In this section, we discuss Riemann solutions of (1.1) with initial data (1.4).

By calculation we can obtain that the eigenvalues of (1.1) are

$$ \lambda_{1}=uv\quad\mbox{and}\quad \lambda_{2}=3uv $$

(2.1)

with the corresponding right-eigenvectors

$$ r_{1}=(u,-v)^{T} \quad\mbox{and}\quad r_{2}=(u,v)^{T}. $$

(2.2)

It is easy to see that \(\nabla\lambda_{1}\cdot r_{1}=0\) and \(\nabla \lambda_{2}\cdot r_{2}\neq0\), where ∇ denotes the gradient with respect to \((u,v)\). Hence \(\lambda_{1}\) is linearly degenerate, \(\lambda_{2}\) is genuinely nonlinear, and system (1.1) is nonstrictly hyperbolic for \(uv=0\). It is easy to obtain that the Riemann invariants of system (1.1) can be taken as

$$ w=uv,\qquad z=v/u. $$

(2.3)

Since equations (1.1) and the Riemann data are invariant under uniform stretching of coordinates

$$ (x,t)\longrightarrow(\kappa x,\kappa t),\quad\kappa\mbox{ is constant}, $$

we consider the self-similar solutions of (1.1) and (1.4)

$$ (u,v) (x,t)=(u,v) (\xi),\quad\xi=x/t. $$

Then the Riemann problem becomes a boundary value problem of ordinary differential equations

$$ \textstyle\begin{cases} -\xi u_{\xi}+(u^{2}v)_{\xi}=0,\\ -\xi v_{\xi}+(uv^{2})_{\xi}=0, \end{cases} $$

(2.4)

with

$$ (u,v) (\pm\infty)=(u_{\pm},v_{\pm}). $$

For smooth solutions, equations (2.4) can be rewritten as

$$ \begin{pmatrix} 2uv-\xi& u^{2} \\ v^{2} & 2uv-\xi \end{pmatrix} \begin{pmatrix} u\\ v \end{pmatrix} _{\xi}=0. $$

(2.5)

It follows from (2.5) that, besides the constant solution, it provides a continuous solution of (2.5) of the form \((u,v)(\xi)\).

Given a left state \((u_{-},v_{-})\), the possible states which can be connected to \((u_{-},v_{-})\) on the right by a rarefaction wave lie on a curve given as follows:

$$ R(u_{-},v_{-}): \quad \textstyle\begin{cases} \xi=\lambda_{2}=3uv,\\ \frac{v}{u}=\frac{v_{-}}{u_{-}},\\ v>v_{-},\qquad u>u_{-}. \end{cases} $$

(2.6)

For a bounded discontinuous solutions, the Rankine–Hugoniot conditions

$$ \textstyle\begin{cases} -\sigma[u]+[u^{2}v]=0,\\ -\sigma[v]+[uv^{2}]=0, \end{cases} $$

(2.7)

hold, where and in what follows, we use the notation \([h]=h_{+}-h_{-}\) with \(h_{-}\) and \(h_{+}\) the values of a function *h* on the left- and right-hand sides of the discontinuity curve, respectively, and *σ* is the velocity of the discontinuity. So, we obtain from (2.7) that

$$ (u_{+}v_{+}-u_{-}v_{-}) (u_{+}v_{-}-u_{-}v_{+})=0, $$

(2.8)

which gives \((u_{+}v_{-}-u_{-}v_{+})=0\) or \((u_{+}v_{+}-u_{-}v_{-})=0\). Thus, for a left state \((u_{-},v_{-})\), \((u_{+}v_{-}-u_{-}v_{+})=0\), and the Lax entropy conditions imply that the possible states can be connected to \((u_{-},v_{-})\) on the right by a shock wave satisfying

$$ S(u_{-},v_{-}):\quad \textstyle\begin{cases} \sigma=\frac{v_{-}}{u_{-}}(u^{2}+u u_{-}+u_{-}^{2}),\\ \frac{v}{u}=\frac{v_{-}}{u_{-}},\\ v< v_{-},\qquad u< u_{-}. \end{cases} $$

(2.9)

For this system (1.1), we can see that the shock wave curves coincide with the rarefaction wave curves.

If \((u_{+}v_{+}-u_{-}v_{-})=0\), then \(u_{+}v_{+}=u_{-}v_{-}\). Since the first characteristic \(\lambda_{1}=uv\) is linearly degenerate, this means that it is a contact discontinuity. Then the possible states that can be connected to \((u_{-},v_{-})\) on the right by a contact discontinuity lie on the curve

$$ J:\quad uv=u_{-}v_{-}. $$

(2.10)

To seek the Riemann solutions based on the value of the left state \((u_{-},v_{-})\), we divide the phase plane \(I=\{(u,v): u\geq0, v\geq0\}\) into three parts: \(\mathcal{A}=\{u>0 \mbox{ and } v>0\}\), \(\mathcal {B}=\{u=0, v\neq0\}\cup\{v=0,u\neq0\}\), and \(\mathcal{C}=\{u=0, v=0\}\).

*Case 1*: \((u_{-},v_{-})\in\mathcal{A}\). In this case, we draw three curves (2.6), (2.9), and (2.10) from the point \((u_{-},v_{-})\) by *R*, *S*, and *J*, respectively. Obviously, the curve *J* has asymptotes \(u=0\) and \(v=0\). So, we divide the phase plane into six domains: \(\mbox{I} \cup\mbox{II} \cup\mbox{III} \cup\mbox{IV} \cup \mathcal{B} \cup\mathcal{C}\); see Fig. 1. Hence we divide this case to four subcases as follows.

*Case 1.1*: \((u_{+},v_{+})\in \mbox{I or II}\). The Riemann solutions can be described as

$$ (u_{-},v_{-})+J+(u_{*},v_{*})+R+(u_{+},v_{+}), $$

(2.11)

where and in what follows, “+” means “followed by”, denotes the state \((u_{-},v_{-})\), and the state \((u_{*},v_{*})\) is determined by

$$ \textstyle\begin{cases} u_{*}v_{*}=u_{-}v_{-},\\ \frac{v_{*}}{u_{*}}=\frac{v_{+}}{u_{+}}, \end{cases} $$

(2.12)

which gives

$$ u_{*}=\sqrt{\frac{u_{+}u_{-}v_{-}}{v_{+}}},\qquad v_{*}= \sqrt{\frac{v_{+}u_{-}v_{-}}{u_{+}}}. $$

(2.13)

Moreover, the rarefaction wave *R* can be expressed as follows:

$$\begin{aligned} R:\quad \textstyle\begin{cases} \xi=\lambda_{2}=3uv,\\ \frac{v}{u}=\frac{v_{+}}{u_{+}},\\ u_{*}\leq u\leq u_{+},\qquad v_{*}\leq v\leq v_{+}. \end{cases}\displaystyle \end{aligned}$$

(2.14)

*Case 1.2*: \((u_{+},v_{+})\in\) III or IV. We can obtain that the corresponding Riemann solutions are

$$ (u_{-},v_{-})+J+(u_{*},v_{*})+S+(u_{+},v_{+}), $$

(2.15)

where the intermediate state \((u_{*},v_{*})\) is described by (2.13), and the shock wave *S* is expressed as follows:

$$ S:\quad \textstyle\begin{cases} \sigma=\frac{v_{*}}{u_{*}}(u_{+}^{2}+u_{+} u_{*}+u_{*}^{2}),\\ \frac{v_{+}}{u_{+}}=\frac{v_{*}}{u_{*}},\\ v_{+}< v_{*},\qquad u_{+}< u_{*}. \end{cases} $$

(2.16)

*Case 1.3*: \((u_{+},v_{+})=(0,0)\in\mathcal{C}\). The Riemann solutions are

$$ (u_{-},v_{-})+S+(u_{+},v_{+}), $$

(2.17)

where the speed of the shock wave *S* is

$$ \sigma=u_{-}v_{-}. $$

(2.18)

*Case 1.4:*. \((u_{+},v_{+})\in\mathcal{B}\). In this case, we can see that

$$ \lambda_{2}^{+}=\lambda_{1}^{+}=0< \lambda_{1}^{-}=u_{-}v_{-}< \lambda_{2}^{-}=3 u_{-}v_{-}, $$

(2.19)

which implies that the solutions cannot be constructed as before. Hence the Riemann solutions containing a weighted *δ*-measure supported on a line should be constructed to establish the existence.

To define the measure solutions, we introduce the following definitions.

### Definition 2.1

*A pair of*
\((u,v)\)
*constitutes a solution of* (1.1) *in the sense of distributions if*

$$ \textstyle\begin{cases} \int_{0}^{+\infty}\int_{-\infty}^{+\infty} \{\varphi_{t}+(uv)\varphi_{x}\}u \,dx \,dt=0,\\ \int_{0}^{+\infty}\int_{-\infty}^{+\infty} \{\varphi_{t}+(uv)\varphi_{x}\}v \,dx \,dt=0, \end{cases} $$

(2.20)

*for all test functions*
\(\varphi(t,x)\in C_{0}^{\infty}(R_{+}\times R)\).

### Definition 2.2

*A two*-*dimensional weighted*
*δ*-*measure*
\(\omega(s)\delta_{\Gamma }\)
*supported on a smooth curve*
\(\Gamma=\{(t(s),x(s)): a< s< b\}\)
*is defined by*

$$ \bigl\langle \omega(s)\delta_{\Gamma},\varphi\bigl(t(s),x(s)\bigr)\bigr\rangle = \int_{a}^{b} \omega(s)\varphi\bigl(t(s),x(s)\bigr)\,ds $$

(2.21)

*for all test functions*
\(\varphi(t,x)\in C_{0}^{\infty}(R\times R)\).

### Definition 2.3

*A pair of*
\((u,v)\)
*is called a delta*-*shock wave solution of* (1.1) *if it is represented in the form*

$$ (u,v) (t,x)= \textstyle\begin{cases} (u_{-},v_{-}),& x< x(t),\\ (u_{\delta},\omega(t)\delta(x-x(t))),& x=x(t),\\ (0,v_{+}),& x>x(t), \end{cases} $$

(2.22)

*or*

$$ (u,v) (t,x)= \textstyle\begin{cases} (u_{-},v_{-}),& x< x(t),\\ (\omega(t)\delta(x-x(t)),v_{\delta}),& x=x(t),\\ (u_{+},0),& x>x(t), \end{cases} $$

(2.23)

*and satisfies Definition*
2.1, *where*
\((u_{l},v_{l})\), \((u_{r},v_{r})\)
*are piecewise smooth bounded solutions of* (1.1), \(\omega (t)\in C^{1}[0,+\infty)\), \(\delta(\cdot)\)
*is the standard Dirac measure supported on the curve*
\(x=x(t)\), *and*
\(\omega(t)\)
*is the weight of the delta*-*shock wave on the state variable*
*u*
*or*
*v*.

Now, we consider the case \((u_{-},v_{-})\in\mathcal{A}\) and \((0,v_{+})\in \mathcal{B}\). Similarly, we also describe the other case \((u_{-},v_{-})\in \mathcal{A}\) and \((u_{+},0)\in\mathcal{B}\).

Following [4, 5, 22, 27, 38, 39], we claim that (2.22) is a delta-shock wave solution of (1.1) in the sense of distributions if it satisfies the following generalized Rankine–Hugoniot condition:

$$ \textstyle\begin{cases} \frac{dx}{dt}=\sigma_{\delta},\\ \sigma_{\delta}[u]=[u^{2}v],\\ \frac{d\omega(t)}{dt}=\sigma_{\delta}[v]-[uv^{2}], \end{cases} $$

(2.24)

with

$$ \sigma_{\delta}=u_{-}v_{-},\qquad\omega(t)=u_{-}v_{-}v_{+} t. $$

(2.25)

In addition to the generalized Rankine–Hugoniot condition, the discontinuity must satisfy the delta-shock wave entropy condition (2.19), which means that all the characteristic lines on both sides of the discontinuity are not outcoming.

*Case 2*. \((u_{-},v_{-})\in\mathcal{B}\). Here, we only consider the case \((u_{-},v_{-})\in\mathcal{B}\) and \(u_{-}=0\). As for the other case \((u_{-},v_{-})\in\mathcal{B}\) and \(v_{-}=0\), we can obtain the same results and omit the details. We divide this case into two subcases; see Fig. 2.

*Case 2.1*. \((u_{+},v_{+})\in\mathcal{B}\cup\mathcal{C}\). The corresponding Riemann solutions are expressed by

$$ (u_{-},v_{-})+J+(u_{+},v_{+}), $$

(2.26)

where the speed of the contact discontinuity \(\sigma_{J}=0\); see Fig. 3.

*Case 2.2*. \((u_{+},v_{+})\in\mathcal{A}\). The Riemann solutions are constructed as

$$ (u_{-},v_{-})+J+(0,0)+R+(u_{+},v_{+}), $$

(2.27)

where the speed of *J* vanishes. The *J* and *R* are formed as a composite wave; see Fig. 4.

So, we complete the construction of the Riemann solutions to system (1.1).