Gevrey regularity for the generalized Kadomtsev-Petviashvili I (gKP-I) equation

1.   Introduction and position of problem

The study of nonlinear wave processes in real media with dispersion, despite the significant progress in this area in recent years, for example, ^[1,2,3] and numerous references in these works are still relevant. This, in particular, concerns the dynamics of oscillations in cases where high-energy particle fluxes occur in the medium, which significantly change such parameters of propagating wave structures, such as their phase velocity, amplitude and characteristic length. In recent and earlier years, a fairly large number of works have been devoted to studies of this kind of relativistic effects (see ^[12,13,14]).

Recently, the great interest on the KP equation has led to the construction and the study of many extensions to the KP equation. These new extended models propelled greatly the research that directly resulted in many promising findings and gave an insight into some novel physical features of scientific and engineering applications. Moreover, lump solutions, and interaction solutions between lump waves and solitons, have attracted a great amount of attentions aiming to make more progress in solitary waves theory. Lump solutions, have been widely studied by researchers for their significant features in physics and many other nonlinear fields^[18,19,20].

Let $u = u(x, y, t), (x, y, t) \in \mathbb{R}^{3}$ and $\alpha \geq 4$. We consider the initial value problem for the generalized Kadomtsev-Petviashvili I equation,

with

This equation belongs to the class of Kadomtsev-Petviashvili equations, which are models for the propagation of long dispersive nonlinear waves which are essentially unidirectional and have weak transverse effects. Due to the asymmetric nature of the equation with respect to the spatial derivatives, it is natural to consider the Cauchy problem for (1.1) with initial data in the anisotropic Sobolev spaces $H^{s_1, s_2} (\mathbb{R}^{2})$, defined by the norm

Many authors have investigated the Cauchy problem for Kadomtsev-Petviashvili equations as in, for instance ^[4,8,16]. Yan et al. ^[17] established the local well-posedness of the Cauchy problem for the Kadomtsev-Petviashvili I equation in anisotropic Sobolev spaces $H^{s_{1}, s_{2}}(\mathbb{R})$ with $s_{1} > -\frac{\alpha-1}{4}$, $s_{2} \geq 0$ with $\alpha\geq4$ and globally well-posed in $H^{s_{1}, 0}(\mathbb{R})$ with $s_{1} > -\frac{(\alpha-1)(3\alpha-4)}{4(5\alpha+3)}$ if $4\leq\alpha\leq5$ also proved that the Cauchy problem is globally well-posed in $H^{s_{1}, 0}(\mathbb{R})$ with $s_{1} > -\frac{\alpha(3\alpha-4)}{4(5\alpha+4)}$ if $\alpha > 5$. The authors in ^[8] proposed the problem

and proved that it is globally well-posed for given data in an anisotropic Gevrey space $G^{\sigma_{1}, \sigma_{2}}(\mathbb{R}^{2}), \sigma_{1}, \sigma_{2}\geq0$, with respect to the norm

With initial data in anisotropic Gevrey space

and $\kappa\geqslant1$, we will consider the problem (1.1). The spaces $G_{s_{1}, s_{2}}^{\sigma_1, \sigma_2, \kappa}$ can be defined as the completion of the Schwartz functions with respect to the norm

where

In addition to the holomorphic extension property, Gevrey spaces satisfy the embeddings $G_{s_{1}}^{\sigma_1, \sigma_2} \hookrightarrow G_{s_{1}'}^{\sigma'_1, \sigma_2'}$ for $s_{1}, s_{1}'\in \mathbb{R}$ and $\sigma_{i}' < \sigma_{i}$ where $G_{s_{1}, 0}^{\sigma_1, \sigma_2, 1} = G_{s_{1}}^{\sigma_1, \sigma_2}$, which follow from the corresponding estimates

The main aim to consider initial data in these spaces is because of the Paley-Wiener Theorem.

Proposition 1.1. Let $\sigma_{1} > 0$, $s\in \mathbb{R}$. Then $f \in G_{x}^{\sigma_{1}}(\mathbb{R})$ if and only if it is the restriction to the to the real line of a function $F$ which is holomorphic in the strip $\lbrace x+iy \in \mathbb{C}:\ | y | < \sigma \rbrace$, and satisfies

Notation

We will also need the full space time Fourier transform denoted by

In both cases, we will denote the corresponding inverse transform of a function

To simplify the notation, we introduce some operators. We first introduce the operator $A_{\kappa}^{\sigma_1, \sigma_2}$, which we define as

Then, we may then define another useful operator

For $x \in \mathbb{R}^{n}$, we denote $\langle x \rangle = (1 + |x|^{2})^{1/2}$. Finally, we write $a \lesssim b$ if there exists a constant $C > 0$ such that $a \leq Cb$, and $a \sim b$ if $a \lesssim b \lesssim a$. If the constant $C$ depends on some quantity $q$, we denote this by $a \lesssim_{q} b$.

Function spaces

Since our proofs rely heavily on the theory developed by Yan et al., let us state the function spaces they used explicitly, so that we can state their useful properties which we will exploit in our modifications of their spaces. The main function spaces they used are the so-called anisotropic Bourgain spaces, adapted to the generalized Kadomtsev-Petviashvili I, whose norm is given by

where

with $m(\xi, \eta) = \xi|\xi|^{\alpha}+\frac{\eta^{2}}{\xi}$.

Furthermore, we will also need a hybrid of the anlytic Gevrey and anisotropic Bourgain spaces, designated $X_{s_{1}, s_{2}, b}^{\sigma_1, \sigma_2, \kappa}(\mathbb{R}^{3})$ and defined by the standard

It is well-known that these spaces satisfy the embedding

Thus, solutions constructed in $X_{s_{1}, s_{2}, b}^{\sigma_1, \sigma_2, \kappa}$ belong to the natural solution space.

When considering local solutions, it is useful to consider localized versions of these spaces. For a time interval $I$ and a Banach space $\mathcal{Y}$, we define the localized space $\mathcal{Y}(I)$ by the norm

2.   Main results

The first result related to the short-term persistence of analyticity of solutions is given in the next Theorem.

Theorem 2.1. Let $s_{1} > -\frac{\alpha-1}{4}, s_{2} \geq 0$, $\alpha\geq 4$, $\sigma_1 \geq 0$, $\sigma_2 \geq 0$ and $\kappa\geq1$. Then for all initial data $f \in G_{s_{1}, s_{2}}^{\sigma_1, \sigma_2, \kappa}$ and $\vert \xi\vert^{-1}\widehat{f}(\xi, \mu)\in L^{2}(\mathbb{R}^{2})$, there exists $\delta = \delta(\parallel f\parallel_{G_{s_{1}, s_{2}}^{\sigma_1, \sigma_2, \kappa}}) > 0$ and a unique solution $u$ of $(1.1)$ on the time interval $[0, \delta]$ such that

Moreover the solution depends continuously on the data $f$. In particular, the time of existence can be chosen to satisfy

for some constants $c_{0} > 0$ and $\gamma > 1$. Moreover, the solution $u$ satisfies

The second main result concerns the evolution of the radius of analyticity for the $x$-direction is given in the next Theorem. Here

Theorem 2.2. Let $\sigma_1 > 0$, $s_{1} > -\frac{\alpha-1}{4}, \alpha = 4, 6, 8, ...$ and assume that $f \in G_{s_{1}}^{\sigma_1, 0}$, $\vert \xi\vert^{-1}\widehat{f}(\xi, \mu)\in L^{2}(\mathbb{R}^{2})$. Then the solution $u$ given by Theorem 2.1 extends globally in time, and for any $T > 0$, we have

with $\rho = \frac{4}{\alpha-1}+\varepsilon$ for $\varepsilon > 0$ when $\alpha = 4$ and $\rho = 1$ when $\alpha = 6, 8, 10, ...$ and the constant $C$ is a positive.

The method used here for proving lower bounds on the radius of analyticity was introduced in ^[15] in the study of the non-periodic KdV equation. It was applied to the the higher order nonlinear dispersive equation in ^[9] and the system of mKdV equation in ^[10].

Our last aim is to show the regularity of the solution in the time. A non-periodic function $\phi(x)$ is the Gevrey class of order $\kappa$ i.e, $\phi(x)\in G^{\kappa}$, if there exists a constant $C > 0$ such that

Here we will show that for $x, y\in \mathbb{R}$, for every $t\in [0, \delta]$ and $j, l, n \in \lbrace 0, 1, 2, \dots \rbrace$, there exist $C > 0$ such that,

i.e, $u(\cdot, \cdot, t)\in G^{\sigma}(\mathbb{R})\times G^{\sigma}(\mathbb{R})$ in $x, y$ and $u(x, y, \cdot)\in G^{(\alpha+1)\kappa}([0, \delta])$ in time variable.

Theorem 2.3. Let $s_{1} > -\frac{\alpha-1}{4}, s_{2} \geq 0$, $\alpha\geq 4$, $\sigma_1 \geq 0$, $\sigma_2 \geq 0$ and $\kappa\geq1$.

If $f \in G_{s_{1}, s_{2}}^{\sigma_1, \sigma_2, \kappa}$ then the solution

given by Theorem 2.1, belongs to the Gevrey class $G^{(\alpha+1)\kappa}$ in time variable.

Corollary 2.4. Let $\sigma_1 > 0, s_{1} > -\frac{\alpha-1}{4}, \alpha = 4, 6, 8, ....$ If $f \in G_{s_{1}}^{\sigma_1, 0}$ then the solution

given by Theorem 2.2, belongs to the Gevrey class $G^{(\alpha+1)}$ in time variable.

The rest of the paper is organized as follows: In section 3, we present all the auxiliary estimates that will be employed in the remaining sections. We prove Theorem 2.1 in subsection 4.1 using the standard contraction method and Theorem 2.2 in subsection 4.2. Finally, in section 5, we prove $G^{(\alpha+1)}$ regularity in time.

3.   Auxiliary estimates

To begin with, let us consider the related linear problem

By Duhamel's principle the solution can be written as

where

We localize it in $t$ by using a cut-off function satisfying $\psi\in C_{0}^{\infty}(\mathbb{R})$, with $\psi = 1$ in $[-1, 1]$ and supp$\psi \subset [-2, 2]$.

We consider the operator $\Phi$ given by

where $\psi_{\delta}(t) = \psi(\frac{t}{\delta})$. To this operator, we apply the following estimates.

Lemma 3.1. (Linear estimate) Let $s_{1}, s_{2}\in \mathbb{R}, -\frac{1}{2} < b'\leq 0 \leq b \leq b'+1, \sigma_1 \geq 0, \sigma_2 \geq 0, \kappa\geq1 \ and \ \delta\in (0, 1).$ Then

Proof. The proofs of (3.3) and (3.4) for $\sigma_1 = \sigma_2 = 0$ can be found in Lemma 2.1 of ^[17]. These inequalities clearly remain valid for $\sigma_1, \sigma_2 > 0$, as one merely has to replace $f$ by $A_{\kappa}^{\sigma_1, \sigma_2}f$, $F$ by $A_{\kappa}^{\sigma_1, \sigma_2}F$.

The final preliminary fact we must state is the following bilinear estimate, which is Lemma 3.1 of ^[17].

Lemma 3.2. (Bilinear estimate in Bourgain space.)

Let $s_{1}\geq-\frac{\alpha-1}{4}+4\alpha\epsilon, s_{2} \geq 0$, $\alpha\geq 4$, $b = \frac{1}{2}+\epsilon$ and $b' = -\frac{1}{2}+2\epsilon$. Then, we have

To this result, we apply the following Lemma, which is a corollary of Lemma 3.2.

Lemma 3.3. (Bilinear estimate in Gevrey-Bourgain space.)

Let $s_{1} > -\frac{\alpha-1}{4}+4\alpha\epsilon, s_{2} \geq 0, \alpha\geq 4, \sigma_1 \geq 0, \sigma_2 \geq 0, \kappa\geq1, b = \frac{1}{2}+\epsilon \ and \ b' = -\frac{1}{2}+2\epsilon.$ Then, we have

Proof. It is not hard to see that

By Lemma 3.2, we get

4.   Proof of main results regarding the existence

4.1.   Local well-posedness in an anisotropic Gevrey space

The above Lemmas will be used without somtimes mention to prove Theorem 2.1.

Lemma 4.1. Let $s_{1} > -\frac{\alpha-1}{4}+4\alpha\epsilon, s_{2} \geq 0, \alpha\geq 4, \sigma_1 \geq 0, \sigma_2 \geq 0, \kappa\geq1, b = \frac{1}{2}+\epsilon, b' = -\frac{1}{2}+2\epsilon \ and \ 0 < \delta < 1.$ Then

and

Proof. Combining Lemma 3.3 and Lemma 3.1 with the fixed point Theorem. We define

Then, we have

We choose $\delta$ such that

We have

Thus, $\Phi(u)$ maps $\mathbb{B}(0, 2C\| f \|_{G_{s_{1}, s_{2}}^{\sigma_1, \sigma_2, \kappa}})$ into $\mathbb{B}(0, 2C\| f \|_{G_{s_{1}, s_{2}}^{\sigma_1, \sigma_2, \kappa}})$ which is a contraction, since

Here we choose $\delta$ such that

We choose the time of existence where

For appropriate choice of $c_{0}$, this will satisfy inequalities (4.1) and (4.2).

From Lemma 4.1, we see that for initial data $f(x, y)\in G_{s_{1}, s_{2}}^{\sigma_1, \sigma_2, \kappa}(\mathbb{R}^{2})$ if the lifespan $\delta = c_{0}/(1 + \| f \|_{X_{s_{1}, s_{2}, b}^{\sigma_1, \sigma_2, \kappa}})^{\frac{1}{\epsilon}}$ then the map $\Phi(u)$ is a contraction on a small ball centered at the origin in $X_{s_{1}, s_{2}, b}^{\sigma_1, \sigma_2, \kappa}$. Hence, the map $\Phi(u)$ has a unique fixed point $u$ in a neighborhood of $0$ with respect to the norm $\|\cdot\|_{X_{s_{1}, s_{2}, b}^{\sigma_1, \sigma_2, \kappa}}$.

The rest of the proof follows the standard argument.

4.2.   Global well-posedness

In this section, we prove Theorem 2.2. The first step is to obtain estimates on the growth of the norm of the solutions. For this end, we need to prove the following approximate conservation law.

Theorem 4.2. Let $\sigma_{1} > 0$ and $\delta$ be as in Theorem 2.1, there exist $b\in (1/2, 1)$ and $C > 0$, such that for any solution $u\in X_{0, b}^{\sigma_1, 0}(I)$ to the Cauchy problem (1.1) on the time interval $I \subset [0, \delta]$, we have the estimate

with $\varrho\in [0, \frac{3}{4})$ if $\alpha = 4$, and $\varrho = 1$ if $\alpha = 6, 8, 10, \dots$.

Before we may show the proof, let us first state some preliminary Lemmas. The first one is an immediate consequence of Lemma 12 in ^[15].

Lemma 4.3. For $\sigma > 0$, $0\leq\theta\leq1$ and $\xi, \xi_{1} \in \mathbb{R}$, we have

This will be used to prove the following key estimate.

Lemma 4.4. Let $N^{\sigma_1, 0}_{1}(u)$ be as in Eq $(1.4)$ for $\sigma_1 \geq 0$ and $\sigma_2 = 0$. Then for $b$ as in Lemma 3.2, we have

with $\varrho\in [0, \frac{3}{4})$ if $\alpha = 4$ and $\varrho = 1$ if $\alpha = 6, 8, 10, ...$

Proof. We first observe that the inequality in Lemma 3.2, is equivalent to

where we denote $\phi(\tau, \xi, \eta) = \langle \tau + m(\xi, \eta)\rangle$. With this, we observe that the left side of the inequality in Lemma 4.4 can be estimated by Lemma 4.3 as

If we apply Lemma 3.2 with $s_1 = -\varrho$, $s_2 = 0$, it will follow, from the comments above, that

Proof of Theorem 4.2. Begin by applying the operator $A_{1}^{\sigma_1, 0}$ to Eq (1.1). If we let $U = A_{1}^{\sigma_1, 0} u$, then Eq (1.1) becomes

where $\alpha = 4, 6, 8, ..$ and $N^{\sigma_1, 0}_{1}(u)$ is defined in Lemma 4.4. Multiplying (4.4) by $U$ and integrating with respect to the spatial variables, we obtain

If we apply integration by parts, we may rewrite the left-hand side as

which can then be rewritten as

By noticing that $\partial_{x}^{j} U(x, t) \rightarrow 0$ as $|x| \rightarrow \infty$ (see ^[15]) we obtaining

Integrating with respect to time yields

Applying Cauchy-Schwarz and the definition of $U$, we obtain

We now apply Lemma 4.4 and the fact that $b = \frac{1}{2}+\epsilon$, we can further estimate this by

as desired.

Proof of Theorem 2.2. With the tools established in the previous subsection, we may begin the proof of Theorem 2.2. Let us first suppose that $T^{*}$ is the supremum of the set of times $T$ for which

If $T^{*} = \infty$, there is nothing to prove, so let us assume that $T^{*} < \infty$. In this case, it suffices to prove that

for all $T > T^{*}$. To show that this is the case, we will use Theorem 2.1 and Theorem 4.2 to construct a solution which exists over subintervals of width $\delta$, using the parameter $\sigma_{1}$ to control the growth of the norm of the solution. We first prove the case $s = 0$ and then we will generalize the case.

4.2.1.   The case $s = 0$

The desired result will follow from the following proposition.

Proposition 4.5. Let $T > 0$, $x = 0$, $0 < \sigma_{1}\leq \sigma_{0}$ and $\delta > 0$ be numbers such that $n \delta \leq T < (n + 1) \delta$. Then the solution $u$ to the Cauchy problem $(1.1)$ satisfies

and

if

for some constant $C > 0$.

Proof. For fixed $T \geq T^{*}$, we will prove, for sufficiently small $\sigma_{1} > 0$, that

We will use the Theorem 2.1 and Theorem 4.2 with the time step

The smallness conditions on $\sigma_{1}$ will be

where $C > 0$ is the constant in Theorem 4.2. Proceeding by induction, we will verify that

for $n\in \lbrace 1, \cdot\cdot\cdot, m+1\rbrace$, where $m\in \mathbb{N}$ is chosen, so that $T \in [m\delta, (m+1)\delta)$. This $m$ does exist, since by Theorem 2.1 and the definition of $T^{*}$, we have

We cover now, the interval $[0, \delta]$, and by Theorem 4.2, we have

where we used that

since $\sigma_{1}\leq \sigma_{0}$. This verifies (4.12) for $n = 1$ and now, (4.13) follows using again

as well as $C\sigma_{1}^{\varrho}\Vert f\Vert_{G_{0}^{\sigma_{0}, 0}}\leq 1.$ Next, assuming that (4.12) and (4.13) hold for some $n \in \lbrace 1, \cdot\cdot\cdot, m\rbrace$, we will prove that they hold for $n+1$. We estimate

verifying (4.12) with $n$ replaced by $n+1$. To get (4.13) with $n$ replaced by $n+1$, it is then enough to have

But this holds by (4.11), since

Finally, the condition (4.11) is satisfied for $\sigma_{1} \in (0, \sigma_{0})$ such that

Thus, $\sigma_{1} = C_{1}T^{-\frac{1}{\varrho}}$, where

4.2.2.   The general case

For general $s$, we have

The case $s = 0$ already being proved, we know that there is a $T_{1} > 0$, such hat

and

where $\varsigma > 0$ depends on $f, \sigma_{0}$ and $\varsigma$. We now conclude that

and

The proof of Theorem 2.2 is now completed.

5.   Gevrey's regularity in time

We follow the methods found in ^[5,6,7,11] to treat the regularity in time in Gevrey sens for unique solution of (1.1).

Proposition 5.1. Let $\delta > 0, s_{1} > -\frac{\alpha-1}{4}$, $s_{2}\geq0$ and

be the solution of (1.1). Then $u$ belong in $x, y$ to $G^{\kappa}$ for all times near the zero. In other words,

for all $(x, y) \in \mathbb{R}^{2} \quad, C > 0, t \in [0, \delta], l, n \in \lbrace 0, 1, ...\rbrace.$

Proof. We have, for any $t \in [0, \delta]$

We observe that

and

This implies that

Thus,

Since $(2l)!^{\kappa} \leq c_{1}^{2l}(l!)^{2\kappa}$ and $(2n)!^{\kappa} \leq c_{2}^{2n}(n!)^{2\kappa}$, for some $c_{1}, c_{2} > 0$, we have for all $l, n \in \{0, 1, 2, \dots \}$

where $C_{0} = \Vert u(\cdot, \cdot, t)\Vert_{G_{s_{1}, s_{2}}^{\sigma_1, \sigma_2, \kappa}}$ and $C_{1} = c_{0}^{2}C_{\sigma_{1}, \sigma_{2}, \kappa}$ and $c_{0} = \max(c_{1}, c_{2})$, which implies that the solution $u$ is analytic in $x, y$ for all time near zero and $s_{1}, s_{2} \geq 0$.

Now, for $-\frac{\alpha-1}{4} < s_{1} < 0$, $s_{2} \geq 0$ and for any $0 < \epsilon < \sigma_{1}$, there exists a positive constant $C = C_{s, \epsilon} > 0$ such that

This implies that if

then

which allows us to conclude that $u$ is in $G^{\kappa}$ in $x, y$ for all $s_{1} > -\frac{\alpha-1}{4}$, $s_{2}\geq0$.

In order to prove Theorem 2.3 it is enough to prove the following result.

Lemma 5.2. For $j, l, n \in \lbrace 0, 1, 2, \dots\rbrace$, the next inequality

holds, where $L = C^{\alpha}+\frac{1}{120^{\kappa}}+\frac{C}{40^{\kappa}}$, $\forall x, y \in \mathbb{R}$, $t \in [0, \delta]$.

In fact, taking $l = n = 0$ we obtain

Proof. We use the induction on $j$ to prove Lemma 5.2.

For $j = 0$ and $l, n\in \lbrace 0, 1, 2, \dots\rbrace$, we have, by (5.1)

For $j = 1$ and $l, n\in \lbrace 0, 1, 2, \dots\rbrace$, we have

The terms of (5.4) can be estimated as

The nonlinear terms are treated as follows

Recalling that for $l \geq p$ and $n \geq k$, we have the next inequality

By (5.7), we have

At this stage, we use the fact that

Then,

From (5.5), (5.6) and (5.9), it follows that

We assume that (5.2) is correct for $j \geq m \geq 1$ where $l, n\in \lbrace 0, 1, 2, \dots \rbrace$ and then we prove it for $m = j+1$ and $l, n\in \lbrace 0, 1, 2, \dots \rbrace$.

We obtain

These terms are estimated as follows

and

The nonlinear terms are treated as follows

Using (5.7) to estimate (5.12)$_1$

We estimate (5.12)$_2$ as

To estimate (5.12)$_3$, we recall that for $j \geq q$, $l\geq p$ and $n \geq k$, we have the next inequality

Then

Finally by using (5.13)-(5.15) we arrive at

for all $(x, y) \in \mathbb{R}^{2}, t \in [0, \delta]$.

The detailed proof of (5.12) for $\kappa = 1$ is given in ^[6].

6.   Conclusions

We have discussed the local well-posedness for a generalized Kadomtsev-Petviashvili I equation in an anisotropic Gevrey space. We proved the existence of solutions using the Banach contraction mapping principle. This was done by using the bilinear estimates in anisotropic Gevrey-Bourgain. We used this local result and a Gevrey approximate conservation law to prove that global solutions exist. These solutions are Gevrey class of order $(\alpha+1)\kappa$ in the time variable. The results of the present paper are new and significantly contribute to the existing literature on the topic.

Acknowledgments

The authors wish to thank deeply the anonymous referee for his/here useful remarks and his/here careful reading of the proofs presented in this paper.

Conflict of interest

The authors declare that they have no conflict of interest.