Analysis of a stochastic two-species Schoener's competitive model with Lévy jumps and Ornstein–Uhlenbeck process

1.   Preliminary

The objective of this work is to deal with the rotation-Camassa-Holm model (RCH)

where $m = V-V_{xx}$,

in which constant $\digamma$ is a parameter to depict the Coriolis effect due to the Earth's rotation. Gui et al. ^[9] derive the nonlinear RCH equation (1.1) (also see ^[3,10]), depicting the motion of the fluid associated with the Coriolis effect.

Recently, many works focus on the study of Eq (1.1). Zhang ^[23] investigates the well-posedness for Eq (1.1) on the torus in the sense of Hadamard if assuming its initial value in the space $H^s$ with the Sobolev index $s > \frac{3}{2}$, and gives a Cauchy-Kowalevski type proposition for Eq (1.1) under certain conditions. It is shown in Gui et al. ^[9] that Eq (1.1) has similar dynamical features with those of Camassa-Holm and irrotational Euler equations. The travelling wave solutions are found and classified in ^[10]. The well-posedness, geometrical analysis and a more general classification of travelling wave solution for Eq (1.1) are carried out in Silva and Freire ^[17]. Tu et al. ^[20] investigate the well-posedness of the global conservative solutions to Eq (1.1).

If $\digamma = 0$ (implying $h_1 = h_2 = 0$), namely, the Coriolis effects disappear, Eq (1.1) becomes the standard Camassa-Holm (CH) model ^[2], which has been investigated by many scholars ^[1,6,7,8,16]. For some dynamical characteristics of the CH, we refer the reader to the references ^{[11,12,13,14,15,22]}.

Motivated by the works made in ^[4,21], in which the $H^1(\mathbb{R})$ global weak solution to the CH model is studied without restricting that the initial value obeys the sign condition, we investigate the rotation-Camassa-Holm equation (1.1) and utilize the viscous approximation technique to handle the existence of global weak solution in $H^1(\mathbb{R})$. As the term $V_{xxx}$ appears in Eq (1.1), it yields difficulties to establish estimates of solutions for the viscous approximation of Eq (1.1) (In fact, using a change of coordinates, Silva and Freire ^[18] eliminate the term $V_{xxx}$ and discuss other dynamical features of Eq (1.1)). The key contribution of this work is that we overcome these difficulties and establish a high order integrable estimate and prove that $\frac{\partial V(t, x)}{\partial x}$ possesses upper bound. These two estimates take key roles in proving the existence of the $H^1(\mathbb{R})$ global weak solution for Eq (1.1) without the sign condition.

This work is structured by the following steps. Definition of the $H^1(\mathbb{R})$ global weak solutions and several Lemmas are given in Section 2. The main conclusion and its proof are presented in Section 3.

2.   Lemmas

We rewrite the initial value problem for the RCH equation (1.1)

Employing operator $\Lambda^{-2} = (1-\partial_x^2)^{-1}$ to multiply Eq (1.1), we have

where

It can be found in ^[9,10,17,18] that

We cite the definition (see ^[4,21]).

Definition 2.1. The solution $V(t, x):[0, \infty) \times \mathbb{R} \rightarrow \mathbb{R}$ is called as a global weak solution to system (2.1) or (2.2) if

(1) $V \in C([0, \infty) \times\mathbb{R}) \cap L^{\infty}\left([0, \infty); H^{1}(\mathbb{R})\right);$

(2) $\|V(t, .)\|_{H^{1}(\mathbb{R})} \leq\left\|V_{0}\right\|_{H^{1}(\mathbb{R})};$

(3) $V = V(t, x)$ obeys (2.2) in the sense of distribution.

Define $\phi(x) = e^{\frac{1}{x^{2}-1}}$ if $|x| < 1$ and $\phi(x) = 0$ if $|x| \geq 1$. Set $\phi_{\varepsilon}(x) = \varepsilon^{-\frac{1}{4}} \phi\left(\varepsilon^{-\frac{1}{4}} x\right)$ with $0 < \varepsilon < \frac{1}{4}$. Assume $V_{\varepsilon, 0} = \phi_{\varepsilon} \star V_{0}$, where $\star$ represents the convolution, we see that $V_{\varepsilon, 0} \in C^{\infty}$ for $V_{0}(x) \in H^{s}, s > 0$. To discuss global weak solutions for Eq (1.1), we handle the following viscous approximation problem:

in which

Utilizing (2.3) and denoting $p_{\varepsilon}(t, x) = \frac{\partial V_{\varepsilon}(t, x)}{\partial x}$ yield

Simply for writing, let $c$ represent arbitrary positive constants (independent of $\varepsilon$).

Lemma 2.1. Let $V_{0} \in H^{1}(\mathbb{R})$. For each number $\sigma \geq 2$, system (2.3) has a unique solution $V_{\varepsilon} \in$ $C\left([0, \infty); H^{\sigma}(\mathbb{R})\right)$ and

which has the equivalent expression

Proof. For parameter $\sigma\geq 2$, we acquire $V_{\varepsilon, 0}\in C([0, \infty); H^\infty(\mathbb{R}))$. Employing the conclusion in ^[5] derives that system (2.3) has a unique solution $V_\varepsilon(t, x) \in C([0, \infty); H^\sigma(\mathbb{R}))$. Using (2.3) arises

Integrating about variable $t$ for both sides of the above identity, we obtain (2.5).

In fact, as $\varepsilon\rightarrow 0$, we have

Lemma 2.2. If $V_0(x) \in H^{1}(\mathbb{R})$, for $A_\varepsilon(t, x)$ and $B_\varepsilon(t, x)$, then

and

where $c = c(\left\|V_{0}\right\|_{H^{1}(\mathbb{R})})$.

Proof. For any function $U(x)$ and the operator $\Lambda^{-2}$, it holds that

and

Utilizing (2.6), (2.11), (2.12), the expression of function $A_\varepsilon(t, x)$ and the Tonelli theorem, we have

and

which derive that (2.7) and (2.8) hold. Utilizing (2.7) and (2.8) yields

and

which complete the proof of (2.9). Furthermore, using (2.4) and (2.6), we have

which finishes the proof of (2.10).

Lemma 2.3. Provided that $0 < \alpha_1 < 1$, $T > 0$, constants $a < b$, then

where constant $c_{1}$ depends on $a, b, \alpha_1, T, k$ and $\left\|V_{0}\right\|_{H^{1}(\mathbb{R})}$.

Proof. We utilize the methods in Xin and Zhang ^[21] to prove this lemma. Let function $g(x)\in C^{\infty}(\mathbb{R})$ and satisfy

Define function $f(\eta): = \eta(|\eta|+1)^{\alpha_1}, \; \eta \in \mathbb{R}$. We note that the function $f$ belongs to $C^1(\mathbb{R})$ except $\eta = 0$. Here we give the expressions of its first and second derivatives as follows:

from which we have

and

Note that

Making use of $g(x)f^{\prime}(p_{\varepsilon})$ to multiply (2.4), from (2.16) and (2.17), integrating over $([0, \infty) \times\mathbb{R})$ by parts, we obtain

Applying (2.15) yields

For $t \geq 0,$ using $0 < \alpha_1 < 1$, (2.14) and the Hölder inequality gives rise to

and

Moreover, we have

Utilizing the Hölder inequality and (2.14) leads to

Using the last part of (2.14), we have

As shown in Lemma 2.2, there exists a constant $c_0 > 0$ to ensure that

Utilizing the second part in (2.14) arises

Applying (2.18)–(2.26) yields

where $c > 0$ relies only on $T > 0, a, b, \alpha_1$ and $\|V_0\|_{H^1(\mathbb{R})}.$ Furthermore, we have

The proof of (2.13) is completed.

Lemma 2.4. For $(t, x) \in(0, \infty) \times\mathbb{R}$, provided that $V_{\varepsilon} = V_{\varepsilon}(t, x)$ satisfy problem (2.3), then

in which positive constant $c = c(\|V_{0}\|_{H^{1}(\mathbb{R})})$.

Proof. Using Lemma 2.2 gives rise to

Assume that $H = H(t)$ satisfies the problem

Due to (2.28), we know that $H = H(t)$ is a supersolution^* of parabolic equation (2.4) associated with initial value $\frac{\partial V_{\varepsilon, 0}}{\partial x}$. Utilizing the comparison principle for parabolic equations arises

^*The supersolution is defined by $\sup\limits_{x\in\mathbb{R}}p_\varepsilon(t, x)$. If there exists a point $(t, x_0)$ such that $\sup\limits_{x\in\mathbb{R}}p_\varepsilon(t, x)) = p_\varepsilon(t, x_0)$, then $\frac{\partial p_\varepsilon (t, x_0)}{\partial x} = 0$ and $\frac{\partial^2p_\varepsilon(t, x_0)}{\partial x^2} < 0$.

We choose the function $F(t): = \frac{2}{t}+\sqrt{2 c}, t > 0$. Since $\frac{d F}{d t}(t)+\frac{1}{2} F^{2}(t)-c =$ $\frac{2 \sqrt{2 c }}{t} > 0$, we conclude

which finishes the proof of (2.27).

Lemma 2.5. There exists a subsequence $\left\{\varepsilon_{j}\right\}_{j \in \mathbb{N}}, \; \varepsilon_{j}\rightarrow 0$ and $V(t, x) \in L^{\infty}\left([0, \infty); H^{1}(\mathbb{R})\right) \cap H^{1}([0, T] \times \mathbb{R}),$ for every $T\geq 0,$ such that

The proof of Lemma 2.5 can be found in Coclite el al. ^[4].

Lemma 2.6. Assume $V_0\in H^1(\mathbb{R})$. Then $\{B_{\varepsilon}(t, x)\}_\varepsilon$ is uniformly bounded in $W_{loc}^{1, 1}([0, \infty) \times\mathbb{R})$. Moreover, there has a sequence $\left\{\varepsilon_{j}\right\}_{j \in N}$, $\varepsilon_j\rightarrow 0$ to guarantee that

where function $B\in L^{\infty}([0, T); W^{1, \infty}(\mathbb{R}))$ and $1 < r < \infty$.

The standard proof of Lemma 2.6 can be found in ^[4]. We omit its proof here.

For conciseness, we use overbars to denote weak limits which are taken in the space $L^{r}[(0, \infty) \times\mathbb{R}) \text { with } 1 < r < 3$.

Lemma 2.7. There exist a sequence $\left\{\varepsilon_{j}\right\}_{j \in N}$ tending to zero and two functions $p \in L_{l o c}^{r}([0, \infty) \times\mathbb{R}), \overline{p^{2}} \in L_{l o c}^{r_1}([0, \infty) \times\mathbb{R})$ such that

for each $1 < r < 3$ and $1 < r_1 < \frac{3}{2}$. In addition, it holds that

Proof. Lemmas 2.1 and 2.2 validate (2.29) and (2.30). The weak convergence in (2.30) ensures the reasonableness of (2.31). Using Lemma 2.5 and (2.29) derives that (2.32) holds.

For conciseness in the following discussion, we denote $\left\{p_{\varepsilon_{j}}\right\}_{j \in N}$, $\left\{V_{\varepsilon_{j}}\right\}_{j \in N}$ and $\left\{B_{\varepsilon_{j}}\right\}_{j \in N}$ by $\left\{p_{\varepsilon}\right\}_{\varepsilon > 0}$, $\left\{V_{\varepsilon}\right\}_{\varepsilon > 0}$ and $\left\{B_{\varepsilon}\right\}_{\varepsilon > 0}$. Assume that $F \in C^{1}(\mathbb{R})$ is an arbitrary convex function with $F^{\prime}$ being bounded, Lipschitz continuous on $\mathbb{R}$. Using (2.29) derives that

Multiplying (2.4) by $F^{\prime}(p_\varepsilon)$ yields

Lemma 2.8. Suppose that $F \in C^{1}(\mathbb{R})$ is a convex function with $F^{\prime}$ being bounded, Lipschitz continuous on $\mathbb{R}$. In the sense of distribution, then

where $\overline{pF(p)}$ and $\overline{F^{\prime}(p) p^{2}}$ represent the weak limits of $p_{\varepsilon} F(p_\varepsilon)$ and $F^{\prime}(p_\varepsilon) p_{\varepsilon}^{2}$ in $L_{\mathit{\text{ loc }}}^{r_1}([0, \infty) \times \mathbb{R}), 1 < r_1 < \frac{3}{2}$, respectively.

Proof. Applying Lemmas 2.5 and 2.7, letting $\varepsilon\rightarrow 0$ in (2.33) and noticing the convexity of function $F$, we finish the proof of (2.34).

Lemma 2.9. ^[4] Almost everywhere in $[0, \infty) \times\mathbb{R},$ it has

where $\eta_{+}: = \eta_{\chi[0, +\infty)}(\eta), \eta_{-}: = \eta_{\chi(-\infty, 0]}(\eta)$, $\eta \in\mathbb{R}$.

Using Lemmas 2.4 and 2.7 leads to

Lemma 2.10. For $t\geq 0, x\in\mathbb{R}$, in the sense of distribution, it holds that

Proof. Making use of (2.4), Lemmas 2.5–2.7, we derive that (2.35) holds by letting $\varepsilon \rightarrow 0$.

Lemma 2.11. Provided that $F \in C^{1}(\mathbb{R})$ is a convex function with $F^{\prime} \in L^{\infty}(\mathbb{R})$, for every $T > 0$, in the sense of distribution, then

Proof. Suppose that $\left\{w_{\delta}\right\}_{\delta}$ is a kind of mollifiers defined in $(-\infty, \infty)$. Let $p_{\delta}(t, x): = \left(p(t, \cdot) \star w_{\delta}\right)(x)$ in which $\star$ denotes the convolution with respect to variable $x$. Using (2.35) yields

Utilizing the assumptions on $F$ and $F^{\prime}$ and letting $\delta \rightarrow 0$ in (2.36), we complete the proof.

Following the ideas in ^[21], we hope that the weak convergence of $p_{\varepsilon}$ should be strong convergence in (2.30). The strong convergence leads to the existence of global weak solution for system (2.1).

Lemma 2.12. ^[4] Assume $V_{0} \in H^{1}(\mathbb{R})$. Then

Lemma 2.13. ^[4] If $V_{0} \in H^{1}(\mathbb{R})$, $L > 0$, then

where

$F_L^+(\rho) = F_L(\rho)\chi_{[0, \infty)}(\rho)$ and $F_L^{-}(\rho) = F_L(\rho)\chi_{(-\infty, 0]}(\rho)$, $\rho\in(-\infty, \infty).$

Lemma 2.14. ^[4] Let $L > 0$. For $F_L(\rho)$ defined in (2.37), then

Lemma 2.15. Assume $V_{0} \in H^{1}(\mathbb{R})$. For almost all $t > 0$, then

Lemma 2.16. Assume $V_{0} \in H^{1}(\mathbb{R})$. For almost all $t > 0$, then

Using Lemmas 2.8 and 2.11–2.14, the proofs of Lemmas 2.15 and 2.16 are analogous to those of Lemmas 4.4 and 4.5 in Tang et al. ^[19]. Here we omit their proofs.

Lemma 2.17. Assume $V_0\in H^1(\mathbb{R})$. Almost everywhere in $[0, \infty)\times(-\infty, \infty)$, it holds that

Proof. Using Lemmas 2.15 and 2.16 arises

Applying Lemma 2.6 drives that there has a constant constant $N > 0$ to ensure

Using Lemmas 2.9 and 2.14 yields

Since the map $\rho \rightarrow \rho_{+}+\left(F_{L}^{-}\right)^{\prime}(\rho)$ is convex, it holds that

Using (2.40) gives rise to

Since $\rho\rightarrow (L+\rho)\chi_{(-\infty, -L)}(\rho)$ is concave, letting $L$ be sufficiently large, we have

Using (2.39)–(2.44) yields

which together with the Gronwall inequality yields

Using the Fatou lemma, Lemma 2.9 and (2.45), sending $L \rightarrow \infty$, it holds that

which finishes the proof of (2.38).

3.   Main result and its proof

Theorem 3.1. Assume that $V_{0}(x) \in H^{1}(\mathbb{R})$. Then system (2.1) has at least a global weak solution $V(t, x)$. Furthermore, this weak solution possesses the features:

(a) For $(t, x) \in[0, \infty)\times\mathbb{R}$, there exists a positive constant $c = c(\|V_{0}\|_{H^{1}(\mathbb{R})})$ such that

(b) If $a, b \in\mathbb{R}, \; a < b$, for any $0 < \alpha_1 < 1$ and $T > 0$, it holds that

where positive constant $c_0$ relies on $\alpha_1, k, T, a, b$ and $\left\|V_{0}\right\|_{H^{1}(\mathbb{R})}$.

Proof. Utilizing (2.3), (2.5) and Lemma 2.5, we derive (1) and (2) in Definition 2.1. From Lemma 2.17, we have

Employing Lemmas 2.5 and 2.6 results in that $V$ is a global weak solution to system (2.2). Making use of Lemmas 2.3 and 2.4 gives rise to inequalities (3.1) and (3.2). The proof is finished.

4.   Conclusions

In this work, we study the rotation-Camassa-Holm (RCH) model (1.1), a nonlinear equation describing the motion of equatorial water waves with the Coriolis effect due to the Earth's rotation. The presence of the term $V_{xxx}$ in the RCH equation leads to difficulties of establishing estimates of solutions for the viscous approximation. To overcome these difficulties, we establish a high order integrable estimate and show that $\frac{\partial V(t, x)}{\partial x}$ possesses an upper bound. Using these two estimates and the viscous approximation technique, we examine the existence of $H^1(\mathbb{R})$ global weak solutions to the RCH equation without the sign condition.

Acknowledgments

The authors are very grateful to the reviewers for their valuable and meaningful comments of the paper.

Conflict of interest

The authors declare no conflicts of interest.