A note on the non-homogeneous initial boundary problem for an Ostrovsky-Hunter type equation

We consider an Ostrovsky-Hunter type equation, which also includes the short pulse equation, or the Kozlov-Sazonov equation. We prove the well-posedness of the entropy solution for the non-homogeneous initial boundary value problem. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method.

When f (u) = u 2 , (1) reads (13) with γ > 0 is known under different names such as the reduced Ostrovsky equation [31,32], the Ostrovsky-Hunter equation [3] and the short-wave equation [21]. It was deduced considering two asymptotic expansions of the shallow water equations, first with respect to the rotation frequency and then with respect to the amplitude of the waves [20,22].
Instead, if γ < 0, (13), known as the Vakhnenko equation [27,42], describes the short-wave perturbations in a relaxing medium when the equations of motion are closed by the dynamic equation of state (see [4,43]).
Moreover, (1) with f (u) = u 2 can be seen as the limit of no high-frequency dispersion (β = 0) of the nonlinear evolution equation known as the Ostrovsky equation [30]. This equation generalizes the Korteweg-deVries equation that corresponds to γ = 0. In [8,15], the authors proved the wellposedness of the entropy solution of the Cauchy problem for (13), while, following [7,38], in [6], the convergence of the solutions of (14) to the entropy solutions of (13) is proven.
In [9,16,20], the authors study the well-posedness for the initial boundary value problem for (13). In particular, in [9], they prove the well-posedness of the entropy solution of non-homogeneous initial boundary value problem for (13), assuming on the initial datum (4) and (6), while, on the boundary datum they assume (10).
The fundamental assumption, which is made in [9], is the following one: Observe that, if we have (13) with γ < 0, [9, Theorem 1.2] is not true, because, from a mathematical point of view, [9, Lemma 2.6] does not hold (see also Lemmas 2.4,3.7.) In this paper, we improve the result of [9], when γ > 0, assuming on the boundary datum (7). Moreover, we extend the results of [9] in the case γ < 0, assuming on the boundary datum (10).
When f (u) = u 3 , (1) reads It was introduced both Kozlov and Sazonov [23] as a model equation describing the nonlinear propagation of optical pulses of a few oscillations duration in dielectric media, and Schäfer and and Wayne [37] as a model equation describing the propagation of ultra-short light pulses in silica optical fibers. Moreover, [2,5,36,34] show that (16) is a particular Rabelo equation which describes pseudospherical surfaces. It also is interesting to remind that equation (16) was proposed earlier in [29] in the context of plasma physic and that the similar equations describe the dynamics of radiating gases [25,39].
In [41], the authors deduce (16) to describe the short pulse propagation in nonlinear metamaterials characterized by a weak Kerr-type nonlinearity in their dielectric response. In particular, they proved that γ κ < 0, or γ κ > 0. (17) Finally, an interpretation of (16) in the context of Maxwell equations is given in [33].
On the other hand, (16) is the limit of no high-frequency dispersion (β = 0) of non-linear equation It was derived by Costanzino, Manukian and Jones [19] in the context of the nonlinear Maxwell equations with high-frequency dispersion. Kozlov and Sazonov [23] show that (18) is an more general equation than (16) to describe the nonlinear propagation of optical pulses of a few oscillations duration in dielectric media. In [10,15], The wellposedness of the Cauchy problem for (16) is proven, while, in [11,12], the authors proved the convergence of the solutions of (18) to the the entropy ones of (16). In [18], the authors prove the convergence of a finite difference scheme to the unique entropy solution of (13) and (16) on a bounded domain with periodic boundary conditions. That result also provides an existence proof for periodic entropy solutions for (13) and (16).
In [13], the authors prove the well-posedness of the entropy solution of nonhomogeneous initial boundary value problem for (16), under the assumption and, assuming on the boundary datum (7). In this paper, under the assumption (7), we extend the result of [13] in the case Moreover, we prove the well-posedness of the entropy solution of non-homogeneous initial boundary value problem for (16) in the case: under the assumption (12).
Observe that, integrating (1) in (0, x), we gain the integro-differential formulation of (1) (see [26,35]) that is equivalent to where One of the main issues in the analysis of (23) is that the equation is not preserving the L 1 norm, as a consequence the nonlocal source term P and the solution u are a priori only locally bounded. Indeed, from (22) and (23) is clear that we cannot have any L ∞ bound without an L 1 bound. Since we are interested in the bounded solutions of (1), some assumptions on the decay at infinity of the initial condition u 0 are needed. The unique useful conserved quantities are In the sense that if u(t, ·) has zero mean at time t = 0, then it will have zero mean at any time t > 0. In addition, the L 2 norm of u(t, ·) is constant with respect to t. Therefore, we require that initial condition u 0 belongs to L 2 ∩ L ∞ and has zero mean. Due to the regularizing effect of the P equation in (23) we have that Therefore, if a map u ∈ L ∞ ((0, T ) × (0, ∞)), T > 0, satisfies, for every convex map η ∈ C 2 (R), in the sense of distributions, then [17, Theorem 1.1] provides the existence of strong trace u τ 0 on the boundary x = 0. Following [1,9,13,15,20], we give the following definition of solution Definition 1.1. Assume (24). We say that u ∈ L ∞ ((0, T ) × (0, ∞)), T > 0, is an entropy solution of the initial-boundary value problem (1), (2) and (3) if for every nonnegative test function φ ∈ C 2 (R 2 ) with compact support, and c ∈ R where u τ 0 (t) is the trace of u on the boundary x = 0.
Following [24], the main result of this paper is the following theorem.
2. The case f (u) = u 2 , κ ∈ R, γ > 0. In this section, we prove Theorem 1.2 in the case Therefore, we consider (22), or (23), with f (u), κ, γ as in (30) Our existence argument is based on passing to the limit in a vanishing viscosity approximation of (23).
Following [13], fix a small number ε > 0 and let u ε = u ε (t, x) be the unique classical solution of the following mixed problem where u ε,0 and g ε are C ∞ (0, ∞) approximations of u 0 and g such that u 0,ε → u 0 , a.e. and in L p (0, ∞), 1 ≤ p < ∞, g ε → g, a.e. and in L p and C 0 is a constant independent on ε.
Let us prove some a priori estimates on u ε denoting with C 0 the constants which depend only on the initial data, and C(T ) the constants which depend also on T .
Arguing as in [ Lemma 2.2. For each t ≥ 0, (33) holds. In particular we get Moreover, for every 0 ≤ t ≤ T .
3. Let us consider the following function We have that Proof. We begin by observing that, integrating on (0, x) the second equation of (31), we get Differentiating (39) with respect to t, we have It follows from (33) and (40) that Integrating on (0, x) the first equation of (31), thanks to (37) and (40), we have It follows from the regularity of u ε that In particular, we have for every 0 ≤ t ≤ T .
Proof. Due to (31) and (64), Since the map solves the equation the comparison principle for parabolic equations implies that In a similar way we can prove that Therefore, which gives (66).
We construct a solution by passing to the limit in a sequence {u ε } ε>0 of viscosity approximations (31). We use the compensated compactness method [40].
In particular, we have P ε k → P a.e. and in L p where and (28) holds.
Proof. Let η : R → R be any convex C 2 entropy function, and q : R → R be the corresponding entropy flux defined by q (u) = −κuη (u). By multiplying the first equation in (31) with η (u ε ) and using the chain rule, we get Therefore, Murat's lemma [28] implies that The L ∞ bound stated in Lemma 2.6, (70), and the Tartar's compensated compactness method [40] give the existence of a subsequence {u ε k } k∈N and a limit function u ∈ L ∞ ((0, T ) × (0, ∞)), T > 0, such that (67) holds.
Our existence argument is based on passing to the limit in a vanishing viscosity approximation of (23).
Following [13], defined γ = −a 2 , fix a small number 0 < ε < 1 and let u ε = u ε (t, x) be the unique classical solution of the following mixed problem where u ε,0 and g ε are C ∞ (0, ∞) approximations of u 0 and g such that and C 0 is a constant independent on ε. Clearly, (71) is equivalent to the integrodifferential problem Let us prove some a priori estimates on u ε denoting with C 0 the constants which depend only on the initial data, and C(T ) the constants which depends also on T . Proof. We begin by observing that ∂ t u ε (t, 0) = g ε (t), being u ε (t, 0) = g ε (t). It follows from (73) that Differentiating (73) with respect to x, we have From (74) and being u ε a smooth solution of (73), an integration over (0, ∞) gives Let us consider the following function Observe that, by (71), (72) and (76), Moreover, Lemma 3.2. For each t > 0, we have that Proof. We begin by observing that (79) and (80) Observe that, by (33), (71) and (73), from (33) and (71), we have that due to (72) and the Young inequality, for every 0 ≤ t ≤ T . In particular, we have that Proof. We begin by observing that from the first equation of (71), (76) and (78), we have Multiplying (91) by 2v ε , (72), (81) and an integration on (0, ∞) give it follows from (87), (93) and the Young inequality that In particular, we obtain that Proof. Let 0 ≤ t ≤ T . Arguing as in Lemma 2.6, we have (94). (94) and the Young inequality give (95).
Lemma 3.5. Fix T > 0. There exists a constant C(T ) > 0, independent on ε, such that In particular, we have that for every 0 ≤ t ≤ T .
Lemma 3.6. Consider the function defined in (37). We have that In particular, we get for every 0 ≤ t ≤ T .
Following [13], fix a small number ε > 0 and let u ε = u ε (t, x) be the unique classical solution of the following mixed problem where u ε,0 and g ε are C ∞ (0, ∞) approximations of u 0 and g such that u 0,ε → u 0 , a.e. and in L p (0, ∞), 1 ≤ p < ∞, g ε → g, a.e. and in L p and C 0 is a constant independent on ε.
Let us prove some a priori estimates on u ε denoting with C 0 the constants which depend only on the initial data, and C(T ) the constants which depend also on T .
Arguing as in Section 2, we have the following result.
Arguing as in Lemma 2.3, we have the following result.

Lemma 4.3.
Consider the function defined in (37). For each t > 0, we have that for every 0 ≤ t ≤ T .
Proof of Theorem 1.2. Arguing as in [9, Theorem 1.1], the proof is concluded.
Observe that (23) with f (u), κ, γ defined in (9) is equivalent to the following one (see [16]): Unlike Sections 2, 3, 4, our existence argument is based on passing to the limit in a vanishing viscosity approximation of (135). Following [16], defined κ = −b 2 , γ = a 2 , fixed 0 < ε < 1, let u ε = u ε (t, x) be the unique classical solution of the following mixed problem where u ε,0 and g ε are C ∞ (0, ∞) approximations of u 0 and g such that and C 0 is a constant independent on ε.
Let us prove some a priori estimates on u ε denoting with C 0 the constants which depend only on the initial data, and C(T ) the constants which depend also on T .
for every 0 ≤ t ≤ T .
We construct a solution by passing to the limit in a sequence {u ε } ε>0 of viscosity approximations (136). We use the compensated compactness method [40].
Proof of Theorem 1.2. Lemma 5.7 give the existence of entropy solution of (135).
Arguing as in [9, Theorem 1.2], the proof is concluded.