On the initial-boundary value problem for a Kuramoto-Sinelshchikov type equation

1.   Introduction

In this paper, we investigate the well-posedness of the classical solutions for the equation:

with $\kappa, \, q, \, \nu, \, \delta, \, \beta\in \mathbb{R}$ and $\beta\ne 0$.

We are interested in the initial-boundary value problem for this equation. More precisely we consider the following boundary conditions

Moreover, we augment (1.1) with the following initial datum:

on which assume

Assuming $q = 0$, (1.1) reads

(1.9) arises in interesting physical situations, for example as a model for long waves on a viscous fluid owing down an inclined plane ^[50]. and to derive drift waves in a plasma ^[20]. (1.9) was derived also independently by Kuramoto ^[30,31,32] as a model for phase turbulence in reaction-diffusion systems and by Sivashinsky ^[47] as a model for plane flame propagation, describing the combined influence of diffusion and thermal conduction of the gas on the stability of a plane flame front.

Equation (1.9) also describes incipient instabilities in a variety of physical and chemical systems ^[7,23,33]. Moreover, (1.9), which is also known as the Benney-Lin equation ^[4,41], was derived by Kuramoto in the study of phase turbulence in the Belousov-Zhabotinsky reaction ^[38].

The dynamical properties and the existence of exact solutions for (1.9) have been investigated in ^{[21,26,28,43,44,51]}. In ^[3,6,22], the control problem for (1.9) with periodic boundary conditions, and on a bounded interval are studied, respectively. In ^[8], the problem of global exponential stabilization of (1.9) with periodic boundary conditions is analyzed. In ^[24], it is proposed a generalization of optimal control theory for (1.9), while in ^[42] the problem of global boundary control of (1.9) is considered. In ^[45], the existence of solitonic solutions for (1.9) is proven. In ^[5,18,48], the well-posedness of the Cauchy problem for (1.9) is proven, using the energy space technique, a priori estimates together with an application of the Cauchy-Kovalevskaya and the fixed point method, respectively. In particular, in ^[18], the well-posedness of the Cauchy problem for (1.1) is proven. In ^[12], following ^[9,10,37,46] it is proven that, when $\nu, \, \delta, \, \beta^2$ go to zero, the solution of (1.9) converges to the unique entropy one of the Burgers equation. Finally, the initial-boundary value problem for (1.9), under the conditions (1.2) is analyzed in ^[39,40], in a quarter plane and in a bounded domain, respectively, using the energy space technique, under appropriate assumptions on $\kappa, \, \nu, \, \delta, \, \beta$.

Taking $q = \nu = \beta = 0$ in (1.1), we have the Korteweg-de Vries equation ^[27]

that has a very wide range of applications, such as magnetic fluid waves, ion sound waves, and longitudinal astigmatic waves.

From a mathematical point of view, in ^[16,19,25], the Cauchy problem for (1.10) is studied, while in ^[29], the author reviewed the travelling wave solutions for (1.10). Moreover, in ^[14,37,46], the convergence of the solution of (1.10) to the unique entropy one of the Burgers equation is proven.

Taking $\kappa = r = \delta = \mu = \beta = 0$, (1.1) becomes

which is known as the modified Korteweg-de Vries equation.

^{[1,2,15,34,35,36]} show that (1.11) is a non-slowly-varying envelope approximation model that describes the physics of few-cycle-pulse optical solitons. In ^[19,25], the Cauchy problem for (1.11) is studied, while, in ^[13,46], the convergence of the solution of (1.11) to the unique entropy solution of the following scalar conservation law

The main result of this paper is the following theorem.

Theorem 1.1. Fix $T > 0$. The initial value problems (1.1)-(1.2)-(1.7), (1.1)-(1.3)-(1.7), (1.1)-(1.4)-(1.7), (1.1)-(1.5)-(1.7), (1.1)-(1.6)-(1.7), admit an unique solution

Moreover, if $u_1$ and $u_2$ are two solutions of the same initial-boundary value problem for (1.1), we have

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

Compared to ^[39], Theorem 1.1 gives the well-posedness of the initial-boundary value problem (1.1)-(1.2)-(1.7), without any additional assumption on the constants. The proof of Theorem 1.1 relies on deriving suitable a priori estimates together with an application of the Cauchy-Kovalevskaya Theorem ^[49].

The paper is organized as follows. In Sections 2, 3, 4, 5, 6, we prove Theorem 1.1 for (1.1)-(1.2)-(1.7), (1.1)-(1.3)-(1.7), (1.1)-(1.4)-(1.7), (1.1)-(1.5)-(1.7), (1.1)-(1.6)-(1.7), respectively.

2.   Proof of the Theorem 1.1 for (1.1)-(1.2)-(1.7)

In this section, we prove Theorem 1.1 for (1.1)-(1.2)-(1.7).

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

Following ^[11,17], we introduce the auxiliary variable:

Observe that

In particular, thanks to (1.1)-(1.2)-(1.7), (2.1) and (2.2),

Moreover, thanks to (1.2) and (2.2), we have that

Again by (1.1)-(1.2)-(1.7), we have the following equation for $v$:

Lemma 2.1. Fix $T > 0$. There exists a constant $C(T) > 0$, such that

for every $0\le t\le T$. In particular, we have that

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

Proof. Let $0\le t\le T$. We begin by observing that, thanks to (2.3),

Therefore, by (2.8), multiplying (2.5) by $2v$, an integration on $(0, \infty)$ gives

Observe that, for each $x\in (0, \infty)$,

Due (1.2), (2.10) and the Young inequality,

It follows from (2.9) that

Thanks to (2.3),

Therefore, by the Young inequality,

Consequently, by (2.11),

By the Gronwall Lemma and (2.4), we have

which gives (2.6).

Finally, we prove (2.7). By (2.6), and (2.12),

Integrating on $(0, t)$, by (2.6), we have

which gives (2.7).

Lemma 2.2. Fix $T > 0$. There exists a constant $C(T) > 0$, such that

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

Proof. Let $0\le t\le T$. We begin by proving (2.13). Observe that, by (2.1),

Consequently, an integration on $(0, \infty)$ gives

Observe that

Due to (1.2), (2.10), (2.17) and the Young inequality,

Therefore, by (2.6) and (2.16),

that is (2.13).

We prove (2.14). By (2.5), we have that

Therefore, an integration on $(0, \infty)$ gives

Due to (1.2), (2.10), (2.17) and the Young inequality,

It follows from (2.18) that

Integrating on $(0, t)$, by (2.7), we have that

which gives (2.14).

Finally, we prove (2.15). By (2.2), we have that

An integration on $(0, \infty)$ gives

Due to (1.2), (2.10), (2.17) and the Young inequality,

It follows from (2.19) that

Integrating on $(0, t)$, by (2.6), we have that

which gives (2.15).

Lemma 2.3. Fix $T > 0$. There exists a constant $C(T) > 0$, such that

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

Proof. Let $0\le t\le T$. We begin by proving (2.20). Thanks to (1.2) and the Young inequality,

Hence,

Integrating on $(0, t)$, by (2.14) and (2.15), we have that

which gives (2.20).

Finally, we prove (2.21). Thanks to (2.13), we have that

(2.21) follows from (2.20) and an integration on $(0, t)$.

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

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

Proof. Let $0\le t\le T$. Consider an positive constant $A$, which will be specified later. Multiplying (1.1)-(1.2)-(1.7) by

we have that

Observe that, thanks to (1.1)-(1.2)-(1.7),

Therefore, thanks to (2.28), an integration of (2.27) on $(0, \infty)$ gives

Due to the Young inequality,

where $D_1$ is a positive constant, which will be specified later. It follows from (2.29) that

Choosing $D_1 = 3$, we have that

Due to (1.2) and the Young inequality,

Consequently, by (2.30),

Thanks to the Young inequality,

It follows from (2.31) that

Choosing

we have that

Integrating on $(0, t)$, by (1.8), (2.15) and (2.21), we get

which gives (2.23).

(2.24) follows from (2.15), (2.23), (2.33) and an integration of (2.32) on $(0, t)$, while (2.22) and (2.23) give (2.25).

Finally, we prove (2.26). Due to (1.2), (2.13), (2.23) and the Hölder inequality,

Hence,

which gives (2.26).

Lemma 2.5. Fix $T > 0$. There exists a constant $C(T) > 0$, such that

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

Proof. Let $0\le t\le T$. Multiplying (1.1)-(1.2)-(1.7) by $2 \partial_t u$, an integration on $(0, \infty)$ gives

Due to (2.23) and the Young inequality,

Consequently, by (2.35),

Choosing $D_2 = \frac{1}{4}$, we have that

Integrating on $(0, t)$, by (2.21) and (2.23), we get

which gives (2.34).

Now, we prove Theorem 1.1.

Proof. Fix $T > 0$. Thanks to Lemmas 2.2, 2.4, 2.5 and the Cauchy-Kovalevskaya Theorem ^[49], we have that $u$ is solution of (1.1)-(1.2)-(1.7) and (1.13) holds.

We prove (1.14). Let $u_1$ and $u_2$ be two solutions of (1.1)-(1.2)-(1.7), which verify (1.13), that is

Then, the function

solves the following initial-boundary value problem:

Observe that, thanks to (2.36),

Therefore, (2.37) is equivalent to the following equation:

Moreover, since $u_1, \, u_2\in L^{\infty}(0, T;H^2(0, \infty))$, we have that

Observe again that, thanks to (2.37),

Therefore, multiplying (2.38) by $2\omega$, thanks to (2.40), an integration on $(0, \infty)$ gives

Due to (2.39) and the Young inequality,

Therefore, by (2.41),

The Gronwall Lemma and (2.37) gives

(1.14) follows from (2.36) and (2.42).

3.   Proof of the Theorem 1.1 for (1.1)-(1.3)-(1.7)

In this section, we prove Theorem 1.1 for (1.1)-(1.3)-(1.7).

Let us prove some a priori estimates on $u$.

Following ^[11,17], we consider the following function:

Observe that

By (1.1)-(1.3)-(1.7) and (3.2),

while, by (1.3) and (3.2), we have (2.4).

Again by (1.1)-(1.3)-(1.7) and (3.2), we have the following equation for $v$.

We prove the following result.

Lemma 3.1. Fix $T > 0$. There exists a constant $C(T) > 0$, such that

for every $0\le t\le T$. In particular, we have (2.7). Moreover,

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

Proof. Let $0\le t\le T$. We begin by observing that, thanks to (3.3),

Consequently, multiplying (3.4) by $2v$, thanks to (3.7), an integration of (3.7) on $(0, \infty)$ gives

Since

thanks to (1.3), (2.10), (3.9) and the Young inequality,

It follows from (3.8) that

Observe that, by the Young inequality,

Consequently, by (3.10),

Observe that, thanks to (3.3),

Therefore, by the Young inequality,

It follows from (3.12) that

By (2.4) and the Gronwall Lemma, we get

which gives (3.5).

We prove (2.7). By (3.5) and (3.13), we have that

Integrating on $(0, t)$, by (3.5), we have (2.7).

Finally, (3.6) follows from (2.7), (3.5), (3.11) and an integration on $(0, t)$.

Lemma 3.2. Fix $T > 0$. There exists a constant $C(T) > 0$, such that

for every $0\le t\le T$. In particular, (2.13), (2.14), (2.15), (2.20), (2.21) hold.

Proof. Let $0\le t\le T$. We prove (3.15). We begin by observing that, thanks (3.2),

Therefore, by (1.3) and the Young inequality,

Integrating on $(0, t)$, by (3.6), we get

which gives (3.15).

Arguing as in Lemma 2.2, we have (2.13), (2.14) and (2.15).

We prove (2.20). Thanks to the Young inequality,

Therefore,

Integrating on $(0, t)$, by (2.14), (2.15) and (3.15), we have (2.20).

Finally, arguing as in Lemma 2.3, we have (2.20).

Lemma 3.3. Fix $T > 0$. There exists a constant $C(T) > 0$, such that

for every $0\le t\le T$. In particular, (2.26) holds.

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

Since, thanks (1.1)-(1.3)-(1.7), $\partial_t u(t, 0) = g'(t)$, multiplying (1.1)-(1.3)-(1.7) by $-2 \partial_{x}^2 u$, thanks to (3.18), an integration on $(0, \infty)$ gives

Due to (1.3) and the Young inequality,

Consequently, by (3.19),

Integrating on $(0, t)$, by (1.8), (2.15) and (2.21), we get

which gives (3.17).

Finally, arguing as in Lemma 2.4, we have (2.26).

Lemma 3.4. Fix $T > 0$. There exists a constant $C(T) > 0$, such that

for ever $0\le t\le T$. In particular, we have (2.25) and

for every $0\le t\le T$. Moreover, (2.34) holds.

Proof. Let $0\le t\le T$. We begin by observing that, thanks to (1.1)-(1.3)-(1.7),

Since, thanks to (1.1)-(1.3)-(1.7), $\partial_t u(t, 0) = g'(t)$, multiplying (1.1)-(1.3)-(1.7) by $2 \partial_{x}^4 u$, thanks (3.22), an integration on $(0, \infty)$ gives

Due to (1.3) and the Young inequality,

It follows from (3.23) that

Thanks to the Young inequality,

Consequently, by (3.24),

Integrating on $(0, t)$, by (1.8), (2.21) and (3.17), we get

which gives (3.20).

We prove (2.25). Thanks to the Hölder inequality,

Therefore, by (3.17) and (3.20),

By (3.16), (3.17) and (3.20), we obtain that

which gives (2.25).

Finally, (3.21) follows from (3.17), (3.20), (3.25) and an integration on $(0, t)$, while arguing as in Lemma 2.5, we have (2.34).

Arguing as in Section 2, we have Theorem 1.1.

4.   Proof of the Theorem 1.1 for (1.1)-(1.4)-(1.7)

In this section, we prove Theorem 1.1 for (1.1)-(1.4)-(1.7).

Let us prove some a priori estimates on $u$.

We begin by proving the following lemma.

Lemma 4.1. Fix $T > 0$. There exists a constant $C(T) > 0$, such that

for every $0\le t\le T$. In particular, we have (2.14), (2.20), (2.21) and

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

Proof. Let $0\le t\le T$. Multiplying (1.1)-(1.4)-(1.7) by $2v$, thanks to (1.1)-(1.4)-(1.7), an integration on $(0, \infty)$ give

Therefore, we have that

Due to the Young inequality,

where $D_3$ is a positive constant, which will be specified later. Consequently, by (4.3),

Choosing $D_3 = \frac{a^2}{2}$, we have that

By the Gronwall Lemma and (1.8), we get

which gives (4.1).

We prove (2.14). Thanks to (1.1)-(1.4)-(1.7), (4.1) and the Young inequality,

(2.14) follows from (4.1) and an integration on $(0, t)$.

We prove (2.20). By (3.16) and (1.1)-(1.4)-(1.7), we have that

Therefore, an integration on $(0, t)$, (2.14) and (4.1) gives (2.20).

Arguing as in Lemma 2.3, we have (2.21).

Finally, we prove (4.2). We begin by observing that, by the Young inequality,

(4.2) follows from (4.1) and an integration on $(0, t)$.

Lemma 4.2. Fix $T > 0$. There exists a constant $C(T) > 0$, such that (2.26) holds. In particular, we have that

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

Proof. Let $0\le t\le T$. Multiplying (1.1)-(1.4)-(1.7) by $-2 \partial_{x}^2 u$, thanks to (1.1)-(1.4)-(1.7), an integration on $(0, \infty)$ gives

Therefore, we have that

Due to the Young inequality,

It follows from (4.7) that

Integrating on $(0, t)$, by (1.8), (2.21) and (4.1), we have that

We prove (2.26). Thanks to (4.1), (4.8) and the Hölder inequality,

Therefore,

which gives (2.26).

(4.5) follows from (2.26) and (4.8).

Finally, we prove (4.6). Thanks to the Young inequality,

(4.1), (4.5) and an integration on $(0, t)$ give (4.6).

Lemma 4.3. Fix $T > 0$. There exists a constant $C(T) > 0$, such that

for every $0\le t\le T$. In particular, (2.25) holds. Moreover, we have (2.34).

Proof. Let $0\le t\le T$. Since, by (1.1)-(1.4)-(1.7), $\partial_{tx}^2 u(t, 0) = 0$, multiplying (1.1)-(1.4)-(1.7) by $2 \partial_{x}^4 u$, thanks to (1.1)-(1.4)-(1.7), an integration on $(0, t)$ gives

Therefore, we have

Due to (2.26), (4.5) and the Young inequality,

It follows from (4.11) that

Integrating on $(0, t)$, by (1.8) and (4.5), we get

which gives (4.10).

(2.25) follows from (4.4), (4.5) and (4.10).

Finally, we prove (2.34). Multiplying (1.1)-(1.4)-(1.7) by $2 \partial_t u$, an integration on $(0, \infty)$ give

Due to (2.26), (4.5), (4.10) and the Young inequality,

Therefore, by (4.12),

Choosing $D_4 = \frac{1}{4}$, we have

An integration on $(0, t)$ and (4.10) give (2.34).

Arguing as in Section 2, we have Theorem 1.1.

5.   Proof of the Theorem 1.1 for (1.1)-(1.5)-(1.7)

In this section, we prove Theorem 1.1 for (1.1)-(1.5)-(1.7).

Let us prove some a priori estimates on $u$.

We begin by proving the following lemma.

Lemma 5.1. Fix $T > 0$. There exists a constant $C(T) > 0$, such that

for every $0\le t\le T$. In particular, we have (2.14), (2.20), (2.21) and (4.2).

Proof. Let $0\le t\le T$. We begin by observing that, thanks to (1.1)-(1.5)-(1.7),

Therefore, multiplying (1.1)-(1.5)-(1.7) by $2u$, thanks to (5.2), an integration on $(0, \infty)$ gives

Due to the Young inequality,

It follows from (5.3) that

Observe that, by the Hölder inequality,

Moreover, by the Young inequality,

where $D_5$ is a positive constant, which will be specified later. Observe that, by the Young inequality,

It follows from (5.5), (5.6) and (5.7) that

that is

Due to the Young inequality,

where $D_6$ is a positive constant, which will be specified later. It follows from (5.8) that

Consequently, by (5.4) and (5.9),

Choosing

we have that

Due to the Young inequality,

It follows from (5.11) that

By the Gronwall Lemma and (1.8), we get

which gives (5.1).

Arguing as in Lemma 4.1, we have (4.2), while (4.2), (5.1), (5.9), (5.10) and an integration on $(0, t)$ give (2.14).

We prove (2.20). By (3.16), (5.7) and (5.10), we have that

(2.14), (5.1) and an integration on $(0, t)$ give (2.20).

Arguing as in Lemma 2.3, we have (2.21), while (3.15) follows from (2.14), (5.1), (5.7), (5.10) and an integration on $(0, t)$.

Lemma 5.2. Fix $T > 0$. There exists a constant $C(T) > 0$, such that (2.26) holds. In particular,

for every $0\le t\le T$. Moreover, we have (2.25),

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

Proof. Let $0\le t\le T$. We begin by observing that, thanks (1.1)-(1.5)-(1.7),

Therefore, multiplying (1.1)-(1.5)-(1.7) by $2 \partial_{x}^4 u$, by (5.15) an integration on $(0, \infty$ gives

Due to the Young inequality,

It follows from (5.16) that

Integrating on $(0, \infty)$, by (1.8), (2.21) and (5.1), we have that

We prove that

We begin by observing that, by (5.1), (5.9) and (5.10),

Due to (5.1) and the Young inequality,

Therefore, by (5.18),

(5.17) and (5.19) give (5.20).

We prove (2.26). Thanks to (5.1), (5.19) and the Hölder inequality, we have (4.9), which gives (2.26).

(5.12) and (5.13) follows from (2.26), (5.17) and (5.18), respectively, while (3.16), (5.7), (5.10) (5.12) and (5.13) gives (2.25).

Finally, we prove (5.14). We begin by observing that, thanks to (1.1)-(1.5)-(1.7),

Therefore, by (5.12) and the Young inequality,

(5.14) follows from (5.12) and an integration on $(0, t$).

Lemma 5.3. Fix $T > 0$. There exists a constant $C(T) > 0$, such that (2.34) holds.

Proof. Let $0\le t\le T$. Arguing as in Lemma 4.3, we have that

Due to the Young inequality,

Therefore, by (5.21), we have

(2.34) follows from (5.12), (5.14) and an integration on $(0, t)$.

Arguing as in Section 2, we have Theorem 1.1.

6.   Proof of the Theorem 1.1 for (1.1)-(1.6)-(1.7)

In this section, we prove Theorem 1.1 for (1.1)-(1.6)-(1.7).

Let us prove some a priori estimates on $u$.

We begin by proving the following lemma.

Lemma 6.1. Fix $T > 0$. There exists a constant $C(T) > 0$, such that

for every $0\le t\le T$. In particular, we have (2.14), (2.20) and (2.21).

Proof. Let $0\le t\le T$. We begin by observing that, thanks to (1.1)-(1.6)-(1.7),

Therefore, multiplying (1.1)-(1.6)-(1.7) by $2v$, thanks to (6.2), an integration on $0, \infty)$ gives

Due to the Young inequality,

Consequently, by (6.3),

Observe that, by (5.5) and (1.1)-(1.6)-(1.7),

Therefore, by the Young inequality,

It follows from (6.4) that

By the Gronwall Lemma and (1.8).

which gives (6.1).

(2.14) follows from (6.1) and an integration on $(0, t)$, while, arguing as in Lemma 2.3, we have (2.20) and (2.21).

Lemma 6.2. Fix $T > 0$. There exists a constant $C(T) > 0$, such that (2.26) holds. In particular,

for every $0\le t\le T$. Moreover, we have (2.24), (2.25) and (2.34).

Proof. Let $0\le t\le T$. Consider an positive constant $B$, which will be specified later. Observing that, since, thanks to (1.1)-(1.6)-(1.7), $\partial_t u(t, 0 = \partial_t \partial_x u(t, 0)$, we have that

Therefore, multiplying (1.1)-(1.6)-(1.7) by $2 \partial_{x}^4 u$, thanks to (6.7), an integration on $(0, \infty)$ gives

Due to the Young inequality,

where $D_7$ is a positive constant, which will be specified later. It follows from (6.8) that

Choosing $D_7 = \frac{1}{4}$, we have that

Observe that by the Young inequality,

Consequently, by (6.9),

Choosing

we have that

Integrating on $(0, t)$, by (1.8), (2.21) and (6.1), we get

We prove (2.26). Thanks to (6.1), (6.10) and the Hölder inequality, we have (4.9), which gives (2.26).

(6.6) follows from (2.26) and (6.10).

Finally, arguing as in Lemma 2.4, we have (2.24) and (2.25), while arguing as in Lemma 5.3, (2.34) holds.

Arguing as in Section 2, we have Theorem 1.1.

Acknowledgments

The authors are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

Conflict of interest

The authors declare that there are no conflict of interest.