Introduction and conclusions

In [2] we investigated the Cauchy problem to the relativistic Euler equation

$$\frac{\partial}{\partial t} \frac{\rho+Pu^2/c^4}{1-u^2/c^2} +... ...ial}{\partial x} \frac{(\rho+P/c^2)u}{1-u^2/c^2}=0, \leqno(rE)$$

$$\frac{\partial}{\partial t}\frac{(\rho+P/c^2)u}{1-u^2/c^2} +\frac{\partial}{\partial x} \frac{P+\rho u^2}{1-u^2/c^2} =0.$$

Here $c$ is a positive constant, the speed of light, and $P$ is a given smooth function of $\rho$ satisfying the assumption

(A): $P(\rho)>0, 0<P'=dP/d\rho < c^2, 0<P''=d^2P/d\rho^2$ for $\rho >0$, and

$$P=A_0\rho^{5/3}(1+\sum_{j=1}^{\infty}A_j(\rho^{2/3}/c^2)^j)$$ (1)

as $\rho \rightarrow 0$. Here $A_0$ is a positive constant and $\sum A_jz^j$ is a power series with positive radius of convergence.

In order to prove the existence of global weak solutions to the Cauchy problem by the theory of compensated compactness, we have to solve the relativistic Euler-Poisson-Darboux equation

$$\eta_{xx}-\eta_{yy}+A(x,y)\eta_y+B(x,y)\eta_x=0, \leqno(rEPD)$$

where the independent variables are

$$x=\frac{c}{2}\log\frac{c+u}{c-u}, \qquad y=\int_0^{\rho} \frac{\sqrt{P'}}{\rho+P/c^2}d\rho$$ (2)

and the unknown function $\eta(x,y)$ is an entropy to $(rE)$. The coefficients of $(rEPD)$ are given by

$$\begin{eqnarray*} A(x,y)&=&\frac{1}{\sqrt{P'}} (1-\frac{P'}{c^2}-\frac{\rho+P/c^... .../c^2}{1-P'u^2/c^4} (1-\frac{P'}{c^2}-\frac{\rho+P/c^2}{2P'}P''). \end{eqnarray*}$$

By the definition (2) and the assumption (1), we see that $A(x,y)$ and $B(x,y)$ are of the form

$$\begin{eqnarray*} A(x,y)&=&\frac{2}{y}+\epsilon ya(\epsilon x^2, \epsilon y^2) ... ...\geq 0}b_{jk} (\epsilon x^2)^j(\epsilon y^2)^k, \\ b_{00}&=& 1. \end{eqnarray*}$$

Here and hereafter we denote

$$\epsilon = 1/c^2.$$ (3)

Introducing the new unknown $V$ by

$$\frac{\partial\eta}{\partial y}=yV, \qquad \eta(x,y)=IV(x,y)= \int_0^y YV(x,Y)dY,$$ (4)

the singularity of $A(x,y)$ in $(rEPD)$ can be eliminated and $(rEPD)$ is reduced to the equation

$$V_{yy}-V_{xx} =\epsilon(yaV_y-\frac{4}{3}xbV_x+2(a+\epsilon y^2D_2a)V -\frac{8}{3}\epsilon xD_2bIV_x).$$

The problem

$$V_{yy}-V_{xx}=\epsilon(yaV_y-\frac{4}{3}xbV_x+ 2(a+\epsilon y^2D_2a)V -\frac{8}{3} \epsilon xD_2bIV_x), \leqno(Q)$$

$$V\vert _{y=0}=0, \qquad V_y\vert _{y=0}=4\phi (x)$$

admits a unique solution $V$ for any smooth $\phi$ given by a formula

$$V(x,y)= \int_{x-y}^{x+y}G(x,y,\xi -x, \epsilon)\phi(\xi)d\xi.$$ (5)

For the proof see [2], Section 5. $G(x,y,z,\epsilon)$ is a smooth function of $\vert x\vert<\infty, y\geq 0, \vert z\vert\leq y$ . Therefore by defining

$$K(x,y,z,\epsilon)=JG(x,y,z,\epsilon) =\int_{\vert z\vert}^yYG(x,Y,z,\epsilon)dY,$$ (6)

we have a formula

$$\eta(x,y)= \int_{x-y}^{x+y}K(x,y,\xi -x,\epsilon) \phi(\xi)d\xi$$ (7)

for solutions of $(rEPD)$. We call this formula the relativistic Darboux formula and $K$ the relativistic Darboux kernel. We know

$$K=(y^2-z^2)(1+O(\epsilon y)).$$ (8)

The purpose of this article is to study the properties of this relativistic Darboux kernel. The motivations are as follows.

First we see that $K(x,y,z,\epsilon)$ tends to the Darboux kernel $K(y,z)=y^2-z^2$ as $\epsilon$ tends to $0$. Therefore the solution $\eta$ of $(rEPD)$ tends to those of the Euler-Poisson-Darboux equation

$$\eta_{uu}-\eta_{yy}+\frac{2}{y}\eta_y=0$$

as $\epsilon$ tends to $0$. But $\eta$ is an entropy and there is a gap between the convergence of entropies and the convergence of solutions $(\rho, u)$ of the Euler equation. We conjecture that weak solutions to $(rE)$ contain a subsequence which converges to a weak solution of the non-relativistic Euler equation

$$\begin{eqnarray*} && \frac{\partial\rho}{\partial t}+ \frac{\partial}{\partial x... ...{\partial t}(\rho u) + \frac{\partial}{\partial x}(\rho u^2+P)=0 \end{eqnarray*}$$

as $\epsilon=1/c^2$ tends to $0$. Actually this is the case if we assume $P=A\rho, A$ being a constant $<c^2$, since the total variations of the solutions obtained by Glimm's scheme in [5] can be estimated uniformly with respect to $\epsilon$. See [4] for a proof. But this conjecture is not yet proved if we assume a more realistic equation of states (A). In order to approach this problem it is necessary to give a particuler account of the dependence of $K$ upon $\epsilon$. We want to find the first order term of the expansion of $K$ with respect to $\epsilon$. Using such detailed informations, we might discuss the properties of weak solutions to the equation with the parameter $\epsilon$ which are obtained by the compensated compactness method. Maybe a general theory cannot be expected and we must consider according to the situations. Second although we can guarantee the existence of weak solutions for any bounded initial data $\rho\vert _{t=0}=\rho^0(x), u\vert _{t=0}=u^0(x)$ such that

$$0\leq \rho^0(x)\leq M, \qquad \vert\frac{c}{2}\log \frac{c+u^0(x)}{c-u^0(x)}\vert\leq M$$

provided that $\epsilon=1/c^2\leq \epsilon_0(M)$, $\epsilon_0(M)$ being a positive number depending upon $M$, but what happens if $\epsilon$ is not so small and $M$ is large? Observing the proof of [2], we find that we use the fact that $K(x,y,z,\epsilon)>0$ on the considered region in which $\vert z\vert<y$. Thus we wonder whether $K>0$ if $\epsilon$ is not so small and $x, y$ are large. In other words the global behavior of the kernel $K$ is of interest. In order to consider this problem, we also should give a particular account of the dependence of $K$ upon $\epsilon$. These are the motivations of this study.

The conclusions of the present study are

Theorem 1   The relativistic Darboux kernel $K(x,y,z,\epsilon)$ is analytic in $\epsilon$ and

$$K(x,y,z,\epsilon)= (y^2-z^2)(1+\sum_{\nu =1}^{\infty} K_{\nu}(x,y,z)\epsilon^{\nu}),$$

where $K_{\nu}(x,y,z)$ is a homogeneous polynomial of $x,y,z$ of order $2\nu$ of the form

$$K_{\nu}=\sum_{i+2j+k=2\nu}K_{ijk}x^iy^{2j}z^k.$$

The power series is convergent for $\vert\epsilon\vert\leq \delta /M^2$, where $\vert x\vert+\vert y\vert\leq M, \vert z\vert\leq \vert y\vert$ and $\delta$ is a positive constant. Particularly

$$K_1=(\frac{5a_{00}}{16}-\frac{1}{4})y^2 -(\frac{a_{00}}{16}+\frac{1}{12})z^2 -\frac{2}{3}xz.$$

Theorem 2   The relativistic Darboux kernel $K(x,y,z,\epsilon)$ is approximated by $K^a(x,y,z,\epsilon)$ given in the following manner. If $A_1<0$, then

$$K^a(x,y,z,\epsilon)= 2\int_{\vert z\vert}^yI_0(\sqrt{\kappa(Y^2-z^2)})\exp(\epsilon( \frac{1}{3}x^2+\alpha Y^2))YdY;$$

If $A_1=0$, then

$$K^a(x,y,z,\epsilon)= -\frac{9}{\epsilon} (\exp(-\frac{\epsil... ... \exp(-\frac{\epsilon}{9}z^2)) \exp(\epsilon(\frac{1}{3}x^2));$$

If $A_1>0$, then

$$K^a(x,y,z,\epsilon)=2 \int_{\vert z\vert}^y J_0(\sqrt{-\kappa(Y^2-z^2)}) \exp(\epsilon(\frac{1}{3}x^2+\alpha Y^2)) YdY.$$

Here $I_0$ is the modified Bessel function of order 0, $J_0$ is the Bessel function of order 0, $\alpha = -(1+7A_1/20A_0)/9$, $\kappa =-(7A_1/30A_0)\epsilon$ and for any smooth $\phi$ the function

$$\eta(x,y)=\int_{x-y}^{x+y}K^a(x,y,\xi-x,\epsilon)\phi(\xi)d\xi$$

satisfies the equation

$$\eta_{xx}-\eta_{yy}+(\frac{2}{y}+\epsilon a_{00}y)\eta_y- \fr... ...y^2)\eta -\epsilon^2\frac{a_{00}^2}{2}I\eta, \leqno(\clubsuit)$$

which is congruent with (rEPD) modulo $O(\epsilon^2)$.