On a model for internal waves in rotating fluids

Forγ, δ > 0, βsatisfyingβ < 0 or 0 < β < 4γ^1/4δ^3/4and

ϕ(k)=γk−βk|k|+δk3,k≠0,ϕ(0)=0,$$\begin{array}{} \displaystyle \phi(k) = \frac{\gamma}{k}-\beta k|k|+\delta k^{3}, k\neq 0, \quad \phi(0) = 0, \end{array}$$(12)

then |ϕ^″(k)| ≥ −2β+8γ^1/4δ^3/4, k ≠ 0.

Let k₀ = (γ/δ)^1/4. One can check that if k < 0 then

ϕ′′(k)≥ϕ′′(−k0)=−2β+8γ1/4δ3/4,$$\begin{array}{} \displaystyle \phi^{\prime\prime}(k)\geq \phi^{\prime\prime}(-k_{0}) = -2\beta+8\gamma^{1/4}\delta^{3/4}, \end{array}$$

while if k > 0 then

ϕ′′(k)≤ϕ′′(k0)=2β−8γ1/4δ3/4.$$\begin{array}{} \displaystyle \phi^{\prime\prime}(k)\leq \phi^{\prime\prime}(k_{0}) = 2\beta-8\gamma^{1/4}\delta^{3/4}. \end{array}$$

These two inequalities, under the hypotheses on β, prove the result.□

Following similar arguments to those of [20, 21, 32] for the case of the Ostrovsky (3) and the RMBO equations, (3) and (4) respectively, we have the following result on well-posedness of (9).

Under the hypotheses of Lemma 1, let f ∈ L²and t > 0. Then

||S(t)f||LtqLxp≤C||f||L2,$$\begin{array}{} \displaystyle ||S(t)f||_{L_{t}^{q}L_{x}^{p}}\leq C||f||_{L^{2}}, \end{array}$$(13)

for someC > 0 and whereq=4θ,p=21−θ,$\begin{array}{} \displaystyle q = \frac{4}{\theta}, p = \frac{2}{1-\theta}, \end{array}$θ ∈ [0, 1].

Recall that a change of variable allows to assume α = 0 in (9). Note that under the hypotheses on β, γ, δ of Lemma 1, the function ϕ in (12) satisfies the conditions (2.1a)-(2.1e) described in [14] and applying Theorem 2.1 of this reference, we have

||Wθ/2(t)f||Ltq(R,Lp)≤C||f||L2,$$\begin{array}{} \displaystyle ||W_{\theta/2}(t)f||_{L_{t}^{q}(\mathbb{R}, L^{p})}\leq C||f||_{L^{2}}, \end{array}$$

for some constant C and where

Ws(t)f(x):=∫−Ωei(kx+ϕ(k)t)|ϕ′′(k)|s/2f^(k)dk,s≥0,$$\begin{array}{} \displaystyle W_{s}(t)f(x): = \int_{-\Omega} e^{i(kx+\phi(k)t)}|\phi^{\prime\prime}(k)|^{s/2}\widehat{f}(k)dk, \quad s\geq 0, \end{array}$$

with Ω = (−∞, 0) ∪ (0, ∞). Now, using Lemma 1, observe that (cf. [20, 21])

||S(t)f||LtqLxp≤c02π||Wθ/2(t)f||Ltq(R,Lp),c0=1−2β+8γ1/4δ3/4,$$\begin{array}{} \displaystyle ||S(t)f||_{L_{t}^{q}L_{x}^{p}}\leq \frac{c_{0}}{2\pi}||W_{\theta/2}(t)f||_{L_{t}^{q}(\mathbb{R}, L^{p})}, \quad c_{0} = \frac{1}{-2\beta+8\gamma^{1/4}\delta^{3/4}}, \end{array}$$

and (13) follows.□

It may be worth emphasizing the particular cases of (13) corresponding to the limiting valuesθ = 0, 1:

||S(t)f||Lt∞Lx2≤C||f||L2,||S(t)f||Lt2Lx∞≤C||f||L2.$$\begin{array}{} \displaystyle ||S(t)f||_{L_{t}^{\infty}L_{x}^{2}}\leq C||f||_{L^{2}}, \quad ||S(t)f||_{L_{t}^{2}L_{x}^{\infty}}\leq C||f||_{L^{2}}. \end{array}$$

Assume thatζis a smooth solution of (8) such thatζ, ζ_x, ζ_xx, ζ_xxx, ζ_xxxx → 0 asx → ±∞. Thenζsatisfies the zero mass condition

I(ζ)=∫−∞∞ζ(x,t)dx=0,t≥0$$\begin{array}{} \displaystyle I(\zeta) = \int_{-\infty}^{\infty}\zeta(x, t)dx = 0, \quad t\geq 0 \end{array}$$(14)

and the time preservation of the momentum and energy

V(ζ)=∫−∞∞ζ(x,t)22dx,E(ζ)=∫−∞∞αζ(x,t)22+F(ζ(x,t))−β2ζ(x,t)Hζx(x,t)+δ2(ζx(x,t))2+γ2((∂x−1ζ)(x,t))2dx,$$\begin{array}{} \displaystyle V(\zeta) = \int_{-\infty}^{\infty}\frac{\zeta(x, t)^{2}}{2}dx, \nonumber\\ \displaystyle E(\zeta) = \int_{-\infty}^{\infty}\left(\alpha\frac{\zeta(x, t)^{2}}{2}+F(\zeta(x, t))-\frac{\beta}{2}\zeta(x, t)\mathscr{H}\zeta_{x}(x, t)+\frac{\delta}{2}(\zeta_{x}(x, t))^{2}+\frac{\gamma}{2}((\partial_{x}^{-1}\zeta)(x, t))^{2}\right)dx, \end{array}$$(15)

whereF^′ = f, F(0) = 0. The functional (15) is the Hamiltonian of (8) with respect to the symplectic structure given by J = −∂/∂x.

in the caseβ < 0, the method (22), (23) can also compute multi-pulses. One of these is shown in Figure 6.

Fig. 6

These computations motivate to study the existence of solitary wave solutions of (8) theoretically. This is developed in the following subsection.

Note that ifψ ∈ Hachieves the minimum (26) for some λ > 0, then there is a Lagrange multiplierμ ∈ ℝ such that I(ψ) = μK(ψ). This means that

−βHψx−(cs−α)ψ−δψxx−γ∂x−2ψ=−μ(p+1)f(ψ).$$\begin{array}{} \displaystyle -\beta \mathscr{H}\psi_{x}-(c_{s}-\alpha)\psi-\delta\psi_{xx}-\gamma\partial_{x}^{-2}\psi = -\mu (p+1)f(\psi). \end{array}$$

Ifμ^′ = (p+1)μ thenφ=(μ′)1p−1ψ$\begin{array}{} \displaystyle \varphi = (\mu^{\prime})^{\frac{1}{p-1}}\psi \end{array}$satisfies (16).

Note that if we integrate (7), with respect toλ, between 0 and t we have

1sF(ts)=tp+1p+1F′(s),$$\begin{array}{} \displaystyle \frac{1}{s}F(t s) = \frac{t^{p+1}}{p+1}F^{\prime}(s), \end{array}$$

which, evaluated at t = 1, implies that

(p+1)F(s)=sF′(s),$$\begin{array}{} \displaystyle (p+1)F(s) = sF^{\prime}(s), \end{array}$$(27)

and, therefore, Fis homogeneous of degree p+1. Some consequences of this are:

The functionalKin (25) is homogeneous of degree p+1. (Note thatIin (24) is homogeneous of degree two.)

They will be used elsewhere.

We denote by G = G(α, β, γ, δ, c_s) the set of solutions of (16). From the homogeneity of I and K, u ∈ G also achieves the minimum

m=m(α,β,γ,δ,cs)=inf{I(u)K(u)2p+1;u∈H,K(u)>0},$$\begin{array}{} \displaystyle m = m(\alpha, \beta, \gamma, \delta, c_{s}) = \inf\{\frac{I(u)}{K(u)^{\frac{2}{p+1}}}; u\in H, K(u) \gt 0\}, \end{array}$$

and thereforeMλ=λ2p+1m.$\begin{array}{} \displaystyle M_{\lambda} = \lambda^{\frac{2}{p+1}}m. \end{array}$If we multiply (8) by ϕ, use (27) and integrate, we have I(φ) = K(φ), in such a way that

G=φ∈H/I(φ)=K(φ)=mp+1p−1$$\begin{array}{} \displaystyle G = \left\{\varphi\in H / I(\varphi) = K(\varphi) = m^{\frac{p+1}{p-1}}\right\} \end{array} $$

Throughout the rest of the paper the following estimates will be used:

||Dx1/2u||2≤ε2||u||2+14ε2||∂xu||2,$$\begin{array}{} \displaystyle ||D_{x}^{1/2}u||^{2}\leq \varepsilon^{2}||u||^{2}+\frac{1}{4\varepsilon^{2}}||\partial_{x}u||^{2}, \end{array}$$(29)

for anyε > 0.

||u||2≤C||∂x−1u||1/2||∂xu||1/2,||∂x−1u||∞≤C||∂x−1u||1/2||u||1/2,$$\begin{array}{} \displaystyle \quad\,\,\,\,\,||u||^{2}\leq C||\partial_{x}^{-1}u||^{1/2}||\partial_{x}u||^{1/2}, \\ \displaystyle ||\partial_{x}^{-1}u||_{\infty}\leq C||\partial_{x}^{-1}u||^{1/2}||u||^{1/2}, \end{array}$$(30)

for some constant C, see e. g. [23].

Assume thatδ, γ > 0 and that one the following conditions holds:

(i)

β < 0, c_s−α < 0.

Then M_λ > 0 forλ > 0.

The proof of Lemma 10 is based on the following estimates of I(u):

In the case of (i):

I(u)≥(−β)∫|Dx1/2u|2dx+γ∫|∂x−1u|2dx+δ∫|∂xu|2dx.$$\begin{array}{} \displaystyle I(u)\geq (-\beta)\int |D_{x}^{1/2}u|^{2}dx+\gamma\int |\partial_{x}^{-1}u|^{2}dx+\delta\int |\partial_{x}u|^{2}dx. \end{array}$$(31)

The proof of (31)-(34) is as follows.

The proof of (31) is trivial since c_s-α < 0.

Once (31)-(34) is proved, we can use (28), (29) and the estimate (14) of [8] to have

λ=K(u)≤C||u||2+||∂xu||2+||∂x−1u||2p+12,$$\begin{array}{} \displaystyle \lambda = K(u)\leq C\left(||u||^{2}+||\partial_{x}u||^{2}+||\partial_{x}^{-1}u||^{2}\right)^{\frac{p+1}{2}}, \end{array}$$(37)

for some constant C. From (37) and (30) we have

λ=K(u)≤C||∂xu||2+||∂x−1u||2p+12,$$\begin{array}{} \displaystyle \lambda = K(u)\leq C\left(||\partial_{x}u||^{2}+||\partial_{x}^{-1}u||^{2}\right)^{\frac{p+1}{2}}, \end{array}$$(38)

for some constant C. Now, using (29) and (30) if necessary, we have that in all the cases (i) to (iv) the right hand side of (38) can be bounded by the right hand side of the corresponding estimate (31) to (34), in such a way that there exists C = C(c_s, p, α, β, γ, δ, ε²) > 0 such that

λ=K(u)≤(I(u))p+12,$$\begin{array}{} \displaystyle \lambda = K(u)\leq (I(u))^{\frac{p+1}{2}}, \end{array}$$

which implies

I(u)≥λC2p+1,$$\begin{array}{} \displaystyle I(u)\geq \left(\frac{\lambda}{C}\right)^{\frac{2}{p+1}}, \end{array}$$

for any u ∈ H. Therefore

Mλ≥λC2p+1>0.$$\begin{array}{} \displaystyle M_{\lambda}\geq \left(\frac{\lambda}{C}\right)^{\frac{2}{p+1}} \gt 0. \end{array}$$

□

Two additional properties will be used to prove the existence result:

From (30) (see also formula (17) in [8], along with (29) of the present paper) we obtain the coercivity of I(u), which is then equivalent to ||u||H2$\begin{array}{} \displaystyle ||u||_{H}^{2} \end{array}$ in all the cases (i) to (iv).

The main result of existence is the following.

Under any of the conditions (i) to (iv) of Lemma 10, letλ > 0 and {u_n}_nbe a minimizing sequence inHforλ. Then there exist subsequences {u_n}_ninH, {y_n}_nin ℝ and u ∈ H such thatu_n(⋅ + y_n) → ustrongly inH. Furthermore, the functionuachieves the minimumI(u) = M_λsubject toK(u) = λ.

The proof is similar to that of other references, see in particular [8, 17, 23].

From coercivity of I, the sequence {u_n} is bounded in H and therefore if we consider the L¹ sequence

ρn=|Dx1/2un|2+|∂x−1un|2+|∂xun|2.$$\begin{array}{} \displaystyle \rho_{n} = |D_{x}^{1/2}u_{n}|^{2}+|\partial_{x}^{-1}u_{n}|^{2}+|\partial_{x}u_{n}|^{2}. \end{array}$$

then ρ_n is bounded in L¹. Thus, there is a subsequence {ρ_n} with

L=limn→∞||ρn||L1,$$\begin{array}{} \displaystyle L = \lim_{n\rightarrow\infty}||\rho_{n}||_{L^{1}}, \end{array}$$

and normalizing (by taking ρ̃_n(x) = Lρ_n(∥ρ_n∥_L¹x)) we may assume ∥ρ_n∥_L¹ = L for all n.

If we apply the Concentration-Compactness Lemma, [22], to ρ_n we have three possibilities:

(a)

Compactness: there exist y_k ∈ ℝ such that for any ε > 0 there is R(ε) > 0 such that for all k

∫|x−yk|≤R(ε)ρkdx≥∫−∞∞ρkdx−ε=L−ε.$$\begin{array}{} \displaystyle \int_{|x-y_{k}|\leq R(\varepsilon)}\rho_{k}dx\geq \int_{-\infty}^{\infty}\rho_{k}dx-\varepsilon = L-\varepsilon. \end{array}$$

The next step is ruling out possibilities (b) and (c). Here the arguments are similar to those of, for example, [17]. Assume that (b) holds. Using (37) and the homogeneity of F, we have

∫|x−y|≤1F(un)dx≤C∫|x−y|≤1|un|p+1dx≤C∫|x−y|≤1ρndxp+12,$$\begin{array}{} \displaystyle \int_{|x-y|\leq 1}F(u_{n})dx\leq C\int_{|x-y|\leq 1}|u_{n}|^{p+1}dx \leq C\left(\int_{|x-y|\leq 1}\rho_{n}dx\right)^{\frac{p+1}{2}}, \end{array}$$

for all y ∈ ℝ and some constant C. By (b) one can choose n(ε) so large that

∫|x−y|≤1F(un)dx≤Cεp−12∫|x−y|≤1ρndx,$$\begin{array}{} \displaystyle \int_{|x-y|\leq 1}F(u_{n})dx \leq C\varepsilon^{\frac{p-1}{2}}\int_{|x-y|\leq 1}\rho_{n}dx, \end{array}$$

for n ≥ n(ε). Summing over intervals centered at even integers y = 2k we have

K(un)≤Cεp−12,n≥n(ε),$$\begin{array}{} \displaystyle K(u_{n})\leq C\varepsilon^{\frac{p-1}{2}}, \quad n\geq n(\varepsilon), \end{array}$$

that is K(u_n) → 0 as n → ∞, which is in contradiction with the assumption that u_n is a minimizing sequence.

Assume now that (c) holds. Define cutoff functions ξ₁, ξ₂ with support on |x| ≤ 2 and |x| ≥ 1/2 respectively and with ξ₁(x) = 1, |x| ≤ 1, ξ₂(x) = 1, |x| ≥ 1. Let us consider

uk,1(x)=ξ1|x−yk|/Ruk(x),uk,2(x)=ξ2|x−yk|/Rkuk(x).$$\begin{array}{} \displaystyle u_{k, 1}(x) = \xi_{1}\left(|x-y_{k}|/R\right)u_{k}(x), \\ \displaystyle u_{k, 2}(x) = \xi_{2}\left(|x-y_{k}|/R_{k}\right)u_{k}(x). \end{array}$$

Then u_k,j, j = 1, 2 satisfy, for k ≥ k₀

I(uk)=I(uk,1)+I(uk,2)+O(ε<!-- ? -->),K(uk)=K(uk,1)+K(uk,2)+O(ε<!-- ? -->).$$\begin{array}{} \displaystyle\,\, I(u_{k}) = I(u_{k, 1})+I(u_{k, 2})+O(\varepsilon), \\ \displaystyle K(u_{k}) = K(u_{k, 1})+K(u_{k, 2})+O(\varepsilon). \end{array}$$

Since u_k is bounded in H, then u_k,1, u_k,2 are bounded in H independently of ε. Therefore K(u_k,1), K(u_k,2) are bounded and then there are subsequences such that

λi(ε)=limk→∞K(uk,i),i=1,2,$$\begin{array}{} \displaystyle \lambda_{i}(\varepsilon) = \lim_{k\rightarrow\infty}K(u_{k, i}), \quad i = 1, 2, \end{array}$$

where λ_i(ε), i = 1, 2 are bounded independently of ε. So there is a sequence ε_j → 0 such that λ_i(ε_j) → λ_i for some λ_i with λ₁+λ₂ = λ. This leads to three possibilities:

λ₁ ∈ (0, λ). Then, by (39)

I(uk)=I(uk,1)+I(uk,2)+O(εj)≥MK(uk,1)+MK(uk,2)+O(εj)=K(uk,1)2p+1+K(uk,2)2p+1M1+O(εj)$$\begin{array}{} \displaystyle I(u_{k}) = I(u_{k, 1})+I(u_{k, 2})+O(\varepsilon_{j}) \geq M_{K(u_{k, 1})}+M_{K(u_{k, 2})}+O(\varepsilon_{j}) = \left(K(u_{k, 1})^{\frac{2}{p+1}}+K(u_{k, 2})^{\frac{2}{p+1}}\right)M_{1}+O(\varepsilon_{j}) \end{array}$$

Taking k → ∞, using that u_k is a minimizing sequence and (39), we have

λ2p+1M1=Mλ≥(λ1(εj))2p+1+(λ2(εj))2p+1M1+O(εj).$$\begin{array}{} \displaystyle \lambda^{\frac{2}{p+1}}M_{1} = M_{\lambda}\geq \left((\lambda_{1}(\varepsilon_{j}))^{\frac{2}{p+1}}+(\lambda_{2}(\varepsilon_{j}))^{\frac{2}{p+1}}\right)M_{1}+O(\varepsilon_{j}). \end{array}$$

And, finally, if j → ∞ then

M1≥λ1λ2p+1+λ2λ2p+1M1>M1,$$\begin{array}{} \displaystyle M_{1}\geq \left(\left(\frac{\lambda_{1}}{\lambda}\right)^{\frac{2}{p+1}}+\left(\frac{\lambda_{2}}{\lambda}\right)^{\frac{2}{p+1}}\right)M_{1} \gt M_{1}, \end{array}$$

which is a contradiction.

So (c) is also ruled out and therefore compactness (a) holds. Now we prove that (a) implies the existence of a minimizer. (Again the arguments are similar to, e. g. [8, 17].) Since u_k is bounded in H, there is a subsequence u_j and u ∈ H such that φ_j = u_j(⋅ +y_j) converges weakly to u in H. Note also that, by Sobolev embedding, u_n is bounded in W^1,q(ℝ) for all q > 2, so the convergence is strong in Wloc1,p+1$\begin{array}{} \displaystyle W^{1, p+1}_{loc} \end{array}$(ℝ), p > 1. Furthermore, by weak lower semicontinuity of I in H, we have

I(u)≤limj→∞I(φj)=Mλ.$$\begin{array}{} \displaystyle I(u)\leq \lim_{j\rightarrow\infty}I(\varphi_{j}) = M_{\lambda}. \end{array}$$(40)

Now we prove that φ_j converges strongly to u in L^p+1. We take σ_j = |φ_j|^p+1. From (39) and compactness (a) of the ρ_k, σ_j also satisfies (a). We take ε > 0 and R₀ > 0 so large that

∫|x|≥R0|u|p+1dx<ε.$$\begin{array}{} \displaystyle \int_{|x|\geq R_{0}}|u|^{p+1}dx \lt \varepsilon. \end{array}$$

By compactness of σ_j, there is j₁(ε) and R(ε) > R₀ such that for j > j₁(ε)

∫|x|≥R(ε)σjdx<ε,$$\begin{array}{} \displaystyle \int_{|x|\geq R(\varepsilon)}\sigma_{j}dx \lt \varepsilon, \end{array}$$(41)

∫|x|≥R(ε)|u|p+1dx≤∫|x|≥R0|u|p+1dx<ε.$$\begin{array}{} \displaystyle \int_{|x|\geq R(\varepsilon)}|u|^{p+1}dx\leq \int_{|x|\geq R_{0}}|u|^{p+1}dx \lt \varepsilon. \end{array}$$(42)

Therefore, if B_ε = B(0, R(ε)), by (41), (42)

∫R∖Bε|φj−u|p+1dx<2p+1ε,$$\begin{array}{} \displaystyle \int_{\mathbb{R}\backslash B_{\varepsilon}}|\varphi_{j}-u|^{p+1}dx \lt 2^{p+1}\varepsilon, \end{array}$$

for j > j₁(ε). On the other hand, the strong convergence in Wloc1,p+1$\begin{array}{} \displaystyle W^{1, p+1}_{loc} \end{array}$(ℝ) implies the existence of j₂(ε) such that

∫Bε|φj−u|p+1dx<ε,$$\begin{array}{} \displaystyle \int_{ B_{\varepsilon}}|\varphi_{j}-u|^{p+1}dx \lt \varepsilon, \end{array}$$

for j > j₂(ε). Finally, if j > max{j₁, j₂} then

∫R|φj−u|p+1dx≤∫Bε|φj−u|p+1dx+∫R∖Bε|φj−u|p+1dx≤(1+2p+1)ε,$$\begin{array}{} \displaystyle \int_{\mathbb{R}}|\varphi_{j}-u|^{p+1}dx\leq \int_{ B_{\varepsilon}}|\varphi_{j}-u|^{p+1}dx+ \int_{\mathbb{R}\backslash B_{\varepsilon}}|\varphi_{j}-u|^{p+1}dx \leq (1+2^{p+1})\varepsilon, \end{array}$$

which implies that φ_j converges strongly to u in L^p+1. Note that since K is locally Lipschitz on L^p+1, [17], the strong convergence implies

K(u)=limj→∞K(φj)=λ.$$\begin{array}{} \displaystyle K(u) = \lim_{j\rightarrow\infty}K(\varphi_{j}) = \lambda. \end{array}$$

Therefore I(u) ≥ M_λ which, along with (40), implies I(u) = M_λ, and u is a minimizer of I subject to K(⋅) = λ. Finally, since I is equivalent to ||⋅||H2,φj$\begin{array}{} \displaystyle ||\cdot ||_{H}^{2}, ~\varphi_{j} \end{array}$ converges weakly to u in H and I(φ_j) → I(u) = M_λ, then φ_j converges strongly to u in H.□

We observe that the conditions in Lemma 10 (or in Theorem 11) seem to be in agreement with those of the limiting caseβ = 0 (Ostrovsky equation) for the existence of solitary waves. More specifically, let us assumeα = 0. Ifβ → 0−, then condition (i) in Lemma 10 implies c_s < c^∗ = 2γδ$\begin{array}{} \displaystyle 2\sqrt{\gamma\delta} \end{array}$, while ifβ → 0+ thenz₊ → c^∗and condition (iv) also reads c_s < c^∗. This coincides with Theorem 2.1 of [18] for the generalized Ostrovsky equation.

A second observation is that the conditions in Lemma 10 seem to be also in agreement with the arguments exposed n [9, 24] to justify the possibility of soliton solutions in the Ostrovsky equation, withα, γ, δ > 0. Linearizing (8) and seeking for plane wave solutions, the corresponding dispersion relation for waves of small amplitude is

ω/k=α+γk2+δk2−β|k|.$$\begin{array}{} \displaystyle \omega/k = \alpha+\frac{\gamma}{k^{2}}+\delta k^{2}-\beta |k|. \end{array}$$(43)

The phase velocity (43) is displayed in Figure 7 forβ < 0. in this case, linear perturbations can only exist within a semibounded range of phase velocities, giving the chance of having solitary waves which are not in resonance with linear perturbations and are not subject to radiative decay.