Abstract
The Kuramoto–Velarde equation describes slow space-time variations of disturbances at interfaces, diffusion–reaction fronts and plasma instability fronts. It also describes Benard–Marangoni cells that occur when there is large surface tension on the interface in a microgravity environment. Under appropriate assumption on the initial data, of the time T, and the coefficients of such equation, we prove the well-posedness of the classical solutions for the Cauchy problem, associated with this equation.
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1 Introduction
In this paper, we investigate the well-posedness of the following Cauchy problem:
with \(\kappa ,\,\nu ,\,\delta ,\,\beta ,\,\gamma ,\,\alpha \in \mathbb {R}\).
On the initial datum, we assume
and one of the following
where
From a physical point of view, Eq. (1.1), known as the Kuramoto–Velarde equation, describes slow space-time variations of disturbances at interfaces, diffusion–reaction fronts and plasma instability fronts [1,2,3]. It also describes Benard–Marangoni cells that occur when there is large surface tension on the interface [4,5,6] in a microgravity environment. This situation arises in crystal growth experiments aboard an orbiting space station, although the free interface is metastable with respect to small perturbations. In particular, the nonlinearities, \(\gamma (\partial _x u)^2\) and \(\alpha u\partial _{x}^2u\), model pressure destabilization effects striving to rupture the interface. Moreover, in [7], (1.1) is deduced to describe the long waves on a viscous fluid flowing down an inclined plane, while, in [8], (1.1) is deduced to model the drift waves in a plasma.
In [9,10,11,12,13,14] (1.1) is used to model the spinodal decomposition of phase separating systems in an external field, while, in [15,16,17], (1.1) is used to describe the spatiotemporal evolution of the morphology of steps on crystal surfaces. Finally, in [18,19,20,21], (1.1) is deduced to describe the growth of thermodynamically unstable crystal surfaces with strongly anisotropic surface tension.
From a mathematical point of view, in [22], the exact solutions for (1.1) are studied, while in [23], the initial boundary problem is analyzed. In [1, 24], the existence of the solitons is proven, while in [25], the existence of traveling wave solutions for (1.1) is analyzed. In [26], the author analyzes the existence of the periodic solution for (1.1), under appropriate assumptions on \(\kappa ,\,\nu ,\,\delta ,\,\beta ,\,\gamma ,\,\alpha \). The well-posedness of the Cauchy problem for (1.1) is proven in [27], using the energy space technique and assuming \(\kappa =0\), and in [28], through a priori estimates together with an application of the Cauchy-Kovalevskaya and choosing
In particular, in [27], the author gives some suitable conditions on \(\nu ,\,\delta ,\,\beta ,\,\gamma ,\,\alpha \), and prove the local well-posedness of (1.1), with \(\kappa =0\). Instead, in [28], under Assumptions (1.2) and (1.8), the authors prove well-posedness of (1.1), for each choose of \(\beta \) and T.
Observe that (1.1) generalizes the following equation:
that (1.9) was also independently deduced by Kuramoto [29,30,31] to describe the phase turbulence in reaction-diffusion systems, and by Sivashinsky [32] to describe plane flame propagation, taking into account the combined influence of diffusion and thermal conduction of the gas on the stability of a plane flame front.
Equation (1.9) can be used to study incipient instabilities in several physical and chemical systems [33,34,35]. Moreover, (1.9), which is also known as the Benney–Lin equation [36, 37], was derived by Kuramoto in the study of phase turbulence in Belousov–Zhabotinsky reactions [38].
The dynamical properties and the existence of exact solutions for (1.9) have been investigated in [39,40,41,42,43,44]. In [45,46,47], the control problem for (1.9) with periodic boundary conditions, and on a bounded interval are studied, respectively. In [48], the problem of global exponential stabilization of (1.9) with periodic boundary conditions is analyzed. In [49], it is proposed a generalization of optimal control theory for (1.9), while in [50] the problem of global boundary control of (1.9) is considered. In [51], the existence of solitonic solutions for (1.9) is proven. In [28, 52,53,54], the well-posedness of the Cauchy problem for (1.9) is proven, using the energy space technique, the fixed point method, a priori estimates together with an application of the Cauchy–Kovalevskaya Theorem and a priori estimates together with an application of the Aubin–Lions Lemma, respectively. Instead, in [55,56,57], the initial-boundary value problem for (1.1) is studied, using a priori estimates together with an application of the Cauchy–Kovalevskaya Theorem, and the energy space technique, respectively. Finally, following [58,59,60], in [61], the convergence of the solution of (1.9) to the unique entropy one of the Burgers equation is proven.
The main result of this paper is the following theorem.
Theorem 1.1
Assuming that (1.2) and one within (1.3), (1.4), (1.5), (1.6) hold, there exists a unique solution u of (1.1), such that
Moreover, if \(u_1\) and \(u_2\) are two solutions of (1.1) in correspondence of the initial data \(u_{1,0}\) and \(u_{2,0}\), we have that
for some suitable \(C>0\), and every \(0\le t\le T\).
Compared to [27], Theorem 1.1 gives some conditions on \(u_0\), \(\beta \) and T to have classical solutions for (1.1), under Assumption (1.2). Moreover, the argument of Theorem 1.1 relies on deriving suitable a priori estimates together with the existence result in [28].
The paper is organized as follows. In Sect. 2, we prove some a priori estimates of (1.1), under Assumptions (1.3), (1.4), (1.5) and (1.6), respectively. Those play a key role in the proof of our main result, which is given in Sect. 3.
2 A priori estimates
In this section, we prove some a priori estimates on u.
We prove the following result.
Lemma 2.1
We have that
for every \(0\le t\le T\), where \(\tau ^2\) is defined in (1.7). In particular, if (1.3) holds, there exists a constant \(C>0\), such that
for every \(0\le t\le T\). Moreover,
for every \(0\le t\le T\).
Proof
Let \(0\le t\le T\) be given. Multiplying (1.1) by \(2u-2\partial _{x}^2u\), an integration on \(\mathbb {R}\) gives
Therefore,
Observe that
Moreover,
Therefore,
Consequently, by (2.8) and (2.9), we have that
Due to the Young inequality,
It follows from (1.3) and (2.7) that
Thanks to the Hölder inequality,
Hence,
Therefore, by the Young inequality,
Consequently, by (2.11),
We define the following function
It follows from (1.7), (2.14) and (2.15) that
Therefore, we have that
Consequently, by (2.15),
Integrating on (0, t), we have that
Hence, by (1.2) and (2.15), we get
which gives (2.1).
Assume (1.3) and we prove (2.2). We begin by observing that, by (2.1),
We assume by contradiction that (2.2) does not hold, i.e.,
Therefore, by (2.18) and (2.19),
It follows from (1.7) and (2.20) that
which contradicts (1.3).
We prove (2.3). Due to (2.2) and (2.13), we have that
Hence,
which gives (2.3).
We prove (2.4) and (2.5). Thanks to (2.2) and (2.14),
Integrating on (0, t), by (1.2), we get
which gives (2.4) and (2.5), respectively.
Finally, we prove (2.6). We begin by observing that [62] [Lemma 2.3] says that
Consequently, by (2.2),
Integrating on (0, t), by (2.4), we have (2.6). \(\square \)
Lemma 2.2
We have that
for every \(0\le t\le T\) and some \(\lambda \in (0,1/4)\). In particular, under Assumption (1.4), we have (2.2), (2.3), (2.4), (2.5) and (2.6).
Proof
Arguing as in Lemma 2.1, we have (2.17). Therefore, by (1.4), (1.7) and (2.17), we have that
Hence,
Thanks to (1.4) and (1.7), an integration on (0, t) gives
Consequently,
(1.2), (2.15) and (2.22) give (2.21).
We assume (1.4) and we prove (2.2). We begin by observing that, by (2.21), we have that
We assume by contradiction that (2.2) does not hold, i.e., we have (2.19). Consequently, by (2.19) and (2.23),
Therefore, to (2.24), we have that
that is
Hence,
which contradicts (1.4).
Finally, arguing as in Lemma 2.1, we have (2.3), (2.4), (2.5) and (2.6). \(\square \)
Lemma 2.3
We have that
for every \(0\le t\le T\). In particular, if (1.5) holds, we have (2.2), (2.3), (2.4), (2.5) and (2.6).
Proof
Arguing as in Lemma 2.1, we have (2.17). Therefore, by (2.17), we have that
Therefore,
Integrating on (0, t), we have that
that is,
(2.25) follows from (1.2) and (2.15).
Assume (1.5) and we prove (2.2). We begin by observing that, by (2.25), we have that
We assume by contradiction that (2.2) does not hold, i.e., we have (2.19). It follows from (2.19) and (2.27) that
Consequently, by (2.28), we have that
which contradicts (1.5).
Finally, arguing as in Lemma 2.1, we have (2.3), (2.4), (2.5) and (2.6). \(\square \)
Lemma 2.4
We have that
for every \(0\le t\le T\), where
In particular, assuming (1.6) and taking
we obtain the following inequality
where
Moreover, (2.2), (2.3), (2.4), (2.5) and (2.6), hold.
Proof
Arguing as in Lemma 2.1, we have (2.17). Therefore, by (2.17) and (2.30), we have that
Hence,
Integrating on (0, t), we have that
Hence,
(1.2), (2.15) and (2.34) give (2.29).
Assume (1.6) and we prove (2.32). We begin by observe that, by (2.29),
We assume by contradiction that (2.2) does not hold, i.e., we have (2.19). It follows from (2.19) and (2.35) that
Therefore, we obtain that
that is
We consider the following function
with
Observe that, by (2.38) and (2.39),
Consequently, by (2.38) and (2.40), (2.37) is verified if
Observe that, by (2.38),
Hence, \(F'(\lambda )>0\) if and only if
that is
Thanks to (2.39), (2.43) is verified when
that is
which contradicts (1.6).
Therefore, if we assume (1.6), (2.37) cannot hold. Observe that, by (2.42),
\(F'(\lambda )\le 0\) when \(\lambda > \frac{1}{4}\), if and only if,
that is
It follows from (2.30) and (2.44) that
\(\lambda \) exist, if
Hence,
that is
which is guaranteed by Assumption (1.6). Therefore, (2.32) holds.
Finally, thanks to (2.32) and (2.33), we have (2.2), while arguing as in Lemma 2.1, we have (2.3), (2.4), (2.5) and (2.6). \(\square \)
Lemma 2.5
Assume that one within (1.3), (1.4), (1.5), (1.6) holds. There exist a constant \(C>0\), such that
for every \(0\le t\le T\). In particular,
Proof
Let \(0\le t\le T\). Multiplying (1.1) by \(2\partial _{x}^4u\), an integration on \(\mathbb {R}\) gives
Therefore, we have that
Due to (2.3) and the Young inequality,
It follows from (2.47) that
Integrating on (0, t), by (1.2), (2.4), (2.5) and (2.6), we get
which gives (2.45).
Finally, we prove (2.46). Thanks to (2.2), (2.45) and the Hölder inequality,
Hence,
which gives (2.46). \(\square \)
Lemma 2.6
Assume that one within (1.3), (1.4), (1.5), (1.6) holds. There exist a constant \(C>0\), such that
for every \(0\le t\le T\).
Proof
Let \(0\le t\le T\). Multiplying (1.1) by \(2\partial _tu\), an integration on \(\mathbb {R}\) gives
Therefore,
Due to the (2.2), (2.3), (2.45) and the Young inequality,
where \(D_1\) is a positive constant, which will specified later. Consequently, by (2.49),
Taking \(D_1=\frac{1}{4}\), we have that
Integrating on (0, t), by (2.5), (2.6) and (2.45), we get
which gives (2.48). \(\square \)
3 Proof of Theorem 1.1
This section devoted to the proof of Theorem 1.1.
Proof of Theorem 1.1
Thanks to Lemmas 2.1, or 2.3, or 2.4, 2.5, 2.6 and the Cauchy-Kovalevskaya Theorem [63], we have that u is solution of (1.1) and (1.10) holds.
We prove (1.11). Let \(u_1\) and \(u_2\) be two solutions of (1.1), which verify (1.10), that is
Then, the function
is the solution of the following Cauchy problem:
Observe the, thanks to (3.1),
Therefore, (3.2) reads
Since, thanks to (3.1),
multiplying (3.2) by \(2\omega \), an integration on \(\mathbb {R}\) gives
Observe that, since \(u_1,\,u_2\in H^2(\mathbb {R})\), for every \(0\le t\le T\), we have that
Consequently, thanks to (3.5), we obtain that
Due to (3.5), (3.6) and the Young inequality,
It follows from (3.4) that
Observe that, by the Hölder inequality,
Hence,
Due to the Young inequality,
Consequently, by (3.7),
Observe that
Therefore, by the Young inequality,
It follows from (3.9) that
The Gronwall Lemma and (3.2) gives
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Coclite, G.M., di Ruvo, L. Well-posedness result for the Kuramoto–Velarde equation. Boll Unione Mat Ital 14, 659–679 (2021). https://doi.org/10.1007/s40574-021-00303-7
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DOI: https://doi.org/10.1007/s40574-021-00303-7