Return range and the cross-section of expected index returns in international stock markets

1.   Introduction

The equation:

was originally derived in ^[14,17] with $f(u) = au^2$ focusing on microbubbles coated by viscoelastic shells. These structures are crucial in ultrasound diagnosis using contrast agents, and the dynamics of individual coated bubbles are explored, taking into account nonlinear competition and dissipation factors such as dispersion, thermal effects, and drag force.

The coefficients $\beta^2$, $\delta$, $\kappa$, and $\gamma^2$ are related to the dissipation, the dispersion, the thermal conduction dissipation, and to the drag force, repsctively.

If $\kappa = \gamma = 0$, we obtain the Kudryashov-Sinelshchikov ^[18] Korteweg-de Vries-Burgers ^[3,20] equation

that models pressure waves in liquids with gas bubbles, taking into account heat transfer and viscosity. The mathematical results on Eq (1.2) are the following:

● analysis of exact solutions in ^[13],

● existence of the traveling waves in ^[2],

● well-posedness and asymptotic behavior in ^[7,11].

If $\beta = 0$, we derive the Korteweg-de Vries equation:

which describes surface waves of small amplitude and long wavelength in shallow water. Here, $u(t, x)$ represents the wave height above a flat bottom, $x$ corresponds to the distance in the propagation direction, and $t$ denotes the elapsed time. In ^{[4,6,10,12,15,16]}, the completele integrability of Eq (1.3) and the existence of solitary wave solutions are proved.

Through the manuscript, we will assume

● on the coefficients

● on the flux $f$, one of the following conditions:

for some positive constant $C_0$;

● on the initial value

The main result of this paper is the following theorem.

Theorem 1.1. Assume Eqs (1.5)–(1.7). For fixed $T > 0$, there exists a unique distributional solution $u$ of Eq (1.1), such that

Moreover, if $u_1$ and $u_2$ are solutions to Eq (1.1) corresponding to the initial conditions $u_{1, 0}$ and $u_{2, 0}$, respectively, it holds that:

for some suitable $C(T) > 0$, and every, $0\le t\le T$.

Observe that Theorem 1.1 gives the well-posedness of (1.1), without conditions on the constants. Moreover, the proof of Theorem 1.1 is based on the Aubin-Lions Lemma ^[5,21]. The analysis of Eq (1.1) is more delicate than the one of Eq (1.2) due to the presence of the nonlinear sources and the very general assumptions on the coefficients.

The structure of the paper is outlined as follows. Section 2 is dedicated to establishing several a priori estimates for a vanishing viscosity approximation of Eq (1.1). These estimates are crucial for proving our main result, which is presented in Section 3.

2.   Vanishing viscosity approximation

To establish existence, we utilize a vanishing viscosity approximation of equation (1.1), as discussed in ^[19]. Let $0 < \varepsilon < 1$ be a small parameter, and denote by $u_\varepsilon\in C^\infty([0, T)\times \mathbb{R})$ the unique classical solution to the following problem ^[1,9]:

where $u_{ \varepsilon, \, 0}$ is a $C^{\infty}$ approximation of $u_0$, such that

Let us prove some a priori estimates on $u_\varepsilon$, denoting with $C_0$ constants which depend only on the initial data, and with $C(T)$ the constants which depend also on $T$.

We begin by proving the following lemma:

Lemma 2.1. Let $T > 0$ be fixed. There exists a constant $C(T) > 0$, which does not depend on $\varepsilon$, such that

for every $0\le t\le T$.

Proof. For $0\le t\le T$. Multiplying equations (2.1) by $2 u_\varepsilon$, and integrating over $\mathbb{R}$ yields

Thus, it follows that

Therefore, applying the Gronwall's lemma and using Eq (2.2), we obtain

which gives Eq (2.3). □

Lemma 2.2. Fix $T > 0$ and assume (1.5). There exists a constant $C(T) > 0$, independent of $\varepsilon$, such that

holds for every $0\le t\le T$.

Proof. Let $0\le t\le T$. Consider $A, \, B$ as two real constants, which will be specified later. Thanks to Eq (1.5), multiplying Eq (2.1) by

we have that

Observe that

Therefore, an integration on $\mathbb{R}$ gives

Taking

we get

Since $0 < \varepsilon < 1$, due to the Young inequality and (2.3),

It follows from Eq (2.8) that

[8, Lemma 2.3] says that

Moreover, we have that

Consequentially, by Eqs (2.9)–(2.11), we have that

An integration on $(0, t)$ and Eqs (2.2) and (2.3) give

Therefore, by Eq (2.3),

We prove Eq (2.4). Thanks to the Hölder inequality,

Hence, we have that

Thanks to Eqs (2.3) and (2.12), we have that

Due to the Young inequality,

By Eq (2.14), we have that

which gives Eq (2.4).

Equation (2.5) follows from Eqs (2.4) and (2.12).

Finally, we prove Eq (2.6). We begin by observing that, from Eqs (2.4) and (2.10), we have

An integration on $(0, t)$ and Eqs (2.5) give Eq (2.6). □

Lemma 2.3. Fix $T > 0$ and assume (1.6). There exists a constant $C(T) > 0$, independent of $\varepsilon$, such that Eq (2.4) holds. Moreover, we have Eqs (2.5) and (2.6).

Proof. Let $0\le t\le T$. Multiplying Eq (2.1) by $-2 \partial_{x}^2 u_\varepsilon$, an integration on $\mathbb{R}$ gives

Therefore, we have that

Due Eqs (1.6) and (2.3) and the Young inequality,

It follows from Eq (2.15) that

Integrating on $(0, t)$, by Eq (2.3), we have that

Thanks to Eqs (2.3), (2.13), and (2.16), we have that

Therefore,

which gives (2.4).

Equation (2.5) follows from (2.4) and (2.16), while, arguing as in Lemma 2.2, we have Eq (2.6).

□

Lemma 2.4. Fix $T > 0$. There exists a constant $C(T) > 0$, independent of $\varepsilon$, such that

for every $0\le t\le T$.

Proof. Let $0\le t\le T$. We begin by observing that,

Thanks to the Hölder inequality,

Hence,

It follows from Eq (2.18) that

Therefore, by Eq (2.5),

An integration on $(0, t)$ and Eq (2.5) gives (2.17). □

3.   Proof of Theorem 1.1

This section is devoted to the proof of Theorem 1.1.

We begin by proving the following result.

Lemma 3.1. Fix $T > 0$. Then,

Consequently, there exist a subsequence $\{ u_{\varepsilon_k} \}_{k\in \mathbb{N}}$ and $u\in L^2_{loc}((0, T)\times \mathbb{R})$ such that

Moreover, $u$ is a solution of Eq (1.1), satisfying Eq (1.8).

Proof. We begin by proving Eq (3.1). To prove Eq (3.1), we rely on the Aubin-Lions Lemma (see ^[5,21]). We recall that

where the first inclusion is compact and the second one is continuous. Owing to the Aubin-Lions Lemma ^[21], to prove Eq (3.1), it suffices to show that

We prove Eq (3.3). Thanks to Lemmas 2.1–2.3,

Therefore,

which gives Eq (3.3).

We prove Eq (3.4). Observe that, by Eq (2.1),

where

Since $0 < \varepsilon < 1$, thanks to Eq (2.5), we have that

Therefore, by Eqs (3.5) and (3.6), we have that

We claim that

Thanks to Eqs (2.4) and (2.5),

Moreover, thanks to Eq (2.3),

We have that

In fact, thanks to Eqs (2.3) and (2.4),

Therefore, Eq (3.4) follows from Eqs (3.7)–(3.10).

Thanks to the Aubin-Lions Lemma, Eqs (3.1) and (3.2) hold.

Consequently, arguing as in [5, Theorem 1.1], $u$ is solution of Eq (1.1) and, thanks to Lemmas 2.1–2.3 and Eqs (2.4), (1.8) holds. □

Proof of Theorem 1.1. Lemma 3.1 gives the existence of a solution of Eq (1.1).

We prove Eq (1.9). Let $u_1$ and $u_2$ be two solutions of Eq (1.1), which verify Eq (1.8), that is,

Then, the function

is the solution of the following Cauchy problem:

Fixed $T > 0$, since $u_1, \, u_2\in H^1(\mathbb{R})$, for every $0\le t\le T$, we have that

We define

and observe that, by Eq (3.13), we have that

Moreover, by Eq (3.11) we have that

Observe that thanks to Eq (3.11),

Thanks to Eqs (3.14) and (3.17), Equation (3.12) is equivalent to the following one:

Multiplying Eq (3.18) by $2\omega$, an integration on $\mathbb{R}$ gives

Therefore, we have that

Due to Eqs (3.13), (3.15) and (3.16) and the Young inequality,

It follows from Eq (3.19) that

The Gronwall Lemma and Eq (3.12) give

Equation (1.9) follows from Eqs (3.11) and (3.20). □

Author contributions

Giuseppe Maria Coclite and Lorenzo Di Ruvo equally contributed to the methodologies, typesetting, and the development of the paper.

Use of AI tools declaration

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

Acknowledgments

Giuseppe Maria Coclite is an editorial boardmember for [Networks and Heterogeneous Media] and was not involved inthe editorial review or the decision to publish this article.

GMC is member of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). GMC has been partially supported by the Project funded under the National Recovery and Resilience Plan (NRRP), Mission 4 Component 2 Investment 1.4 -Call for tender No. 3138 of 16/12/2021 of Italian Ministry of University and Research funded by the European Union -NextGenerationEUoAward Number: CN000023, Concession Decree No. 1033 of 17/06/2022 adopted by the Italian Ministry of University and Research, CUP: D93C22000410001, Centro Nazionale per la Mobilità Sostenibile, the Italian Ministry of Education, University and Research under the Programme Department of Excellence Legge 232/2016 (Grant No. CUP - D93C23000100001), and the Research Project of National Relevance "Evolution problems involving interacting scales" granted by the Italian Ministry of Education, University and Research (MIUR Prin 2022, project code 2022M9BKBC, Grant No. CUP D53D23005880006). GMC expresses its gratitude to the HIAS - Hamburg Institute for Advanced Study for their warm hospitality.

Conflict of interest

The authors declare there is no conflict of interest.