1) converge to the corresponding two-dimensional compressible Euler equations

Literature review. The vanishing viscosity limit for the compressible Navier-Stokes equations to the Euler equations with basic wave patterns has been extensively investigated. In ^[2], the zero-dissipation limit problem for the compressible and isentropic Navier-Stokes equations to the corresponding Euler equations with rarefaction wave solutions was considered. The solution to the compressible isentropic Navier-Stokes equations with shock data converges to the inviscid shock as the viscosity tends to zero and was derived in ^[3]. Concerning the Riemann solution to the Euler equations, which consists of the superposition of basic waves, such a vanishing viscosity limit problem was also investigated. This includes the superposition of a shock wave and a rarefaction wave in ^[4], and the superposition of rarefaction waves and contact discontinuity in ^[5]. For more results on the vanishing viscosity limit problem, further references can be found in ^[6,7,8,9] and the references therein. Up to now, all the results mentioned above are related to one-dimensional case. However, in thr high-dimensional case, the vanishing viscosity limit of the two-dimensional compressible and isentropic Navier-Stokes equations to the Euler equations with a planar rarefaction wave solution was studied in ^[10]; the three-dimensional compressible Navier-Stokes-Fourier equations were further considered in ^[11]; and the two dimensional full compressible Navier-Stokes equations were derived in ^[12]. The planar rarefaction wave to the two-dimensional compressible and isentropic Navier-Stokes equations was given in ^[13]; subsequently, the planar rarefaction wave for the three-dimensional full, compressible Navier–Stokes equations with the heat conductivities in an infinitely long flat nozzle domain was investigated in ^[14]. Very recently, the vanishing viscosity limit to the planar rarefaction wave with vacuum for the three-dimensional full compressible Navier-Stokes equations with temperature-dependent transport coefficients was presented in ^[15]. Nowadays, the nonlinear stability for the radiative hydrodynamics has been studied very thoroughly. The composite wave of two viscous shock waves for the one-dimensional radiative Euler equations was given in ^[16]; then the rarefaction wave case was addressed in ^[17] for the inflow problem and in ^[18] for the outflow problem.

We consider the two-dimensional radiative hydrodynamics (1.1) with the following initial data:

and the far field conditions of the solutions in the $x_1$-direction:

where $\rho_\pm > 0, u_{1\pm}, \theta_\pm$ are the prescribed constants. The periodic boundary condition is imposed on $x_2\in\mathbb{T}$ for the solution $(\rho, u_1, u_2, \theta)(t, x_1, x_2)$ to (1.1), and the end states $(\rho_\pm, u_{1\pm}, \theta_\pm)$ are connected by the rarefaction wave solution to the Riemann problem of the corresponding one-dimensional compressible Euler system:

with the Riemann initial data

With the above assumptions in hand, we could expect that as $\epsilon \to 0$, the solutions to the compressible Navier-Stokes equations (1.1), (1.5), and (1.6) converge to the corresponding planar rarefaction wave for the two-dimensional compressible Euler equations (1.4) with the following Riemann initial data:

Direct calculations reveal that the solution $(\rho, u_1, \theta)$ to the Euler system (1.7) has three distinct eigenvalues

What's more, three corresponding right eigenvectors are denoted by $\gamma_i(\rho, u_1, S)$, it has

and

Furthermore, the two $i$-Riemann invariants $\Sigma_i^{j}(\rho, u_1, S)\quad (i = 1, 3, j = 1, 2)$ to the Euler system (1.7) can be given by

satisfying $\nabla _{(\rho, u_1, S)}\Sigma_i^{j}(\rho, u_1, S)\cdot\gamma _i(\rho, u_1, S)\equiv 0 \quad (i = 1, 3, j = 1, 2)$ for all $\rho > 0, u_1$ and $S$.

In this paper, we mainly consider the 3-rarefaction wave to the Euler systems (1.7) and (1.8). Actually, we also can manage the 1-rarefaction wave similarly without any substantial difference. For the given right state $(\rho_+, u_{1+}, \theta_+)$ with $\rho_+ > 0, \theta_+ > 0$, we know that when $(\rho_-, u_{1-}, \theta_-)\in R_3(\rho_+, u_{1+}, \theta_+)$, where

the Euler systems (1.7) and (1.8) admit a 3-rarefaction wave $(\rho ^r, u_1^r, \theta ^r)(\frac{x_1}{t})$. For more details about basic waves, interested readers can refer to^[19,20]. Subsequently, the above analysis allows us to define the planar rarefaction wave solution to the two-dimensional compressible Euler equations (1.4) with initial data (1.5) and (1.6) as $(\rho ^r, {\boldsymbol{u}}^r, \theta ^r)(\frac{x_1}{t}) = (\rho ^r, u_1^r, 0, \theta ^r)(\frac{x_1}{t})$.

Our main result is given as follows:

Theorem 1.1. Let $(\rho ^r, {\boldsymbol{u}}^r, \theta ^r)(\frac{x_1}{t})$ be the planar 3-rarefaction wave to the two-dimensional compressible Euler system (1.4) with Riemann initial data (1.5). Then there exists a constant $\epsilon_0 > 0$ such that for any $\epsilon \in (0, \epsilon_0)$, we are able to construct smooth solutions $(\rho ^{\epsilon}, {\boldsymbol{u}}^{\epsilon}, \theta ^{\epsilon})$ up to any arbitrarily large but fixed time $T$ for the system (1.1) satisfying

In addition, for each small constant $\underline{T} > 0$, there is an independent constant $C_{\underline{T}, T} > 0$ of $\epsilon$, such that

where $\omega = \frac{1}{4}$.

Moreover, letting $\epsilon \rightarrow 0$, it holds that the smooth solution $(\rho ^{\epsilon}, {\boldsymbol{u}}^{\epsilon}, \theta ^{\epsilon})$ converges to the planar rarefaction wave fan $(\rho ^r, {\boldsymbol{u}}^r, \theta ^r)(\frac{x_1}{t})$ point wisely, except for $(0, 0)$, and

We make some comments on our analysis. Based on the observations of cancellations between the flux terms and viscosity terms for full compressible Navier–Stokes equations in ^[12], this method can also be used in the full system of hydrodynamic equations coupled with a nonlinear elliptic equation. Moreover, we will introduce the new cancellations considering the radiative term. Our concern lies in the convergence rate from the compressible radiative hydrodynamics to the planar rarefaction wave solution for the corresponding Euler equations. Instead of specifying the order of scaling, we provide a broad range of orders. Subsequently, under our chosen scaling setting, we derive a rate in (1.11) through elaborate energy estimates. This demonstrates that our result serves as a generalization of ^[11,12] to radiative hydrodynamics.

The rest of the paper is organized as follows: in section 2, we present the smooth approximate rarefaction wave to the Euler equation, along with an introduction to the hyperbolic wave and the solution profile. Section 3 focuses on reformulating the system for the perturbation of the solution to the radiative hydrodynamics around the solution profile, which consists of the approximate rarefaction wave and the hyperbolic wave. Subsequently, we provide proof of the main result.

Notation. The notations listed below will be widely used in this paper. The standard Sobolev space with the norm $\|\cdot\|_k$ is denoted by $H^k(\mathbb{R}\times \mathbb{T})$ and $H^k(\mathbb{R}\times \mathbb{T}_\epsilon) (k \ge 0, k \in \mathbb{Z}),$ where $\mathbb{T}_\epsilon: = \mathbb{R}/\frac{1}{\epsilon^\alpha}\mathbb{Z}$ is the scaled torus. We set $H^0(\mathbb{R}\times \mathbb{T}_\epsilon) = L^2(\mathbb{R}\times \mathbb{T}_\epsilon)$, $H^0(\mathbb{R}\times \mathbb{T}) = L^2(\mathbb{R}\times \mathbb{T})$, and $\|\cdot \| = \|\cdot \|_0$. We additionally choose $C$ as a general positive constant that is independent of $T, \epsilon$ and $\delta$, and $C_T$ as a positive constant that is independent of $\epsilon$ and $\delta$ but only relies on $T$.

2.   Construction of profile

In this section, we construct the approximate rarefaction wave to the Euler system (1.7) through using the inviscid Burgers' equation, and then the hyperbolic wave will be constructed. We can refer to ^[2,14,21] for details. Last, we construct the solution profile that combines the rarefaction wave and the hyperbolic wave.

2.1.   Smooth approximate rarefaction wave

The Riemann problem for the inviscid Burgers' equation is formulated as:

If $w_- < w_+$, then (2.1) has a rarefaction wave $w^r(x_1, t) = w^r(x_1/t)$ given by

Based on the fact that the rarefaction wave can only be Lipschitz continuous, we follow the method adopted in ^[1] to develop the approximation rarefaction wave with the smooth solution to the Burgers' equation,

where $\delta > 0$ is a small constant depending on the viscosity parameter $\epsilon$. The following lemma is to show the properties of the solution $w(t, x_1)$ to (2.3), which will be frequently used in our analysis (see ^[2]).

Lemma 2.1. Assume $w_+ > w_-$ and set $\tilde{w} = w_+-w_-$, then the problem (2.3) has a unique smooth global solution $w(t, x_1)$ such that

(1) $w_- < w(t, x_1) < w_+$, $w_{x_1} > 0$, for $x_1\in \mathbb{R}$ and $t\ge0, \delta > 0$;

(2)The following estimates hold for all $t\ge0, \delta > 0$, $p\in[1, +\infty]$, $k(\geq 2)\in \mathbb{N}^+$,

(3)There exists a constant $\delta_0\in(0, 1)$ such that for $\delta\in(0, \delta_0]$ and $t > 0$,

Denote $w_{\pm} = \lambda_3(\rho_{\pm}, u_{1\pm}, \theta_{\pm})$, the 3-rarefaction wave $(\rho^r, u_1^r, \theta^r)(x_1, t) = (\rho^r, u_1^r, \theta^r)({x_1}/{t})$ to the Riemann problems (1.7) and (1.8) is given by

where $\Sigma _3^{(j)}$ is the 3-Riemann invariant defined in (1.10). According to the smooth approximate rarefaction wave $(\bar{\rho}, \bar{u}_1, \bar{\theta})(t, x_1)$ of the 3-rarefaction wave fan $(\rho^r, u_1^r, \theta^r)(\frac{x_1}{t})$ can be formulated as

where $w(t, x_1)$ is the smooth solution to the Burgers' equation in (2.3). The $(\bar{\rho}, \bar{u}_1, \bar{\theta})$ satisfy the following equations:

with the initial values $(\bar{\rho}, \bar{u}_1, \bar{\theta})(0, x_1): = (\bar{\rho}_0, \bar{u}_{10}, \bar{\theta}_0)(x_1)$. Furthermore, we can associate to the solution of (2.5) the following quantity (see ^[22])

Based on the Lemma 2.1, we utilize the results of the properties on $w(t, x_1)$, which were obtained in ^[21].

Lemma 2.2. The smooth approximate 3-rarefaction wave $(\bar{\rho}, \bar{u}_1, \bar{\theta})$ constructed in (2.4) satisfies the following properties:

(1) $\bar{u}_{1x_1} = \frac{2}{\gamma+1}w_{x_1} > 0$ for all $x_1\in \mathbb{R}$ and $t\ge0,$

(2)The following estimates hold for all $t\ge0, \delta > 0$, $p\in[1, +\infty]$, $k(\geq 2)\in \mathbb{N}^+$,

(3)There exists a constant $\delta_0\in(0, 1)$ such that for $\delta\in(0, \delta_0]$ and $t > 0$,

2.2.   Hyperbolic wave

To compensate for the error arising from the dissipation terms, the hyperbolic wave was introduced in ^[11] and ^[12]. We also develop hyperbolic wave $(d_1, d_2, d_3)(t, x_1)$ corresponding to the rarefaction wave $(\bar{\rho}, \bar{u_1}, \bar{\theta})$, at the same time we neglect the hyperbolic wave $d_4( = 0)$ with respect to $\bar{q_1}$.

where $\bar{m}_1: = \bar{\rho}\bar{u}_1,$ $\bar{\mathcal{E}}: = \bar{\rho}\bar{E} = \bar{\rho}(\frac{R}{\gamma-1}\bar{\theta}+\frac{1}{2}\bar{u}_1^2)$ represent the momentum and the total energy of the approximate rarefaction wave, respectively. The first three lines of (2.6) are linear hyperbolic systems, which can be rewritten as

where

with three distinct eigenvalues $\bar{\lambda } _{i} = \bar{\lambda }_i(\bar{\rho }, \bar{u}_1, S_{\pm }) = \bar{u}_1+(-1)^{\frac{i+1}{2} } \sqrt[]{\bar{p}_{\bar{\rho }}(\bar{\rho }, S_{\pm })} (i = 1, 3), \bar{\lambda } _{2} = \bar{\lambda }_2(\bar{\rho }, \bar{u}_1, S_{\pm }) = \bar{u}_1$ and the corresponding left and right eigenvectors $\bar{l } _{i} = \bar{l }_i(\bar{\rho }, \bar{u}_1, S_{\pm }), \bar{r } _{i} = \bar{r }_i(\bar{\rho }, \bar{u}_1, S_{\pm }) (i = 1, 2, 3)$ satisfying

where $\bar{L} = (\bar{l}_1, \bar{l}_2, \bar{l}_3) ^{\top}, \bar{R} = (\bar{r}_1, \bar{r}_2, \bar{r}_3)$ and $\mathbb{I}$ is the 3 × 3 identity matrix.

Define

then it has

and we find that $(D_1, D_2, D_3)$ satisfies the following system:

Based on the fact that the 3-Riemann invariant is a constant on the 3-rarefaction wave curve, we obtain

We use (2.8) to rewrite (2.7) as

We easily find that $D_1$ and $D_2$ are decoupled from $D_3$ in (2.8). This enables us to solve the linear hyperbolic system (2.9) on the finite time interval $[0, T]$. Furthermore, we borrow the result in ^[11] to get the following estimates for the hyperbolic wave $(d_1, d_2, d_3)$.

Lemma 2.3. There exists a positive constant $C_T$ independent of $\delta$ and $\epsilon$, such that

In particular, it holds that

2.3.   Approximate solution profile

We set the approximate solution profile $(\tilde{\rho}, \tilde{u}_1, \tilde {\theta})$ of the two-dimensional radiative hydrodynamics as

then the approximate wave profile $(\tilde{\rho}, \tilde{u}_1, \tilde {\theta })$ satisfies

with the initial data

3.   Proof of Theorem 1.1

We denote the perturbation around the approximate wave profile

by

where

and

is the solution to the problem (1.1) with the following initial data:

For simplicity, the superscript $\epsilon$ in $\left (\rho ^\epsilon, {\boldsymbol{u}}^\epsilon, \theta ^\epsilon, {\boldsymbol{q}}^\epsilon \right)$ will be omitted. We use (1.1) and (2.11) to derive the equations for the perturbation $(\phi, \Psi, \zeta, Q):$

with initial data

where

and the initial perturbation is chosen to satisfy

Our aim is to find a solution $(\phi, \Psi, \zeta, Q)$ to (3.4)–(3.6) in the space $X(0, \frac{T}{\epsilon^{\alpha}}),$ which is defined to be

with $0\le \tau _1\le \frac{T}{\epsilon ^\alpha}.$

Proposition 3.1. There exists a positive constant $\epsilon_0 < 1$ such that if $0 < \epsilon\le \epsilon_0,$ then the perturbation problems (3.4)–(3.6) admits a unique solution $(\phi, \Psi, \zeta, Q)\in X (0, \frac{T}{\epsilon ^{\alpha }})$ satisfying

and

for some constant $C_T$ is independent of $\epsilon, \delta,$ but may depend on $T$.

According to the above estimate (3.7), we further derive that

Under the setting $\delta = \epsilon^{\omega}|\ln\epsilon|$ and (3.42), we derive that the upper bound in (3.9) is $C_T\epsilon^\omega|\ln\epsilon|^2$; thus, the proof of Theorem 1.1 is completed.

In order to prove Proposition 3.1, we will perform the analysis under a priori assumption

Proposition 3.2. ($A$ $priori$ estimate). Assume that the problems (3.4)–(3.6) admit a solution $(\phi, \Psi, \zeta, Q)\in X(0, \tau _1(\epsilon))$ for some $\tau_1(\epsilon)(>0)$. Then there exists a positive constant $\epsilon_1$ which is independent of $\epsilon, \delta$ and $\tau_1(\epsilon)$, such that if $0 < \epsilon\le \epsilon_1$ and the a priori Assumptions (3.10), then we have the following uniform estimate:

and

where the constant $C_T$ is independent of $\epsilon, \delta,$ but may depend on $T$.

Before performing the energy estimates, we use the $a$ $priori$ assumption to derive that the density and temperature are uniformly bounded from below and above. Choosing $\frac{\epsilon}{\delta^{\frac{3}{2}}}$ and $\epsilon$ small enough, then it holds

and

Moreover, we use Sobolev's inequality and $a$ $priori$ assumption to get

Zero-order energy estimate for the perturbation system is formulated as follows.

Lemma 3.1. There exists a positive constant $C_T$ such that

Proof. Define $V(x) = x-\mathrm{ln}x-1$, then the entropy and entropy flux are given by:

Under the assumption (3.13), there is a positive $C$ such that $\frac{1}{C}|\phi, \Psi, \zeta|^2\leq \eta\leq C|\phi, \Psi, \zeta|^2$. Straightforward calculations lead to

Multiplying the fourth equation of (3.4) by $\frac{a\epsilon^{\alpha-1}}{4 \theta\tilde {\theta}^3}Q$, and using the notation $\mathcal{P}: = (\tilde \theta-\bar{\theta })$, we have

Actually, the term $\mathrm{div}Q\frac{\zeta}{\theta}$ in (3.16) cannot be estimated directly. Our strategy involves using cancellation, which combines (3.16) and (3.17) together.

Here, we present some preparatory work before performing detailed estimates. Specifically, we use the definition of the approximate solution profile $(\tilde \rho, \tilde u_1, \tilde \theta)$ to rewrite the last two terms on the left side of Eq (3.16). The estimation we conduct thereafter focuses on extracting the dissipation part.

Due to the positive quantity $\bar{u}_{1y_1}$, term $D_2$ serves as the dissipation. Terms $D_{1}$, $D_3$, and $D_4$ consist of approximated waves and their solutions. By appropriately choosing parameters $\epsilon$ and $\delta$, we then utilize the smallness of the wave's length to bound them effectively.

We integrate $D_1$ over $[0, \tau']\times \mathbb{R} \times \mathbb{T}_\epsilon$ with $\tau'\leq \tau_1(\epsilon)$ to get that

Then, integrating $D_3$ over $[0, \tau']\times \mathbb{R} \times \mathbb{T}_\epsilon$, we can obtain

For any $\frac{1}{2} < \tilde{\iota} < 1$, and $\iota\in \left(\frac{1}{2\tilde\iota}, 1\right)$, $z_1, z_2\sim 1$, we consider the following function

It is easy to check that

So we obtain

Based on the above analysis, it holds that

and

Moreover, by integrating the above inequality over the interval $[0, \tau']\times \mathbb{R} \times \mathbb{T}_\epsilon$, we easily find that the last two terms on the right-hand side can be bounded by $C_T\frac{\epsilon}{\delta^{\frac{5}{2}}}\sup_{0\leq \tau\leq \tau_1(\epsilon)}\|(\phi, \zeta)\|^2$.

Collecting the above estimates together, we choose $\frac{\epsilon}{\delta^{\frac{3}{2}}}\ll 1$ to deduce that

where

Based on the above analysis, we combine (3.16) and (3.17); and then integrate the resulting equation over $[0, \tau']\times \mathbb{R} \times \mathbb{T}_\epsilon$ to deduce that

According to the scaling argument in (3.2), we use Lemmas 2.2 and 2.3 to deduce that

and

For any fixed time $T$, it follows from the Holder's inequality and Lemmas 2.2 and 2.3 that

Moreover, we use Lemmas 2.2 and 2.3 to obtain that

As for the term $\mathcal{I}_5$, it contains similar terms as $\mathcal{I}_{i}$ with $1\leq i\leq 4$. Therefore, it satisfies similar estimates, and for the sake of simplicity, we omit the details.

By using the definition of $\tilde{\theta }$, $a$ $priori$ Assumption (3.10), and (3.13), we have

where we used Sobolev's inequality and $a$ $priori$ assumption $\|\zeta\|_{L^\infty}\leq C\|\zeta\|_{2}\leq C\epsilon^{1-\alpha}$ in the second line. Moreover, in the last line of the preceding estimate, we employed the expression

which consist of different waves.

Similarly, it holds that

By using Holder's inequality, Sobolev's inequality, Young's inequality and $a$ $priori$ Assumption (3.10), we have

and

Substituting the above estimates into (3.18) and taking $\epsilon$ and $\frac{\epsilon }{\delta ^{5/2}}$ ($\ll1$) suitably small, we can complete the proof of the (3.14) in Lemma 3.1. □

Lemma 3.2. There exists a positive constant $C_T$ such that

Proof. Applying the operator $\nabla$ to the first equation of (3.4) and then multiplying the resulting equation by $\frac{R\theta }{\rho }\nabla \phi$ yields

Multiplying the second equation of (3.4) by $-\Delta \Psi$ results in

We multiply the third equation of (3.4) by $-\frac{1}{\theta }\Delta \zeta$ to get

where

Applying the operator $\nabla$ to the fourth equation of (3.4) and then multiplying the resulting equation by $\frac{a\epsilon ^{1-\alpha } }{4 \theta \tilde {\theta}^{3}}\nabla Q$ we obtain

Now, we add Eqs (3.20)–(3.24) together. Then, we integrate the resulting equation over the domain $[0, \tau']\times \mathbb{R}\times \mathbb{T}_\epsilon$ to obtain

where we used the cancellations in the flux terms.

Now we are in a position to estimate the right-hand terms in Eq (3.25). For simplicity, we will focus on estimating some typical terms, while others can be treated similarly.

Straightforward calculations give that

Using the definition of $\bar q$, and Lemmas 2.2 and 2.3, it holds that

where we used Sobolev inequalities

and

We use the definition of $\tilde{u}_1$ and Lemmas 2.2 and 2.3 to get that

Similarly, it holds that

Using the definition of $\tilde \theta$, it holds that

When it comes to the estimate for $J_4(\tau, y)$, we just focus on the following two terms, and others can be well controlled. Specifically, we have

where we have used the facts $\|\nabla\zeta\|\le\|\nabla\zeta\|^{1/2}\|\nabla\zeta\|_1^{1/2}$ and the $a$ $priori$ Assumption (3.10).

Moreover, the standard calculations give that

Furthermore, the remaining terms in Eq (3.25) can be estimated in a similar manner, and we will omit the details for the sake of simplicity.

To close the estimate of (3.25), we need to have the estimate of $\int\limits_{0}^{\tau'}\|\nabla \phi \|^2\mathrm{d}\tau$. Following similar procedures in Lemma 3.2 ^[12], and keeping in mind the different scalings, we have

Thus, plugging the above estimates into (3.25) and taking$\frac{\epsilon^{2\alpha-1}}{\delta}, \frac{\epsilon^\alpha}{\delta^2}$ and $\epsilon$ suitably small, we can complete the proof of (3.19). □

Lemma 3.3. There exists a positive constant $C_T$ such that

Proof. We apply the operator $\nabla^2$ to the first equation of (3.4), then multiply it by $R\nabla^2\psi$ to obtain

Next, we are dividing the second equation of (3.4) by $\rho$, applying the operator $\nabla$ and then multiplying the resulting equation by $-\frac{\rho ^2}{\theta }\nabla \Delta \Psi$ to obtain

where

We divide the third equation of (3.4) by $\rho$, and apply the operator $\nabla$ to the resulting equation, then multiply it by $-\frac{\rho ^2}{\theta ^2}\nabla \Delta \zeta$ to deduce that

where

and

Applying the operator $\nabla^2$ to the fourth equation of (3.4) and then multiplying the resulting equation by $\frac{a\rho\epsilon ^{\alpha-1 } }{4 \theta^2 \tilde {\theta}^{3}}\nabla Q$, we have

Now we add (3.28)–(3.31) together and integrate the resulting equation over $[0, \tau']\times \mathbb{R}\times \mathbb{T}_\epsilon$ to obtain

Now, we focus on estimating the terms on the right-hand side of Eq (3.32). We will begin by estimating some typical terms, and we anticipate that the remaining terms can be treated similarly. We use the $a$ $priori$ Assumption (3.10) and Sobolev's inequality to obtain

Similarly, using Hölder's inequality, Sobolev's inequality, and Young's inequality, we obtain

It follows from Lemmas 2.2 and 2.3 that

Using $a$ $priori$ Assumption (3.10) and Sobolev's inequality, we have

and

Based on Young's inequality, Lemma 2.2, Lemma 2.3, and $a$ $priori$ Assumption (3.10), it holds that

and

For $\mathcal{M}(\tau, y)$, we will focus on several terms that may cause difficulties.

Integration by parts gives that

then, we obtain that

Direct calculations give that

Then, we will only present estimates for the first two terms, while the remaining terms are treated in the same fashion.

By Sobolev's inequality and the a priori Assumption (3.10), we obtain

and

By Lemmas 2.2 and 2.3, it holds that

In order to close the estimate of (3.32), we also introduce the estimate of $\int\limits_{0}^{\tau'}\|\nabla^2 \phi \|^2\mathrm{d}\tau$. Following the same procedures as in Lemma 3.4 of ^[12], and considering the new scaling argument introduced here, we finally derive that

By incorporating the above estimates into (3.32); and combining them with (3.33), complete the proof of (3.27). □

Lemma 3.4. There exists a positive constant $C_T$ such that

Proof. It remains to show the energy estimate of $Q$.

• Zero-order energy estimate of $Q$:

We multiply the fourth equation of (3.4) with $Q$ and integrate the resulting equation over $\mathbb{R}\times \mathbb{T}_\epsilon$ to get

Using Lemmas 2.2 and 2.3, we have

and

Thus, for any $\tau\in [0, \tau_1(\epsilon)]$, we plug the above estimates into (3.35), then use (3.14) and (3.19) to obtain

• First-order energy estimate of $Q$:

Applying the operator $\nabla$ to the fourth equation of (3.4), then multiplying with $\nabla Q$ and integrating the resulting equation over $\mathbb{R}\times \mathbb{T}_\epsilon$, we have

Direct calculations give that

Furthermore, we use Lemmas 2.2 and 2.3 to derive that

and

so we have

Obviously, it holds that

Thus, for any $\tau\in [0, \tau_1(\epsilon)]$, inserting the above estimates into (3.37), then using (3.19) and (3.27), we obtain

• Second-order energy estimate of $Q$:

Last, we apply the operator $\nabla^2$ to the fourth equation of (3.4), multiply with $\nabla^2 Q$, and integrate the resulting equation over $\mathbb{R}\times \mathbb{T}_\epsilon$ to obtain

We divide the $\mathcal{W}_{3, 1}$ into two terms.

We use Lemmas 2.2 and 2.3 to obtain

and

Then plugging the above estimates into $\mathcal{W}_{3, 1}$, we have

Also, it holds

Thus, for any $\tau\in [0, \tau_1(\epsilon)]$, inserting the above estimates into (3.39), then using (3.19) and (3.27), we obtain

Finally, we combine (3.36), (3.38), and (3.40) together to end the proof of (3.34). □

Collecting Lemmas 3.1–3.4 together, choosing $\delta = \epsilon^{\omega}|\ln\epsilon|$ with

such that

we finally close the $a$ $priori$ Assumption (3.10) and obtain $a$ $priori$ estimates (3.11) and (3.12). Eventually, Proposition 3.1 is proved.

4.   Conclusions

In this paper, we establish the vanishing viscosity limit for the two-dimensional radiative hydrodynamics system, demonstrating that its solutions converge to the planar rarefaction wave solution of the corresponding compressible Euler equations as the viscosity parameter $\epsilon\to 0$. Further, we develop new cancelation mechanisms to address the radiative term in the system. Instead of specifying the order of scaling, we provide a broad range of orders. Using carefully designed scaling relationships and performing meticulous energy estimates, we establish the explicit convergence rate presented in (1.11). This shows that our result serves as a generalization of ^[11,12] to radiative hydrodynamics.

Use of Generative-AI tools declaration

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

Conflict of interest

The author declare no conflict of interest.