1 Introduction

Consider the nonlinear partial differential equation

$$\begin{aligned} V_{t}-V_{txx}+kV_{x}+mVV_{x}= \frac{9}{2}V_{x}V_{xx}+\frac{3}{2}VV_{xxx}, \quad (t,x)\in\mathbb{R}_{+}\times\mathbb{R}, \end{aligned}$$
(1)

where \(m>0\) and k are constants. Assume that \(V_{0}(x)=V(0,x)\) is an initial value to Eq. (1). We establish the inequality

$$\begin{aligned} c_{1} \Vert V_{0} \Vert _{L^{2}(\mathbb{R})}\leq \Vert V \Vert _{L^{2}(\mathbb{R})}\leq c_{2} \Vert V_{0} \Vert _{L^{2}(\mathbb {R})}, \end{aligned}$$
(2)

where \(c_{1}>0\) and \(c_{2}>0\) are constants independent of t.

If \(k=-1\) and \(m=\frac{3}{2}\), then Eq. (1) becomes the Fornberg–Whitham equation [1, 2]

$$\begin{aligned} V_{t}-V_{txx}-V_{x}+\frac{3}{2}VV_{x}= \frac{9}{2}V_{x}V_{xx}+\frac {3}{2}VV_{xxx}, \quad(t,x)\in\mathbb{R}_{+}\times\mathbb{R}. \end{aligned}$$
(3)

Recently, Holmes and Thompson [3] proved the well-posedness of Eq. (3) in the Besov space in the periodic and nonperiodic cases and established a Cauchy–Kowalevski-type theorem for Eq. (3) to show the existence and uniqueness of analytic solutions. The blow-up criterion for the solutions is given in [3]. Using several estimates derived from the Fornberg–Whitham equation itself and the conclusions in [4], Haziot [5] found sufficient conditions on the initial data to guarantee the wave breaking of solutions of Eq. (3). Gao et al. [6] proved the \(L^{1}\) local stability of strong solutions of Eq. (3).

We know that the dynamic properties of the Fornberg–Whitham model are related to those of the Cammassa–Holm equation[7], Degasperis–Processi equation [8], and Novikov equation[9], which have peakon solutions (see[10–13]). Other dynamical properties of the Camassa–Holm, Degasperis–Processi, and Novikov equations can be found in [14–21] and the references therein.

We write the Cauchy problem for Eq. (1):

$$\begin{aligned} \left \{ \textstyle\begin{array}{l}V_{t}-V_{txx}=-kV_{x}-mVV_{x}+\frac{9}{2}V_{x}V_{xx}+\frac {3}{2}VV_{xxx}\\ \phantom{V_{t}-V_{txx}}= -kV_{x}-(\frac{m}{2}V^{2})_{x}+\frac{3}{4}\partial_{xxx}^{3}V^{2},\\ V(0,x)=V_{0}(x), \end{array}\displaystyle \right . \end{aligned}$$
(4)

which is equivalent to

$$\begin{aligned} \left \{ \textstyle\begin{array}{l} V_{t}+\frac{3}{2}VV_{x}+\partial_{x}Q(t,x)=0,\\ V(0,x)=V_{0}(x), \end{array}\displaystyle \right . \end{aligned}$$
(5)

where \(m>0\) is a constant, \(\varLambda=(1-\partial^{2}_{x})^{\frac{1}{2}}\), and \(Q(t,x)= [kV+(\frac{m}{2}-\frac{3}{4})V^{2}(t,x) ]\).

Motivated by the desire to further investigate the Fornberg–Whitham equation (3), the objective of this work is to establish the existence and uniqueness of entropy solutions for Eq. (1). Using the viscous approximation techniques and assuming that the initial value \(V_{0}(x)\) belongs to the space \(L^{1}(\mathbb{R})\cap L^{\infty}(\mathbb{R})\), we prove the well-posedness of the entropy solutions. The novelty is that we derive a new \(L^{2}(\mathbb{R})\) conservation law for Eq. (1). The ideas for obtaining our main result come from those in [22].

The structure of this paper is as follows. In Sect. 2, we establish several estimates for the viscous approximations of problem (5), and in Sect. 3, we present our main results and their proofs.

2 Estimates of viscous approximations

Set

$$\begin{aligned} \phi(x)=\left \{ \textstyle\begin{array}{l@{\quad}l} e^{\frac{1}{x^{2}-1}}, & \vert x \vert < 1,\\ 0, & \vert x \vert \geq1, \end{array}\displaystyle \right . \end{aligned}$$

\(\phi_{\varepsilon}(x)=\varepsilon^{-\frac{1}{4}}\phi (\varepsilon^{-\frac{1}{4}}x)\) with \(0<\varepsilon<1\), and \(V_{0,\varepsilon}=\phi_{\varepsilon}\star V_{0}=\int_{\mathbb{R}}\phi _{\varepsilon}(x-y)V_{0}(y)\,dy\). We have \(V_{0,\varepsilon}\in C^{\infty}\) for any \(V_{0}\in H^{s}\) with \(s\geq0\).

For conciseness in this paper, we let c denote an arbitrary positive constant, which is independent of parameter ε and time t.

For a smooth function \(V_{0,\varepsilon}\) and \(s\geq0\), we have

$$\begin{aligned} & \Vert V_{0,\varepsilon} \Vert _{L^{p}(\mathbb{R})}\leq c \Vert V_{0} \Vert _{L^{p}(\mathbb{R})} \quad\textrm{for }1\leq p< \infty , \\ & V_{0,\varepsilon}\rightarrow V_{0}\quad(\varepsilon\rightarrow 0)\textrm{ in } L^{p}(\mathbb{R})\textrm{ for } 1\leq p< \infty , \\ & \Vert V_{0,\varepsilon} \Vert _{H^{q}}\leq c \Vert V_{0} \Vert _{H^{s}} \quad\textrm{if } q\leq s. \end{aligned}$$

For problem (4), we will discuss the limiting behavior of a sequence of smooth functions \(\{V_{\varepsilon}\} _{\varepsilon>0}\), where each function \(V_{\varepsilon}\) satisfies the viscous problem

$$\begin{aligned} \left \{ \textstyle\begin{array}{l}\partial_{t}V_{\varepsilon}-\partial_{txx}^{3}V_{\varepsilon}+k\partial_{x}V_{\varepsilon}+mV_{\varepsilon}\partial_{x}V_{\varepsilon}\\ \quad=\frac{9}{2}V_{\varepsilon}\partial_{xxx}^{3}V_{\varepsilon}+\frac {3}{2}\partial_{x}V_{\varepsilon}\partial_{xx}^{2}V_{\varepsilon}+\varepsilon \partial_{xx}^{2}V_{\varepsilon}-\varepsilon\partial_{xxxx}^{4}V_{\varepsilon},\quad(t,x)\in\mathbb {R}_{+}\times\mathbb{R},\\ V_{\varepsilon}(0,x)=V_{0,\varepsilon}(x), \quad x\in\mathbb{R}, \end{array}\displaystyle \right . \end{aligned}$$
(6)

or, in the equivalent form,

$$\begin{aligned} \left \{ \textstyle\begin{array}{l}\partial_{t}V_{\varepsilon}+\frac{3}{4}\partial _{x}(V_{\varepsilon}^{2}) +\partial_{x}Q_{\varepsilon}(t,x)=\varepsilon\partial_{xx}^{2}V_{\varepsilon},\\ Q_{\varepsilon}(t,x)=\varLambda^{-2} [kV_{\varepsilon}+(\frac{m}{2}-\frac {3}{4})V^{2}_{\varepsilon}],\\ V_{\varepsilon}(0,x)=V_{0,\varepsilon}(x), \end{array}\displaystyle \right . \end{aligned}$$
(7)

where

$$\begin{aligned} Q_{\varepsilon}(t,x)=\frac{1}{2} \int_{\mathbb{R}}e^{-|x-y|} \biggl[kV_{\varepsilon}(t,y)+ \biggl(\frac{m}{2}-\frac{3}{4}\biggr)V^{2}_{\varepsilon}(t,y) \biggr]\,dy. \end{aligned}$$
(8)

Lemma 2.1

If\(V_{0}\in L^{2}(\mathbb{R})\), then for any fixed\(\varepsilon>0\), there exists a unique global smooth solution\(V_{\varepsilon}=V_{\varepsilon}(t,x)\)to the Cauchy problem (6) belonging to\(C([0,\infty); H^{s}(\mathbb{R}))\)with\(s\geq0\).

Proof

Using Theorem 2.3 in [23], we directly get the result of this lemma. □

Now we give the following lemma, which plays a key role in our investigation of Eq. (1).

Lemma 2.2

Suppose that\(V_{\varepsilon}\)is a solution of problem (7), \(V_{0}\in L^{2}(\mathbb{R})\), and\(t>0\). Then

$$\begin{aligned}& c_{1} \Vert V_{0} \Vert _{L^{2}(\mathbb{R})}\leq \bigl\Vert V_{\varepsilon}(t,\cdot) \bigr\Vert _{L^{2}(\mathbb{R})}\leq c_{2} \Vert V_{0} \Vert _{L^{2}(\mathbb{R})}, \end{aligned}$$
(9)
$$\begin{aligned}& \varepsilon \int_{0}^{t} \bigl\Vert \partial_{x}V_{\varepsilon}( \tau,\cdot ) \bigr\Vert ^{2}_{L^{2}(\mathbb{R})}\,d\tau\leq c_{3} \Vert V_{0} \Vert ^{2}_{L^{2}(\mathbb{R})}, \end{aligned}$$
(10)

where\(c_{1}\), \(c_{2}\), and\(c_{3}\)are positive constants independent ofεandt.

Proof

Let \(g_{\varepsilon}=(\frac{2m}{3}-\partial _{xx}^{2})^{-1}V_{\varepsilon}\). We have

$$\begin{aligned} \frac{2m}{3}g_{\varepsilon}-\partial_{xx}^{2}g_{\varepsilon}=V_{\varepsilon}. \end{aligned}$$
(11)

Multiplying the first equation of problem (7) by \(g_{\varepsilon}-\partial_{xx}^{2}g_{\varepsilon}\) and integrating over \(\mathbb{R}\) yields

$$\begin{aligned} & \int_{\mathbb{R}}\partial_{t}V_{\varepsilon}\bigl(g_{\varepsilon}-\partial _{xx}^{2}g_{\varepsilon}\bigr)\,dx- \varepsilon \int_{\mathbb{R}}\partial_{xx}^{2}V_{\varepsilon}\bigl(g_{\varepsilon}-\partial_{xx}^{2}g_{\varepsilon}\bigr)\,dx \\ & \quad=-\frac{3}{2} \int_{\mathbb{R}}V_{\varepsilon}\partial _{x}V_{\varepsilon}\bigl(g_{\varepsilon}-\partial_{xx}^{2}g_{\varepsilon}\bigr)\,dx- \int_{\mathbb{R}}\partial_{x}Q_{\varepsilon}(t,x) \bigl(g_{\varepsilon}-\partial _{xx}^{2}g_{\varepsilon}\bigr)\,dx. \end{aligned}$$
(12)

We have

$$\begin{aligned}[b] & \int_{\mathbb{R}}\partial_{t}V_{\varepsilon}\bigl(g_{\varepsilon}-\partial _{xx}^{2}g_{\varepsilon}\bigr)\,dx- \varepsilon \int_{\mathbb{R}}\partial_{xx}^{2}V_{\varepsilon}\bigl(g_{\varepsilon}-\partial_{xx}^{2}g_{\varepsilon}\bigr)\,dx \\ &\quad= \int_{\mathbb{R}}\biggl(\frac{2m}{3}\partial_{t}g_{\varepsilon}- \partial _{txx}^{3}g_{\varepsilon}\biggr) \bigl(g_{\varepsilon}-\partial_{xx}^{2}g_{\varepsilon}\bigr)\,dx -\varepsilon \int_{\mathbb{R}}\biggl(\frac{2m}{3}\partial_{xx}^{2}g_{\varepsilon}-\partial_{xxxx}^{4}g_{\varepsilon}\biggr) \bigl(g_{\varepsilon}-\partial _{xx}^{2}g_{\varepsilon}\bigr)\,dx\hspace{-18pt} \\ &\quad= \int_{\mathbb{R}}\biggl(\frac{2m}{3}g_{\varepsilon}\partial_{t}g_{\varepsilon}-g_{\varepsilon}\partial_{txx}^{3}g_{\varepsilon}-\frac{2m}{3} \partial _{t}g_{\varepsilon}\partial_{xx}^{2}g_{\varepsilon} +\partial_{xx}^{2}g_{\varepsilon}\partial_{txx}^{3}g_{\varepsilon}\biggr)\,dx \\ &\quad \quad-\varepsilon \int_{\mathbb{R}}\biggl(\frac{2m}{3}g_{\varepsilon}\partial _{xx}^{2}g_{\varepsilon}-\frac{2m}{3} \bigl(\partial_{xx}^{2}g_{\varepsilon}\bigr)^{2}-g_{\varepsilon}\partial_{xxxx}^{4}g_{\varepsilon}+\partial_{xx}^{2}g_{\varepsilon}\partial_{xxxx}^{4}g_{\varepsilon}\biggr)\,dx \\ &\quad= \int_{\mathbb{R}}\biggl(\frac{2m}{3}g_{\varepsilon}\partial_{t}g_{\varepsilon}-\biggl(\frac{2m}{3}+1 \biggr)g_{\varepsilon}\partial_{txx}^{3}g_{\varepsilon}+ \partial _{xx}^{2}g_{\varepsilon}\partial_{txx}^{3} g_{\varepsilon}\biggr)\,dx \\ &\quad\quad-\varepsilon \int_{\mathbb{R}}\biggl(\frac{2m}{3}g_{\varepsilon}\partial_{xx}^{2}g_{\varepsilon}-\biggl( \frac{2m}{3}+1\biggr)g_{\varepsilon}\partial _{xxxx}^{4}g_{\varepsilon}+ \partial_{xx}^{2}g_{\varepsilon}\partial_{xxxx}^{4}g_{\varepsilon}\biggr)\,dx \\ &\quad= \int_{\mathbb{R}}\biggl(\frac{2m}{3}g_{\varepsilon}\partial_{t}g_{\varepsilon}+\biggl(\frac{2m}{3}+1\biggr) \partial_{x}g_{\varepsilon}\partial_{tx}^{2}g_{\varepsilon}+\partial_{xx}^{2}g_{\varepsilon}\partial_{txx}^{3} g_{\varepsilon}\biggr)\,dx \\ &\quad\quad-\varepsilon \int_{\mathbb{R}}\biggl(-\frac{2m}{3}\partial _{x}g_{\varepsilon}\partial_{x}g_{\varepsilon}- \biggl(\frac{2m}{3}+1\biggr)\partial _{xx}^{2}g_{\varepsilon}\partial_{xx}^{2}g_{\varepsilon}-\partial _{xxx}^{3}g_{\varepsilon}\partial_{xxx}^{3}g_{\varepsilon}\biggr)\,dx \\ &\quad=\frac{1}{2}\frac{d}{dt} \int_{\mathbb{R}} \biggl(\frac {2m}{3}g_{\varepsilon}^{2}+ \biggl(\frac{2m}{3}+1\biggr) (\partial_{x}g_{\varepsilon})^{2}+\bigl(\partial_{xx}^{2}g_{\varepsilon}\bigr)^{2} \biggr)\,dx \\ & \quad\quad+\varepsilon \int_{\mathbb{R}} \biggl(\frac{2m}{3}(\partial _{x}g_{\varepsilon})^{2}+\biggl(\frac{2m}{3}+1 \biggr) \bigl(\partial_{xx}^{2}g_{\varepsilon}\bigr)^{2}+\bigl(\partial_{xxx}^{3}g_{\varepsilon}\bigr)^{2} \biggr)\,dx.\end{aligned} $$
(13)

For the right-hand side of (13), integrating by parts and using (11) result in

$$\begin{aligned} &-\frac{3}{2} \int_{\mathbb{R}}V_{\varepsilon}\partial_{x}V_{\varepsilon}\bigl(g_{\varepsilon}-\partial_{xx}^{2}g_{\varepsilon}\bigr)\,dx- \int_{\mathbb{R}}\partial_{x}Q_{\varepsilon}(t,x) \bigl(g_{\varepsilon}-\partial _{xx}^{2}g_{\varepsilon}\bigr)\,dx \\ &\quad=-\frac{3}{2} \int_{\mathbb{R}}V_{\varepsilon}\partial_{x}V_{\varepsilon}\bigl(g_{\varepsilon}-\partial_{xx}^{2}g_{\varepsilon}\bigr)\,dx+ \int_{\mathbb{R}}\bigl(Q_{\varepsilon}-\partial_{xx}^{2}Q_{\varepsilon}\bigr) (t,x)\partial_{x}g_{\varepsilon}\,dx \\ &\quad=-\frac{3}{2} \int_{\mathbb{R}}V_{\varepsilon}\partial_{x}V_{\varepsilon}\bigl(g_{\varepsilon}-\partial_{xx}^{2}g_{\varepsilon}\bigr)\,dx+ \int_{\mathbb{R}} \biggl[kV_{\varepsilon}+\biggl( \frac{m}{2}- \frac{3}{4}\biggr)V^{2}_{\varepsilon}\biggr] \partial_{x}g_{\varepsilon}\,dx \\ &\quad=\frac{3}{4} \int_{\mathbb{R}}\partial_{x}\bigl(V_{\varepsilon}^{2} \bigr)\partial _{xx}^{2}g_{\varepsilon}\,dx+ \frac{m}{2} \int_{\mathbb{R}}V^{2}_{\varepsilon}\partial_{x}g_{\varepsilon}\,dx+k \int_{\mathbb{R}}\biggl(\frac{2m}{3}g_{\varepsilon}- \partial_{xx}g_{\varepsilon}\biggr)\partial_{x}g_{\varepsilon}\,dx \\ &\quad=\frac{3}{4} \int_{\mathbb{R}}\partial_{x}\bigl(V_{\varepsilon}^{2} \bigr) \biggl[\frac {2m}{3}g_{\varepsilon}-V_{\varepsilon}\biggr]\,dx+\frac{m}{2} \int_{\mathbb {R}}V^{2}_{\varepsilon}\partial_{x}g_{\varepsilon}\,dx+0 \\ &\quad=-\frac{3}{4} \int_{\mathbb{R}}V_{\varepsilon}^{2} \partial_{x}V_{\varepsilon}\,dx =0. \end{aligned}$$
(14)

From (12), (13), and (14) we conclude that

$$\begin{aligned}[b] &\frac{2m}{3} \Vert g_{\varepsilon} \Vert _{L^{2}}^{2}+ \biggl(\frac {2m}{3}+1\biggr) \Vert \partial_{x} g_{\varepsilon} \Vert _{L^{2}}^{2}+ \bigl\Vert \partial_{xx}^{2}g_{\varepsilon}\bigr\Vert _{L^{2}}^{2} \\ & \quad\quad{}+2\varepsilon \int_{0}^{t} \biggl(\frac{2m}{3} \Vert \partial _{x}g_{\varepsilon} \Vert _{L^{2}}^{2}+ \biggl(\frac{2m}{3}+1\biggr) \bigl\Vert \partial _{xx}^{2}g_{\varepsilon}\bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \partial _{xxx}^{3}g_{\varepsilon}\bigr\Vert _{L^{2}}^{2} \biggr)\,d\tau \\ & \quad=\frac{2m}{3} \bigl\Vert g_{\varepsilon}(0,\cdot) \bigr\Vert _{L^{2}}^{2}+\biggl(\frac{2m}{3}+1 \biggr) \bigl\Vert \partial_{x}g_{\varepsilon}(0,\cdot ) \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \partial_{xx}^{2}g_{\varepsilon}(0, \cdot ) \bigr\Vert _{L^{2}}^{2}.\end{aligned} $$
(15)

Using the smoothness of the function \(V_{0,\varepsilon}\), we have

$$\begin{aligned} \bigl\Vert g_{\varepsilon}(0,\cdot) \bigr\Vert _{L^{2}}, \bigl\Vert \partial _{x}g_{\varepsilon}(0,\cdot) \bigr\Vert _{L^{2}}, \bigl\Vert \partial _{xx}^{2}g_{\varepsilon}(0, \cdot) \bigr\Vert _{L^{2}}\leq c \Vert V_{0,\varepsilon} \Vert _{L^{2}}\leq c \Vert V_{0} \Vert _{L^{2}}. \end{aligned}$$

It follows from (11) that

$$\begin{aligned} \bigl\Vert V_{\varepsilon}(t,\cdot) \bigr\Vert _{L^{2}(\mathbb{R})}^{2} =& \int _{\mathbb{R}} \biggl(-\partial_{xx}^{2}g_{\varepsilon}+ \frac {2m}{3}g_{\varepsilon}\biggr)^{2}\,dx \\ =& \int_{\mathbb{R}}\bigl(\partial_{xx}^{2}g_{\varepsilon}\bigr)^{2}\,dx-\frac {4m}{3} \int_{\mathbb{R}}g_{\varepsilon}\partial_{xx}^{2}g_{\varepsilon}\,dx+\frac{4m^{2}}{9} \int_{\mathbb{R}}g_{\varepsilon}^{2}\,dx \\ =& \int_{\mathbb{R}}\bigl(\partial_{xx}^{2}g_{\varepsilon}\bigr)^{2}\,dx+\frac {4m}{3} \int_{\mathbb{R}}(\partial_{x}g_{\varepsilon})^{2} \,dx+\frac {4m^{2}}{9} \int_{\mathbb{R}}g_{\varepsilon}^{2}\,dx. \end{aligned}$$
(16)

Using (15) and (16), we derive that there exist constants \(c_{1}\) and \(c_{2}\) such that

$$\begin{aligned} c_{1} \Vert V_{0} \Vert _{L^{2}(\mathbb{R})}\leq \Vert V_{\varepsilon} \Vert _{L^{2}(\mathbb{R})}\leq c_{2} \Vert V_{0} \Vert _{L^{2}(\mathbb {R})} \end{aligned}$$
(17)

and

$$\begin{aligned} \varepsilon \int_{0}^{t} \Vert \partial_{x}V_{\varepsilon} \Vert _{L^{2}}^{2}\,d\tau \leq&2\varepsilon \int_{0}^{t} \biggl( \bigl\Vert \partial _{xxx}^{3}g_{\varepsilon}\bigr\Vert _{L^{2}}^{2}+2\biggl(\frac{2m}{3} \biggr)^{2}\varepsilon \Vert \partial_{x} g_{\varepsilon} \Vert _{L^{2}}^{2} \biggr)\,d\tau \\ \leq&\varepsilon c \int_{0}^{t} \biggl(\frac{2m}{3} \Vert \partial_{x} g_{\varepsilon} \Vert _{L^{2}}^{2}+ \biggl(\frac{2m}{3}+1\biggr) \bigl\Vert \partial_{xx}^{2} g_{\varepsilon}\bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \partial_{xxx}^{3}g_{\varepsilon}\bigr\Vert _{L^{2}}^{2} \biggr)\,d\tau \\ \leq&\varepsilon c \bigl( \bigl\Vert g_{\varepsilon}(0,\cdot) \bigr\Vert _{L^{2}}^{2}+ \bigl\Vert \partial_{x}g_{\varepsilon}(0, \cdot) \bigr\Vert _{L^{2}}^{2} + \bigl\Vert \partial_{xx}^{2}g_{\varepsilon}(0,\cdot) \bigr\Vert _{L^{2}}^{2} \bigr) \\ \leq& c \Vert V_{0,\varepsilon} \Vert _{L^{2}}^{2} \\ \leq& c \Vert V_{0} \Vert _{L^{2}}^{2}. \end{aligned}$$
(18)

The proof of Lemma 2.2 follows from (17) and (18). □

Letting \(\varepsilon=0\) in the proof of Lemma 2.2, for Eq. (1), we obtain inequality (2).

Using Lemma 2.2, we give the following conclusion for the term \(Q_{\varepsilon}(t,x)\).

Lemma 2.3

If\(V_{0}\in L^{2}(\mathbb{R})\), then

$$\begin{aligned}& \bigl\Vert Q_{\varepsilon}(t,\cdot) \bigr\Vert _{L^{1}(\mathbb{R})},\quad \bigl\Vert \partial_{x}Q_{\varepsilon}(t,\cdot) \bigr\Vert _{L^{1}(\mathbb {R})}\leq c \bigl( \Vert V_{0} \Vert _{L^{2}}+ \Vert V_{0} \Vert _{L^{2}}^{2} \bigr), \end{aligned}$$
(19)
$$\begin{aligned}& \Vert Q_{\varepsilon} \Vert _{L^{\infty}(\mathbb{R}_{+}\times\mathbb {R})},\quad \Vert \partial_{x}Q_{\varepsilon} \Vert _{L^{\infty}(\mathbb {R}_{+}\times\mathbb{R})}\leq c \bigl( \Vert V_{0} \Vert _{L^{2}}+ \Vert V_{0} \Vert _{L^{2}}^{2}\bigr). \end{aligned}$$
(20)

Proof

We have

$$\begin{aligned} Q_{\varepsilon}(t,x)=\frac{1}{2} \int_{\mathbb{R}}e^{-|x-y|} \biggl[kV_{\varepsilon}(t,y)+ \biggl(\frac{m}{2}-\frac{3}{4}\biggr)V^{2}_{\varepsilon}(t,y) \biggr]\,dy \end{aligned}$$
(21)

and

$$\begin{aligned} \partial_{x}Q_{\varepsilon}(t,x)=\frac{1}{2} \int_{\mathbb {R}}e^{-|x-y|}sign(x-y) \biggl[kV_{\varepsilon}(t,y)+ \biggl(\frac{m}{2}-\frac {3}{4}\biggr)V^{2}_{\varepsilon}(t,y) \biggr]\,dy. \end{aligned}$$
(22)

Using the Schwarz inequality leads to

$$\begin{aligned} \biggl\vert \int_{\mathbb{R}}e^{-\frac{1}{2} \vert x-y \vert }V_{\varepsilon}(t,y)\,dy \biggr\vert \leq& \biggl( \int_{\mathbb{R}}e^{- \vert x-y \vert }\,dy \biggr)^{\frac{1}{2}} \biggl( \int_{\mathbb {R}}V^{2}_{\varepsilon}(t,y)\,dy \biggr)^{\frac{1}{2}} \\ \leq& c \Vert V_{0} \Vert _{L^{2}}. \end{aligned}$$
(23)

Utilizing the Tonelli theorem and (23), we get

$$\begin{aligned} \int_{\mathbb{R}} \biggl\vert \int_{\mathbb{R}}e^{- \vert x-y \vert }V_{\varepsilon}(t,y)\,dy \biggr\vert \,dx =& \int_{\mathbb{R}} \biggl\vert \int_{\mathbb{R}}e^{-\frac {1}{2} \vert x-y \vert }V_{\varepsilon}(t,y)\,dy \biggr\vert e^{-\frac{1}{2} \vert x-y \vert }\,dx \\ \leq& \Vert V_{0} \Vert _{L^{2}} \int_{\mathbb{R}}e^{-\frac {1}{2} \vert x-y \vert }\,dx\leq c \Vert V_{0} \Vert _{L^{2}} \end{aligned}$$
(24)

and

$$\begin{aligned} \int_{\mathbb{R}} \biggl\vert \int_{\mathbb{R}}e^{- \vert x-y \vert }V^{2}_{\varepsilon}(t,y)\,dy \biggr\vert \,dx =& \int_{\mathbb{R}} \biggl\vert \int_{\mathbb{R}}e^{- \vert x-y \vert }V^{2}_{\varepsilon}(t,y)\,dy \biggr\vert \,dx \\ \leq& c \Vert V_{0} \Vert ^{2}_{L^{2}}. \end{aligned}$$
(25)

From (21)–(25) and Lemma 2.2 we derive that (19) and (20) hold. The proof is finished. □

If \(V_{0}\in L^{1}(\mathbb{R})\cap L^{\infty}(\mathbb{R})\), then we derive \(V_{0}\in L^{2}(\mathbb{R})\).

Lemma 2.4

If\(V_{0}\in L^{1}(\mathbb{R})\cap L^{\infty}(\mathbb{R})\), then

$$\begin{aligned} \bigl\Vert V_{\varepsilon}(t,\cdot) \bigr\Vert _{L^{\infty}}\leq \Vert V_{0} \Vert _{L^{\infty}}+c t \bigl( \Vert V_{0} \Vert _{L^{2}}+ \Vert V_{0} \Vert _{L^{2}}^{2} \bigr). \end{aligned}$$
(26)

Proof

Using the first equation of problem (7), we have

$$\begin{aligned} \partial_{t}V_{\varepsilon}+\frac{3}{2}V_{\varepsilon}\partial_{x}V_{\varepsilon}-\varepsilon\partial_{xx}V_{\varepsilon}=-\partial_{x}Q_{\varepsilon}. \end{aligned}$$
(27)

Applying Lemma 2.3 yields

$$\begin{aligned} \Vert \partial_{x}Q_{\varepsilon} \Vert _{L^{\infty}(\mathbb{R}_{+}\times \mathbb{R})} \leq c \bigl( \Vert V_{0} \Vert _{L^{2}}+ \Vert V_{0} \Vert _{L^{2}}^{2} \bigr). \end{aligned}$$
(28)

Setting \(K(t)=\| V_{0}\|_{L^{\infty}(\mathbb{R})}+ct (\| V_{0}\|_{L^{2}}+\| V_{0}\|_{L^{2}}^{2} )\), we get

$$\begin{aligned} \frac{dK}{dt}=c \bigl( \Vert V_{0} \Vert _{L^{2}}+ \Vert V_{0} \Vert _{L^{2}}^{2} \bigr). \end{aligned}$$
(29)

Since \(\| V_{\varepsilon}(0,x)\|_{L^{\infty}(\mathbb {R})}\leq K(0)\), using the comparison principle, we derive that (26) holds. □

Applying Lemma 2.4 and the methods presented in [22], we obtain the following result.

Lemma 2.5

(Oleinik-type estimate)

Let\(V_{0}\in L^{1}(\mathbb {R})\cap L^{\infty}(\mathbb{R})\)and\(T>0\). Then

$$\begin{aligned} \partial_{x} V_{\varepsilon}(t,x)\leq\frac{1}{t}+C_{T}, \quad x\in R, 0< t\leq T, \end{aligned}$$
(30)

where the constant\(C_{T}\)depends onT.

We omit the proof of this lemma since it is similar to that of Lemma 2.11 in [22].

We state the concepts of weak solution and entropy weak solution (see [22, 24]).

Definition 2.6

(Weak solution)

A function \(V: \mathbb {R}_{+}\times\mathbb{R}\rightarrow\mathbb{R}\) is called a weak solution of the Cauchy problem (5) if

  1. (i)

    \(V\in L^{\infty}(\mathbb{R}_{+}; L^{2}(\mathbb{R}))\), and

  2. (ii)

    \(\partial_{t}V+\frac{3}{4}\partial_{x}(V^{2})+\partial_{x}Q(t,x)=0\) in \(D'([0,\infty)\times\mathbb{R})\), that is, for all \(f\in C_{c}^{\infty}([0,\infty)\times\mathbb{R})\), we have the identity

    $$\begin{aligned} \int_{R_{+}} \int_{\mathbb{R}} \biggl(V\partial_{t}f+ \frac{3V^{2}}{4}\partial _{x}f-\partial_{x}Q(t,x)f \biggr)\,dx\,dt + \int_{\mathbb {R}}V_{0}(x)f(0,x)\,dx=0. \end{aligned}$$
    (31)

Definition 2.7

(Entropy weak solution)

We call a function \(V: \mathbb{R}_{+}\times\mathbb{R}\rightarrow\mathbb{R}\) an entropy weak solution of Cauchy problem (5) if

  1. (i)

    V is a weak solution in the sense of Definition 2.6,

  2. (ii)

    \(V\in L^{\infty}([0,T]\times\mathbb{R})\) for any \(T>0\), and

  3. (iii)

    for any convex \(C^{2}\) entropy function \(\eta: \mathbb {R}\rightarrow\mathbb{R}\) with corresponding entropy flux \(q: \mathbb {R}\rightarrow\mathbb{R}\) defined by \(q'(V)=\frac{3}{4}\eta'(V)V\), we have

    $$\begin{aligned} \partial_{t}\eta(V)+\partial_{x}q(V)+ \eta'(V)\partial_{x}Q\leq0 \quad\textrm {in } D'\bigl([0,\infty)\times\mathbb{R}\bigr), \end{aligned}$$
    (32)

    that is, for all \(f\in C_{c}^{\infty}([0,\infty)\times\mathbb{R})\), \(f(t,x)\geq0\), we have

    $$\begin{aligned} \int_{\mathbb{R}_{+}} \int_{\mathbb{R}} \bigl(\eta(V)\partial_{t} f+q(V) \partial_{x}f-\eta'(V)\partial_{x}Qf \bigr)\,dx\,dt+ \int_{\mathbb{R}}\eta \bigl(V_{0}(x)\bigr)f(0,x)\,dx\geq0. \end{aligned}$$
    (33)

Remark 2.8

As stated by Coclite and Karsen [22], by a standard argument we get that the Kruzkov entropies/entropy fluxes

$$\begin{aligned} \eta(V)= \vert V-k_{1} \vert ,\qquad q(V):=\frac{3}{4}\operatorname{sign}(V-k_{1}) \bigl(V^{2}-k_{1}^{2}\bigr), \end{aligned}$$
(34)

where \(k_{1}\) is an arbitrary constant, satisfy (33).

3 Main results

We state the following \(L^{1}(\mathbb{R})\) stability result of entropy weak solutions for Eq. (1).

Theorem 3.1

(\(L^{1}\)-stability)

Assume that\(V_{1}(t,x)\)and\(V_{2}(t,x)\)are two entropy weak solutions of problem (5) with initial data\(V_{01}\in L^{1}(\mathbb{R})\cap L^{\infty}(\mathbb{R})\)and\(V_{02}\in L^{1}(\mathbb{R})\cap L^{\infty}(\mathbb{R})\), respectively. Let\(T>0\)be the maximal existence time of solutions\(V_{1}(t,x)\)and\(V_{2}(t,x)\). Then

$$\begin{aligned} \bigl\Vert V_{1}(t,\cdot)-V_{2}(t,\cdot) \bigr\Vert _{L^{1}(\mathbb{R})}\leq Ce^{Ct} \int_{-\infty}^{\infty} \bigl\vert V_{01}(x)-V_{02}(x) \bigr\vert \,dx, \quad t\in[0, T], \end{aligned}$$
(35)

whereCdepends on\(V_{01}\in L^{1}(\mathbb{R})\cap L^{\infty}(\mathbb {R})\)and\(V_{02}\in L^{1}(\mathbb{R})\cap L^{\infty}(\mathbb{R})\)andT.

The proof of Theorem 3.1 is the standard argument presented in Gao et al. [6]. We omit its proof.

We employ the compensated compactness method in [25, 26] to discuss the strong convergence of a subsequence of the viscosity approximations.

Lemma 3.2

Let\(\{V_{\varepsilon}\}_{\varepsilon>0}\)be a family of functions defined on\((0,\infty)\times\mathbb{R}\)such that

$$\Vert V_{\varepsilon} \Vert _{L^{\infty}}\leq C_{T}, $$

where the constant\(C_{T}>0\)depends onT, and the family

$$\bigl\{ \partial_{t}\eta(V_{\varepsilon})+\partial_{x}q(V_{\varepsilon}) \bigr\} _{\varepsilon>0} $$

is compact in\(H^{-1}_{\mathrm{loc}}((0,\infty)\times\mathbb{R})\)for any convex\(\eta\in C^{2}(\mathbb{R})\), where\(q(V)=aV\eta'(V)\)with constant\(a>0\). Then there exist a sequence\(\{\varepsilon_{n}\}_{n\in N}\), \(\varepsilon_{n}\rightarrow0\), and a function\(V\in L^{\infty}((0, T)\times\mathbb{R})\), \(T>0\), such that

$$\begin{aligned} V_{\varepsilon_{n}}\rightarrow V\quad\textrm{a.e. and in } L^{p}_{\mathrm{loc}} \bigl((0, \infty)\times\mathbb{R}\bigr), 1\leq p< \infty. \end{aligned}$$

Lemma 3.2 can be found in [25] or [26].

Lemma 3.3

([27])

Suppose thatΩis a bounded open subset of\(R^{H}\), \(H\geq2\). Assume that the sequence\(\{M_{n}\} _{n=1}^{\infty}\)of distributions is bounded in\(W^{-1, \infty}(\varOmega )\)and

$$M_{n}=M_{n}^{(1)}+M_{n}^{(2)}, $$

where\(\{M_{n}^{(1)}\}_{n=1}^{\infty}\)lies in a compact subset of\(H_{\mathrm{loc}}^{-1}(\varOmega)\), and\(\{M_{n}^{(2)}\}_{n=1}^{\infty}\)lies in a bounded subset of\(L^{1}_{\mathrm{loc}}(\varOmega)\). Then\(\{M_{n}\} _{n=1}^{\infty}\)lies in a compact subset of\(H^{-1}_{\mathrm{loc}}(\varOmega)\).

Lemma 3.4

Let\(V_{0}\in L^{1}(\mathbb{R})\cap L^{\infty}(\mathbb{R})\). Then there exists a subsequence\(V_{\varepsilon_{n}}\), \(n\in 1,2,3,\dots\), of\(\{V_{\varepsilon}\}_{\varepsilon>0}\)and a limit function

$$\begin{aligned} V\in L^{\infty}\bigl(\mathbb{R}_{+}; L^{2}(\mathbb{R})\bigr) \cap L^{\infty}\bigl((0,T);L^{\infty}\cap L^{1}( \mathbb{R})\bigr) \end{aligned}$$
(36)

such that

$$\begin{aligned} V_{\varepsilon_{k}}\rightarrow V \quad\textrm{in } L^{p}\bigl((0,T] \times\mathbb {R}\bigr), \forall p\in[1,\infty). \end{aligned}$$
(37)

Proof

Suppose that \(\eta:\mathbb{R}\rightarrow\mathbb{R}\) is an arbitrary convex \(C^{2}\) entropy function that is compactly supported, and \(q: \mathbb{R}\rightarrow\mathbb{R}\) is the corresponding entropy flux defined by \(q'(V)=\frac{3}{4}\eta'(V)V\). We set

$$\begin{aligned} \partial_{t}\eta(V_{\varepsilon})+\partial_{x} q(V_{\varepsilon})=M_{\varepsilon}^{(1)}+M_{\varepsilon}^{(2)}, \end{aligned}$$
(38)

where

$$\begin{aligned} \left \{ \textstyle\begin{array}{l} M_{\varepsilon}^{(1)}=\varepsilon\partial^{2}_{xx}\eta(V_{\varepsilon}),\\ M_{\varepsilon}^{(2)}=-\varepsilon\eta''(V_{\varepsilon})(\partial _{x}V_{\varepsilon})^{2}-\eta'(V_{\varepsilon})\partial_{x}Q_{\varepsilon}(t,x). \end{array}\displaystyle \right . \end{aligned}$$
(39)

We claim that

$$\begin{aligned} \left \{ \textstyle\begin{array}{l} M_{\varepsilon}^{(1)}\rightarrow0\quad\textrm{in } H^{-1}([0,T]\times \mathbb{R}), T>0,\\ M_{\varepsilon}^{(2)} \quad\textrm{is uniformly bounded in } L^{1}([0,T]\times\mathbb{R}). \end{array}\displaystyle \right . \end{aligned}$$
(40)

Using Lemmas 2.2–2.5 yields

$$\begin{aligned} & \bigl\Vert \varepsilon\partial_{xx}^{2} \eta(V_{\varepsilon}) \bigr\Vert _{H^{-1}(\mathbb{R}_{+}\times\mathbb{R})}\leq\sqrt{\varepsilon }c \bigl\Vert \eta' \bigr\Vert _{L^{\infty}} \Vert V_{0} \Vert _{L^{2}(\mathbb {R})}\rightarrow0, \end{aligned}$$
(41)
$$\begin{aligned} & \bigl\Vert \varepsilon\eta''(V_{\varepsilon}) (\partial_{x}V_{\varepsilon})^{2} \bigr\Vert _{L^{1}(\mathbb{R}_{+}\times\mathbb{R})}\leq c \bigl\Vert \eta '' \bigr\Vert _{L^{\infty}(\mathbb{R})} \Vert V_{0} \Vert _{L^{2}(\mathbb {R})}, \end{aligned}$$
(42)
$$\begin{aligned} & \bigl\Vert \eta'(V_{\varepsilon}) \bigr\Vert _{L^{1}((0,T)\times\mathbb {R})}\leq c \bigl\Vert \eta' \bigr\Vert _{L^{\infty}(\mathbb{R})} \Vert V_{0} \Vert _{L^{2}(\mathbb{R})}. \end{aligned}$$
(43)

Therefore we know that (40) holds. Using Lemmas 3.2 and 3.3, we confirm that there exists a subsequence \(\{V_{\varepsilon_{n}}\}\) and a limit function V satisfying (36) such that, as \(n\rightarrow \infty\),

$$\begin{aligned} &V_{\varepsilon_{n}}\rightarrow V\quad\textrm{in } L^{p}_{\mathrm {loc}}( \mathbb{R}_{+}\times\mathbb{R})\textrm{ for any } p\in[1, \infty), \end{aligned}$$
(44)
$$\begin{aligned} &\textrm{and}\quad V_{\varepsilon_{n}}\rightarrow V \quad\textrm{a.e. in } \mathbb{R}_{+}\times\mathbb{R}. \end{aligned}$$
(45)

Using Lemma 2.5, from (44) and (45) we obtain (37). The proof is finished. □

Lemma 3.5

If\(V_{0}\in L^{1}(\mathbb{R})\cap L^{\infty }(\mathbb{R})\), then there exists a function\(Q(t,x)= [kV+ (\frac {m}{2}-\frac{3}{4})V^{2}(t,x) ]\)such that

$$\begin{aligned} Q_{\varepsilon_{n}}\rightarrow Q\quad\textrm{in } L^{p}\bigl([0,T ); W^{1,p}(\mathbb{R})\bigr), T>0,\forall p\in[1, \infty), \end{aligned}$$
(46)

where the sequence\(\varepsilon_{n}\), and the functionVare constructed in Lemma 3.4.

We omit the proof of Lemma 3.5 since it is similar to that of Lemma 4.4 in [22].

Theorem 3.6

Let\(V_{0}\in L^{1}(\mathbb{R})\cap L^{\infty }(\mathbb{R})\). Then there exists at least one entropy weak solution to problem (5).

Proof

If \(f\in C_{c}^{\infty}(\mathbb{R}_{+}\times\mathbb{R})\), then from (31) we get

$$\begin{aligned} \int_{\mathbb{R}_{+}} \int_{\mathbb{R}} \biggl(V_{\varepsilon}\partial _{t}f+\frac{3}{4}V_{\varepsilon}^{2} \partial_{x}f-\partial_{x}Q_{\varepsilon}f+ \varepsilon V_{\varepsilon}\partial_{xx}^{2}f \biggr)\,dx\,dt+ \int_{\mathbb{R}} V_{0,\varepsilon}f(0,x)\,dx=0. \end{aligned}$$
(47)

Using Lemmas 3.4, we make sure that the function V presented in Lemma 3.4 is a weak solution of problem (5) in the sense of Definition 2.6. We have to verify that V satisfies the entropy inequalities in Definition 2.7. Let \(\eta\in C^{2}(\mathbb{R})\) be a convex entropy with flux q defined by \(q'(V)=\frac{3}{4}V\eta'(V)\). Using the convexity of η and problem (7) results in

$$\begin{aligned} \partial_{t}\eta(V_{\varepsilon})+\partial_{x}q(V_{\varepsilon})+ \eta '(V_{\varepsilon})\partial_{x}Q_{\varepsilon}= \varepsilon\partial_{xx}^{2}\eta (V_{\varepsilon}) - \varepsilon\eta{''}(V_{\varepsilon}) (\partial_{x}V_{\varepsilon})^{2} \leq \varepsilon\partial_{xx}^{2}\eta(V_{\varepsilon}). \end{aligned}$$
(48)

Thus by Lemmas 3.4 and 3.5 it follows that the entropy inequality holds. The proof is finished. □

From Theorems 3.1 and 3.6 we have the following:

Theorem 3.7

Let\(V_{0}\in L^{1}(\mathbb{R})\cap L^{\infty}(\mathbb{R})\). Then the Cauchy problem (5) has a unique entropy weak solution in the sense of Definition 2.7.