Abstract
We study the large time asymptotics of solutions to the Cauchy problem for the fractional modified Korteweg-de Vries equation
where \(\alpha \in \left[ 4,5\right) ,\) and \(\left| \partial _{x}\right| ^{\alpha }={\mathcal {F}}^{-1}\left| \xi \right| ^{\alpha }{\mathcal {F}}\) is the fractional derivative. The case of \(\alpha =3\) corresponds to the classical modified KdV equation. In the case of \(\alpha =2\) it is the modified Benjamin–Ono equation. Our aim is to find the large time asymptotic formulas for the solutions of the Cauchy problem for the fractional modified KdV equation. We develop the method based on the factorization techniques which was started in our previous papers. Also we apply the known results on the \({\textbf{L}}^{2}\)—boundedness of pseudodifferential operators.
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1 Introduction
We study the large time asymptotics of solutions to the Cauchy problem for the fractional modified Korteweg-de Vries equation
where \(\alpha \in \left[ 4,5\right) \) and \(w_{0}\) is a real-valued known function. The case of \(\alpha =3\) corresponds to the classical modified KdV equation. In the case of \(\alpha =2,\) (1.1) is the modified Benjamin–Ono equation. Fractional KdV Eq. (1.1) was proposed in paper [16] as a good toy model to understand the influence of a weak dispersion on the dynamics of a scalar conservation law. Indeed, when \(\alpha =\frac{1}{2},\) Eq. (1.1) is reminiscent for large frequencies, to a modified Whitham equation \(\partial _{t}w+{\mathcal {K}} \partial _{x}w=\partial _{x}\left( w^{3}\right) ,\) where the Fourier multiplier operator \({\mathcal {K}}\) has symbol \(K\left( \xi \right) =\sqrt{\frac{\tanh \xi }{\xi }},\) which corresponds to the full dispersion of gravity waves with finite depth. So that the dispersion in (1.1), when \(\alpha =\frac{1}{2}\) is that of purely gravity waves with infinite depth.
The case of \(\alpha \in \left( 2,4\right) \) was considered in paper [9]. As far as we know the large time asymptotics for the Cauchy problem (1.1) with \(\alpha \in \left[ 4,5\right) \) was not studied previously. In the present paper we fill this gap and develop the factorization technique for (1.1) which was started in papers [6,7,8, 10, 13], to obtain the sharp time decay estimates of solutions. Also we apply the known theorems on the \({\textbf{L}}^{2}\)—boundedness of the pseudodifferential operators.
We have the conservation law \(\int _{{\mathbb {R}}}w\left( t,x\right) dx=\int _{{\mathbb {R}}}w_{0}\left( x\right) dx\) for any \(t>0.\) We assume that the total mass of the initial data vanishes \(\int _{{\mathbb {R}}}w_{0}\left( x\right) dx=0,\) then by the conservation of the total mass we obtain \(\int _{{\mathbb {R}}}w\left( t,x\right) dx=\int _{{\mathbb {R}}}w_{0}\left( x\right) dx=0\) for all \(t>0.\) In this case taking the antiderivative, we introduce the new dependent variable \(u=\partial _{x}^{-1}w=\int _{-\infty } ^{x}w\left( t,x^{\prime }\right) dx^{\prime }\) to obtain the potential form of the fractional modified Korteweg-de Vries equation
where \({\mathcal {L}}=\partial _{t}+\frac{1}{\alpha }\left| \partial _{x}\right| ^{\alpha -1}\partial _{x}\) and \(u_{0}\left( x\right) =\int _{-\infty }^{x}w_{0}\left( x^{\prime }\right) dx^{\prime }.\)
To state our results precisely we introduce Notation and Function Spaces. We denote the Lebesgue space by \({\textbf{L}}^{p}=\left\{ \phi \in {\textbf{S}}^{\prime };\left\| \phi \right\| _{{\textbf{L}}^{p}} <\infty \right\} \), where the norm \(\left\| \phi \right\| _{{\textbf{L}} ^{p}}=\left( \int _{{\mathbb {R}}}\left| \phi \left( x\right) \right| ^{p}dx\right) ^{\frac{1}{p}}\) for \(1\le p<\infty \) and \(\left\| \phi \right\| _{{\textbf{L}}^{\infty }}=ess.\sup _{x\in {\textbf{R}}}\left| \phi \left( x\right) \right| \) for \(p=\infty \). The weighted Sobolev space is
\(m,s\in {\textbf{R}},1\le p\le \infty ,\) \(\left\langle x\right\rangle =\sqrt{1+x^{2}},\left\langle i\partial _{x}\right\rangle =\sqrt{1-\partial _{x}^{2}}.\) We also use the notations \({\textbf{H}}^{m,s}={\textbf{H}}_{2}^{m,s},\) \({\textbf{H}}^{m}={\textbf{H}}^{m,0}\) shortly, if it does not cause any confusion. Let \({\textbf{C}}({\textbf{I}};{\textbf{B}})\) be the space of continuous functions from an interval \({\textbf{I}}\) to a Banach space \({\textbf{B}}.\) Different positive constants might be denoted by the same letter C.
Define the dilation operator \({\mathcal {D}}_{t}\phi =t^{-\frac{1}{2}}\phi \left( \frac{x}{t}\right) ,\) the scaling operator \(\left( {\mathcal {B}}\phi \right) \left( x\right) =\phi \left( \mu \left( x\right) \right) ,\) \(\mu \left( x\right) =x^{\frac{1}{\alpha -1}},\) the multiplication factor \(M=e^{it\frac{\alpha -1}{\alpha }\left| \eta \right| ^{\alpha }},\) the symbol \(\Lambda \left( \xi \right) =\frac{1}{\alpha }\xi \left| \xi \right| ^{\alpha -1}.\) Let \(\theta \) be the Heaviside function \(\theta \left( \xi \right) =1\) for \(\xi >0\) and \(\theta \left( \xi \right) =0\) for \(\xi \le 0.\)
Theorem 1.1
Assume that the initial data are such that \(u_{0} \in {\textbf{H}}^{3}\cap {\textbf{H}}^{2,1},\) with sufficiently small norm \(\left\| u_{0}\right\| _{{\textbf{H}}^{3}\cap {\textbf{H}}^{2,1}}\). Then there exists a unique global solution \(u\in {\textbf{C}}\left( \left[ 0,\infty \right) ;{\textbf{H}}^{3}\cap {\textbf{H}}^{2,1}\right) \) of the Cauchy problem (1.2). Moreover there exists a unique modified final state \(W_{+},\) such that \(\left| \xi \right| ^{\frac{\alpha -2}{2}} W_{+}\left( \xi \right) \in {\textbf{L}}^{\infty }\) and the asymptotic formula
is valid for \(t\rightarrow \infty \) uniformly with respect to \(x\in {\mathbb {R}},\) for some \(\delta >0\).
Remark 1.1
Large time asymptotics (1.3) can be written explicitly in the following form
We note that the main term of asymptotics differs from the corresponding linear case by the logarithmic oscillation.
Remark 1.2
The proof of Theorem 1.1 shows that the modified final state \(W_{+}\) can be found as a large-time limit. However this definition is not constructive. Therefore one should develop a method for finding \(W_{+}\) in terms of \(u_{0}\) on the basis of perturbation theory with respect to a parameter characterizing the smallness of the initial data \(u_{0}\) (see the approach developed in [14], also see a discussion of an algorithm for the approximate calculation of \(W_{+}\) in terms of \(u_{0}\) in paper [12]).
As a consequence of Theorem 1.1 using a relation \(w=u_{x}\) we get a result concerning well-posedness of the solution w of the Cauchy problem (1.1).
Corollary 1.1
Assume that the total mass of the initial data vanishes \(\int _{{\mathbb {R}} }w_{0}\left( x\right) dx=0.\) Suppose that the initial data are such that \(w_{0}\in {\textbf{H}}^{2}\cap {\textbf{H}}^{1,1},\) with sufficiently small norm \(\left\| w_{0}\right\| _{{\textbf{H}}^{2}\cap {\textbf{H}}^{1,1}}\). Then there exists a unique global solution \(w\in {\textbf{C}}\left( \left[ 0,\infty \right) ;{\textbf{H}}^{2}\cap {\textbf{H}}^{1,1}\right) \) of the Cauchy problem (1.1). Moreover there exists a unique modified final state \(W_{+},\) such that \(\left| \xi \right| ^{\frac{\alpha -2}{2}}W_{+} \in {\textbf{L}}^{\infty }\) and the asymptotics
is valid for \(t\rightarrow \infty \) uniformly with respect to \(x\in {\mathbb {R}},\) for some \(\delta >0\).
Remark 1.3
The above large time asymptotics can be written explicitly as follows
We organize the rest of the paper as follows. In Sect. 2 we state the factorization formula for (1.2), \({\textbf{L}}^{2}\)—estimates for pseudodifferential operators and prepare \({\textbf{L}}^{\infty }\) and \({\textbf{L}}^{2}\)—estimates for defect operator \({\mathcal {Q}}\) and its adjoint operator \({\mathcal {Q}}^{*}\). Section 3 is devoted to the proof of a-priori estimates of local solutions u of the Cauchy problem (1.2) in the following norm
where \(\widehat{\varphi }\left( t\right) =\) \({{\mathcal {F}}}{{\mathcal {U}}}\left( -t\right) u\left( t\right) ,\) \(\gamma >0\) is small and depends on the size of the data. Finally, we prove Theorem 1.1 in Sect. 4.
2 Preliminaries
The proofs in this section are omitted since they can be obtained in the same way as in our previous papers [1, 13].
2.1 Factorization techniques
Denote the symbol \(\Lambda \left( \xi \right) =\frac{1}{\alpha }\xi \left| \xi \right| ^{\alpha -1},\) then the free evolution group has the form \({\mathcal {U}}\left( t\right) ={\mathcal {F}}^{-1}e^{-it\Lambda \left( \xi \right) }{\mathcal {F}}.\) We write
where \({\mathcal {D}}_{t}\phi =t^{-\frac{1}{2}}\phi \left( \frac{x}{t}\right) \) is the dilation operator. In the integral \(\int _{{\mathbb {R}}}e^{it\left( x\xi -\Lambda \left( \xi \right) \right) }\phi \left( \xi \right) d\xi ,\) the stationary points are defined by the equation \(x=\Lambda ^{\prime }\left( \xi \right) =\left| \xi \right| ^{\alpha -1}\ge 0,\) therefore we find that there are two stationary points \(\xi =\pm \mu \left( x\right) ,\) where \(\mu \left( x\right) =x^{\frac{1}{\alpha -1}},\) \(x>0.\) We note that there are no any stationary point in the region \(x<0\). For convenience we extend \(\mu \left( x\right) \) for all \(x\in {\mathbb {R}}\) by the odd continuation \(\mu \left( x\right) =x\left| x\right| ^{\frac{1}{\alpha -1}-1}.\) Thus we have \(\frac{\mu }{\left| \mu \right| }\Lambda ^{\prime }\left( \mu \right) =x\) for all \(x\in {\mathbb {R}}\). We write
where the scaling operator \(\left( {\mathcal {B}}\phi \right) \left( x\right) =\phi \left( \mu \left( x\right) \right) \). We assume that \({\mathcal {F}} ^{-1}\phi \) is a real-valued function, then \(u={\mathcal {U}}\left( t\right) {\mathcal {F}}^{-1}\phi \) is a real-valued function and \(\phi \left( -\xi \right) =\overline{\phi \left( \xi \right) }.\) We also have \(\Lambda \left( \xi \right) =-\Lambda \left( -\xi \right) ,\) hence
Thus denoting the multiplication factor \(M=e^{-it\left( \frac{\eta }{\left| \eta \right| }\Lambda \left( \eta \right) -\left| \eta \right| \Lambda ^{\prime }\left( \eta \right) \right) }=e^{it\frac{\alpha -1}{\alpha }\left| \eta \right| ^{\alpha }},\) the phase function \(S\left( \xi ,\eta \right) =\Lambda \left( \xi \right) -\frac{\eta }{\left| \eta \right| }\Lambda \left( \eta \right) -\frac{\eta }{\left| \eta \right| }\Lambda ^{\prime }\left( \eta \right) \left( \xi -\eta \right) \) and the defect operator
we obtain the factorization formula for the evolution group \({\mathcal {U}} \left( t\right) {\mathcal {F}}^{-1}\phi =2{\text {Re}}{\mathcal {D}} _{t}{\mathcal {B}}M{\mathcal {Q}}\phi .\) Also we need the representation for the inverse evolution group
since \(x=\frac{\mu }{\left| \mu \right| }\Lambda ^{\prime }\left( \mu \right) ,\) \(\frac{dx}{d\mu }=\left| \Lambda ^{\prime \prime }\left( \mu \right) \right| ,\) where the inverse dilation operator \({\mathcal {D}} _{t}^{-1}\phi =t^{\frac{1}{2}}\phi \left( xt\right) ,\) the inverse scaling operator \(\left( {\mathcal {B}}^{-1}\phi \right) \left( \mu \right) =\phi \left( x\right) \). Then denoting the adjoint defect operator
we obtain the factorization formula for the inverse evolution group \({{\mathcal {F}}}{{\mathcal {U}}}\left( -t\right) \phi ={\mathcal {Q}}^{*}{\overline{M}} {\mathcal {B}}^{-1}{\mathcal {D}}_{t}^{-1}\phi .\)
Define the new dependent variable \(\widehat{\varphi }\left( t\right) =\) \({{\mathcal {F}}}{{\mathcal {U}}}\left( -t\right) u\left( t\right) \). Since \({{\mathcal {F}}}{{\mathcal {U}}} \left( -t\right) {\mathcal {L}}=\partial _{t}{{\mathcal {F}}}{{\mathcal {U}}}\left( -t\right) ,\) where \({\mathcal {L}}=\partial _{t}+i\Lambda \left( -i\partial _{x}\right) ,\) applying the operator \({{\mathcal {F}}}{{\mathcal {U}}}\left( -t\right) \) to equation (1.2) we get \(\partial _{t}\widehat{\varphi }={{\mathcal {F}}}{{\mathcal {U}}}\left( -t\right) u_{x}^{3}.\) Then since \(u_{x}=2{\text {Re}}{\mathcal {D}} _{t}{\mathcal {B}}Mv,\) \(v={\mathcal {Q}}i\xi \widehat{\varphi },\) we obtain
By the definition of the operator \({\mathcal {Q}}^{*}\left( t\right) \) we obtain \({\mathcal {Q}}^{*}\)\(\left( t\right) M^{k}\phi =e^{it\Omega _{k+1} }{\mathcal {D}}_{k+1}{\mathcal {Q}}^{*}\left( \left( k+1\right) t\right) \phi ,\) where \(\Omega _{k+1}=\Lambda \left( \xi \right) -\left( k+1\right) \Lambda \left( \frac{\xi }{k+1}\right) \) for \(k\ne -1.\) We denote below \(\Omega =\Omega _{3}.\) Then we find \({\mathcal {Q}}^{*}\left( t\right) M^{2}v^{3}=e^{it\Omega }{\mathcal {D}}_{3}{\mathcal {Q}}^{*}\left( 3t\right) v^{3},\)
since \(\Omega _{-1}=\Lambda \left( \xi \right) +\Lambda \left( -\xi \right) =0\) and
since \(\Omega _{-3}=\Lambda \left( \xi \right) +3\Lambda \left( -\frac{\xi }{3}\right) =\Lambda \left( \xi \right) -3\Lambda \left( \frac{\xi }{3}\right) =\) \(\Omega _{3},\) where \({\mathcal {D}}_{-1}\phi =e^{-i\frac{\pi }{2}}\phi \left( -x\right) \) and \({\mathcal {D}}_{-3}\phi =\frac{1}{\sqrt{3}}e^{-i\frac{\pi }{2} }\phi \left( -\frac{x}{3}\right) .\) Thus we get the following equation
Finally we mention some important identities. We have \({\mathcal {A}} _{1}{\mathcal {Q}}={\mathcal {Q}}i\xi \) and \(i\xi {\mathcal {Q}}^{*}\) \(={\mathcal {Q}} ^{*}{\mathcal {A}}_{1}\), where \({\mathcal {A}}_{1}={\overline{M}}{\mathcal {A}}_{0}M\), and \({\mathcal {A}}_{0}=\frac{1}{t\Lambda ^{\prime \prime }\left( \eta \right) }\partial _{\eta }.\) The operator \({\mathcal {J}}={\mathcal {U}}\left( t\right) x{\mathcal {U}}\left( -t\right) =x-t\Lambda ^{\prime }\left( -i\partial _{x}\right) ,\) plays a crucial role in the large time asymptotic estimates. Note that \({\mathcal {J}}\) commutes with \({\mathcal {L}}=\partial _{t}+i\Lambda \left( -i\partial _{x}\right) ,\) namely, \(\left[ {\mathcal {J}},{\mathcal {L}} \right] =0\). The symbol \(\Lambda \left( \xi \right) \) satisfies the identity \(\xi \partial _{\xi }\Lambda =\xi \left| \xi \right| ^{\alpha -1} =\alpha \Lambda .\) Hence we have the commutator \(\left[ \widehat{{\mathcal {P}} },e^{-it\Lambda \left( \xi \right) }\right] =0\) with \(\widehat{{\mathcal {P}} }=\alpha t\partial _{t}-\xi \partial _{\xi }.\) Also we define the operator \({\mathcal {P}}=\alpha t\partial _{t}+\partial _{x}x\). The commutator relation \(\left[ {\mathcal {L}},{\mathcal {P}}\right] =\alpha {\mathcal {L}}\) holds. Using \(u\left( t\right) ={\mathcal {U}}\left( t\right) {\mathcal {F}}^{-1} \widehat{\varphi }={\mathcal {F}}^{-1}e^{-it\Lambda \left( \xi \right) } \widehat{\varphi },\) we get
Also we have the identity \({\mathcal {P}}=\alpha t{\mathcal {L}}+\partial _{x}{\mathcal {J}}\) since by a direct calculation \(\alpha t{\mathcal {L}} +\partial _{x}{\mathcal {J}}=\alpha t\partial _{t}+\partial _{x}x.\)
2.2 Estimates for the operator \({\mathcal {Q}}\) in the uniform norm
Define the cut off functions \(\chi _{j}\left( x\right) \in {\textbf{C}} ^{4}\left( {\mathbb {R}}\right) ,\) such that \(\chi _{1}\left( x\right) =1\) for \(\frac{2}{3}\le x\le 2\), \(\chi _{1}\left( x\right) =0\) for \(x\le \frac{1}{3}\) or \(x\ge 3,\) also \(\chi _{2}\left( x\right) =1-\chi _{1}\left( x\right) .\) Denote the operators
for \(\eta >0,\) \(k=1,2.\) Define the kernel
for \(\eta >0.\) We change \(\xi =\eta y\), then we get
We get \(\left| A_{j}\left( t,\eta \right) \right| \le Ct^{\frac{1}{2}}\theta \left( \eta \right) \left| \eta \right| ^{1+j},\) if \(\left| \eta \right| \le t^{-\frac{1}{\alpha }}.\) To compute the asymptotics of the kernel \(A_{j}\left( t,\eta \right) \) for large \(t\left| \eta \right| ^{\alpha }\) we apply the stationary phase method (see [5], p. 110)
for \(t\rightarrow +\infty ,\) where the stationary point \(y_{0}\) is defined by the equation \(g^{\prime }\left( y_{0}\right) =0.\) By virtue of formula (2.2) with \(g\left( y\right) =-S\left( y,1\right) ,\) \(f\left( y\right) =\chi _{1}\left( y\right) y^{j},\) \(y_{0}=1,\) we get
for \(t\left| \eta \right| ^{\alpha }\rightarrow +\infty .\) Hence \(\left| A_{j}\left( t,\eta \right) \right| \le Ct^{\frac{1}{2} }\theta \left( \eta \right) \left| \eta \right| ^{1+j}\left\langle t\left| \eta \right| ^{\alpha }\right\rangle ^{-\frac{1}{2}}\) for all \(t\ge 1,\) \(\eta \in {\mathbb {R}}\).
In the next lemma we find the estimate for the defect operator \({\mathcal {Q}}\) in the uniform metrics.
Lemma 2.1
Let \(4\le \alpha <5\). Then the estimate
is true for all \(t\ge 1,\) \(\eta >0,\) where \(j=1,2,3,\) \(\delta \ge 0.\)
Consider the estimate of \({\mathcal {Q}}_{2}\left( t\right) .\)
Lemma 2.2
Let \(4\le \alpha <5\). Then the estimate
is true for all \(t\ge 1,\) \(\eta >0,\) where \(j=1,2,3,\) \(\delta \ge 0.\)
Next we find the estimate of \({\mathcal {Q}}\left( t\right) \) for \(\eta \le 0.\)
Lemma 2.3
Let \(4\le \alpha <5\). Then the estimate
is true for all \(t\ge 1,\) \(\eta \le 0,\) where \(j=1,2,3,\) \(\delta \ge 0.\)
As a consequence of Lemma 2.1, Lemma 2.2 and Lemma 2.3 we obtain the following result. Define the norm \(\left\| \phi \right\| _{{\textbf{Y}}}=\left\| \left\langle \xi \right\rangle ^{2}\phi \right\| _{{\textbf{L}}^{\infty }}+t^{-\frac{1}{2\alpha }}\left\| \left\langle \xi \right\rangle ^{2}\partial _{\xi }\phi \right\| _{{\textbf{L}} ^{2}}.\) Denote \(\widetilde{\eta }=\eta t^{\frac{1}{\alpha }}.\)
Corollary 2.1
The estimates
and
are true.
2.3 \({\textbf{L}}^{\infty }\) - estimates for the adjoint defect operator \({\mathcal {Q}}^{*}\)
Denote the operators
for \(\xi \ne 0\). Denote the adjoint kernel
for \(\xi \ne 0\). We change \(\eta =\xi y\), then we get
We have \(\left| A^{*}\left( t,\xi \right) \right| \le Ct^{\frac{1}{2}}\left| \xi \right| ^{\alpha -1},\) if \(\left| \xi \right| \le t^{-\frac{1}{\alpha }}.\) By virtue of formula (2.2) with \(g\left( y\right) =S\left( 1,y\right) ,\) \(f\left( y\right) =\chi _{1}\left( y\right) y^{\alpha -2},\) \(y_{0}=1,\) we find
for \(t\left| \xi \right| ^{\alpha }\rightarrow +\infty .\) Hence
for \(t\ge 1,\) \(\xi \in {\mathbb {R}}\).
In the next lemma we estimate the adjoint defect operator \({\mathcal {Q}} _{1}^{*}\) in the uniform metrics.
Lemma 2.4
Let \(4\le \alpha <5\). Then the estimate
is true for all \(t\ge 1,\) \(\xi \in {\mathbb {R}}\), where \(\delta \ge 0.\)
Next we estimate \({\mathcal {Q}}_{2}^{*}\phi .\)
Lemma 2.5
Let \(4\le \alpha <5\). Then the estimate
is true for all \(t\ge 1,\) \(\xi \in {\mathbb {R}}\), where \(0\le \delta <\alpha -\frac{1}{2}.\)
2.4 Estimates for pseudodifferential operators
There are many papers devoted to the \({\textbf{L}}^{2}\) - estimates of pseudodifferential operators (see, e.g. [2,3,4, 11]). Consider the following weighted defect operator
Lemma 2.6
Let \(h\left( t,\xi ,\eta \right) \) satisfy the estimate
\(\sup _{\xi ,\eta \in {\mathbb {R}},t\ge 1} \left| \left\{ \widetilde{\eta }\right\} ^{-\nu }\left\langle \widetilde{\eta }\right\rangle ^{\nu }\left( \eta \partial _{\eta }\right) ^{k}h\left( t,\xi ,\eta \right) \right| \le C\) for \(k=0,1,2,\) with some \(\nu \in \left( 0,1\right) .\) Then there exists a positive constant C such that the inequality \(\left\| \sqrt{\Lambda ^{\prime \prime }}{\mathcal {V}}_{h}\phi \right\| _{{\textbf{L}}^{2}}\le C\left\| \phi \right\| _{{\textbf{L}}^{2}}\) holds.
Similarly we estimate the \({\textbf{L}}^{2}\) - norm of the adjoint weighted defect operator
Lemma 2.7
Let \(h\left( t,\xi ,\eta \right) \) satisfy the estimate
\(\sup _{\xi ,\eta \in {\mathbb {R}},t\ge 1}\left| \left\{ \widetilde{\eta }\right\} ^{-\nu }\left\langle \widetilde{\eta }\right\rangle ^{\nu }\left( \eta \partial _{\eta }\right) ^{k}h\left( t,\xi ,\eta \right) \right| \le C\) for \(k=0,1,2\) with some \(\nu \in \left( 0,1\right) .\) Then there exists a positive constant C such that the inequality \(\left\| {\mathcal {V}} _{h}^{*}\phi \right\| _{{\textbf{L}}^{2}}\le C\left\| \sqrt{\Lambda ^{\prime \prime }}\phi \right\| _{{\textbf{L}}^{2}}\) holds.
2.5 Estimate for derivative of the defect operator
Lemma 2.8
The estimate
is true for all \(t\ge 1,\) where \(\nu >0\) is small.
2.6 Estimates for the derivative of \({\mathcal {Q}}^{*}\)
Define the cut off functions \(\chi _{3}\left( x\right) ,\chi _{4}\left( x\right) \in {\textbf{C}}^{4}\left( {\mathbb {R}}\right) ,\) such that \(\chi _{3}\left( x\right) =1\) for \(x\ge 3\), \(\chi _{3}\left( x\right) =0\) for \(x\le 2,\) also \(\chi _{4}\left( x\right) =1-\chi _{3}\left( x\right) .\) Denote
where \(\widetilde{\xi }=\xi t^{\frac{1}{\alpha }}.\) In the next lemma we obtain the estimates of the derivative of the adjoint defect operator \(\partial _{\xi }{\mathcal {Q}}_{5}^{*}\phi .\)
Lemma 2.9
The estimate
is true for all \(t\ge 1\), where \(0<\nu <1.\)
We now estimate the adjoint defect operators \(\xi {\mathcal {Q}}_{5}^{*} \eta ^{-1}\phi \) and \(\frac{1}{\xi }{\mathcal {Q}}_{6}^{*}\eta \phi .\)
Lemma 2.10
The estimate
is true for all \(t\ge 1\), where \(0<\nu <1,\) \(j=0,1.\)
We also have the estimate for \({\mathcal {Q}}_{5}^{*}\phi .\)
Lemma 2.11
The estimate
is true for all \(t\ge 1\), where \(0<\nu <1.\)
3 A-priori estimates
3.1 Estimates for the nonlinearity
In the next lemma we calculate the asymptotic representation for the nonlinearity \({{\mathcal {F}}}{{\mathcal {U}}}\left( -t\right) \partial _{x}\left( \left| u\right| ^{2}u\right) .\) Define the norm
where \(\widehat{\varphi }\left( t\right) =\) \({{\mathcal {F}}}{{\mathcal {U}}}\left( -t\right) u\left( t\right) ,\) \(\gamma >0\) is small.
Lemma 3.1
Suppose that \(\left\| u\right\| _{{\textbf{X}}_{T}}\le C\varepsilon .\) Then the asymptotic representation
is true for all \(t\ge 1\), \(\xi \in {\mathbb {R}},\) with some \(\delta _{1}>0.\)
Proof
By equation (2.1) we have
We define the cut off function \(\chi _{5}\left( x\right) \in {\textbf{C}} ^{4}\left( {\mathbb {R}}\right) ,\) such that \(\chi _{5}\left( x\right) =1\) for \(\left| x\right| \le 1\) and \(\chi _{5}\left( x\right) =0\) for \(\left| x\right| \ge 2,\) \(\chi _{6}\left( x\right) =1-\chi _{5}\left( x\right) .\) We represent
By Corollary 2.1 we have \(\left| v\right| \le C\varepsilon t^{\frac{\alpha -4}{2\alpha }}\left\langle \widetilde{\eta }\right\rangle ^{-\frac{\alpha -4}{2}}\le C\varepsilon \left| \eta \right| ^{2-\frac{\alpha }{2}}\) in the domain \(\left| \eta \right| \le 2t^{-\frac{1}{\alpha }}.\) Then we estimate the first term as follows
Next we find the asymptotics of the second term. By virtue of Lemma 2.4 and Lemma 2.5 with \(\delta =3\left( \frac{\alpha }{2}-2\right) -\nu ,\) we get
We use the estimate \(\left| v\right| \le C\varepsilon t^{\frac{1}{2}-\frac{2}{\alpha }}\left\langle \widetilde{\eta }\right\rangle ^{2-\frac{\alpha }{2}}\) for \(\left| \eta \right| \ge t^{-\frac{1}{\alpha }}\). Then, we find
By Lemma 2.8 we have
Hence using \(\left| v\right| \le C\varepsilon t^{\frac{1}{2}-\frac{2}{\alpha }}\left\langle \widetilde{\eta }\right\rangle ^{2-\frac{\alpha }{2}} \) for \(\left| \eta \right| \ge t^{-\frac{1}{\alpha }},\) we get
Thus we find
with some \(\delta _{1}>0.\) We use the asymptotics of the kernel \(A^{*}\), so that
Also by Lemma 2.1 and Lemma 2.2 with \(j=1,\) \(\delta =\frac{\alpha -2}{6},\) and by the asymptotics of the kernel \(A_{1}\) we find
Hence we have
Similarly we get
Also \(\left\langle \xi \right\rangle ^{-\frac{1}{2}}\overline{{\mathcal {D}} _{-1}{\mathcal {Q}}^{*}\left( t\right) \left| v\right| ^{2} v}=O\left( \varepsilon ^{3}t^{-\delta _{1}}\right) \) and \(\left\langle \xi \right\rangle ^{-\frac{1}{2}}\overline{{\mathcal {D}}_{-3}{\mathcal {Q}}^{*}\left( 3t\right) v^{3}}=O\left( \varepsilon ^{3}t^{-\delta _{1}}\right) \) since \({\mathcal {D}}_{-1}\theta \left( \xi \right) =0\) in the domain \(\xi >0,\) so that we have
and similarly \(\left| \xi \right| ^{\delta }{\mathcal {D}}_{-3}v=O\left( \varepsilon t^{-\delta _{1}}\right) .\) Hence we get the result of the lemma. Lemma 3.1 is proved. \(\square \)
3.2 Estimate for the derivative \(\partial _{\xi }\widehat{\varphi }\)
We have \({\mathcal {Q}}^{*}={\mathcal {Q}}_{5}^{*}+{\mathcal {Q}}_{6}^{*}.\) Define \(\widehat{\varphi }_{1}\) and \(\widehat{\varphi }_{2}\) such that
and
where \(\Phi _{1}={\mathcal {D}}_{3}{\mathcal {Q}}_{5}^{*}\left( 3t\right) v^{3}+\overline{{\mathcal {D}}_{-3}{\mathcal {Q}}_{5}^{*}\left( 3t\right) v^{3}},\) \(\Phi _{2}={\mathcal {Q}}_{5}^{*}\left( t\right) \left| v\right| ^{2}v+\overline{{\mathcal {D}}_{-1}{\mathcal {Q}}_{5}^{*}\left( t\right) \left| v\right| ^{2}v},\) \(\Phi _{3}={\mathcal {D}} _{3}{\mathcal {Q}}_{6}^{*}\left( 3t\right) v^{3}+\overline{{\mathcal {D}} _{-3}{\mathcal {Q}}_{6}^{*}\left( 3t\right) v^{3}},\) \(\Phi _{4} ={\mathcal {Q}}_{6}^{*}\left( t\right) \left| v\right| ^{2}v+\overline{{\mathcal {D}}_{-1}{\mathcal {Q}}_{6}^{*}\left( t\right) \left| v\right| ^{2}v}.\) Then we have \(\widehat{\varphi }=\) \(\widehat{\varphi }_{1}+\widehat{\varphi }_{2},\) since \(v={\mathcal {Q}}\left( i\xi \left( \widehat{\varphi }_{1}+\widehat{\varphi }_{2}\right) \right) .\)
In the next lemma we find a priori estimate of \(\partial _{\xi }\widehat{\varphi }_{1}\left( t,\xi \right) \).
Lemma 3.2
Suppose that \(\left\| u\right\| _{{\textbf{X}}_{T}}\le C\varepsilon .\) Then the estimate \(\left\| \partial _{\xi }\widehat{\varphi }_{1}\right\| _{{\textbf{L}}^{2}}\le C\varepsilon t^{\frac{1}{2\alpha }}\) is true for all \(t\in \left[ 1,T\right] .\)
Proof
Let us prove the following estimates \(\left\| \partial _{\xi }\widehat{\varphi }_{1}\right\| _{{\textbf{L}}^{2}}\le C\varepsilon t^{\frac{1}{2\alpha }}\) and \(\left\| \frac{1}{\xi }\widehat{{\mathcal {P}}} \widehat{\varphi }_{2}\right\| _{{\textbf{L}}^{2}}\le C\varepsilon t^{\frac{1}{2\alpha }},\) where \(\widehat{{\mathcal {P}}}=\alpha t\partial _{t} -\xi \partial _{\xi }\). Then it follows that
since by Lemma 2.10 and by the estimate of Corollary 2.1\(\left| v\right| \le C\varepsilon t^{\frac{\alpha -4}{2\alpha }}\left\langle \widetilde{\eta }\right\rangle ^{-\frac{\alpha -4}{2}}\left\langle \eta \right\rangle ^{-\frac{1}{2}}\) we have
Differentiating Eq. (3.1) we get
where \(R_{1}=t^{-1}e^{it\Omega }\partial _{\xi }\Phi _{1}+3t^{-1}\partial _{\xi }\Phi _{2}.\)
The derivative \(\partial _{\xi }{\mathcal {Q}}_{5}^{*}\) was estimated in Lemma 2.9
By Corollary 2.1 we have \(\left| v\right| \le C\varepsilon t^{\frac{\alpha -4}{2\alpha }}\left\langle \widetilde{\eta }\right\rangle ^{-\frac{\alpha -4}{2}}\le C\varepsilon \left| \eta \right| ^{2-\frac{\alpha }{2}}\) in the domain \(\left| \eta \right| \le 2t^{-\frac{1}{\alpha }}.\) Then
Also by Corollary 2.1 we find \(\left| v\right| \le C\varepsilon t^{\frac{\alpha -4}{2\alpha }}\left\langle \widetilde{\eta }\right\rangle ^{-\frac{\alpha -4}{2}}\left\langle \eta \right\rangle ^{-\frac{1}{2}}\) in the domain \(\left| \eta \right| \ge t^{-\frac{1}{\alpha }}.\) Hence
since \(4\le \alpha <5\). Next we use Lemma 2.8
and Corollary 2.1\(\left| v\right| \le C\varepsilon t^{\frac{\alpha -4}{2\alpha }}\left\langle \widetilde{\eta }\right\rangle ^{-\frac{\alpha -4}{2}}\left\langle \eta \right\rangle ^{-\frac{1}{2}},\ \)to estimate
since \(4\le \alpha <5\) and \(\nu >0\) is small. Therefore we obtain\(\left\| R_{1}\right\| _{{\textbf{L}}_{\xi }^{2}}\le Ct^{-1}\left\| \partial _{\xi }\Phi _{1}\right\| _{{\textbf{L}}_{\xi }^{2}}+Ct^{-1}\left\| \partial _{\xi }\Phi _{2}\right\| _{{\textbf{L}}_{\xi }^{2}}\le C\varepsilon ^{3}t^{\frac{1}{2\alpha }-1}.\) We need to transform the last term \(i\Omega ^{\prime }e^{it\Omega }\Phi _{1}\) in Eq. (3.3). Denote
and
So that \({\mathcal {Q}}_{5}^{*}={\mathcal {Q}}_{7}^{*}+{\mathcal {Q}}_{8}^{*}\) and \(\Phi _{1}=\Phi _{5}+\Phi _{6},\) where \(\Phi _{5}={\mathcal {D}}_{3} {\mathcal {Q}}_{7}^{*}\left( 3t\right) v^{3}+\overline{{\mathcal {D}} _{-3}{\mathcal {Q}}_{7}^{*}\left( 3t\right) v^{3}},\) \(\Phi _{6} ={\mathcal {D}}_{3}{\mathcal {Q}}_{8}^{*}\left( 3t\right) v^{3}+\overline{{\mathcal {D}}_{-3}{\mathcal {Q}}_{8}^{*}\left( 3t\right) v^{3}}.\) The term \(R_{2}=i\Omega ^{\prime }e^{it\Omega }\Phi _{6}\) can be estimated easily as follows
Next we represent
We have
and
where \(h_{5}\left( t,\xi ,\eta \right) =\frac{i\eta \Omega ^{\prime }\left( \xi \right) }{G_{1}\left( \xi ,\eta \right) }\chi _{3}\left( \widetilde{\eta }\right) \chi _{3}\left( \frac{\eta }{\xi }\right) \) and \(h_{6}\left( t,\xi ,\eta \right) =\frac{i\eta \Omega ^{\prime }\left( \xi \right) }{G_{2}\left( \xi ,\eta \right) }\chi _{3}\left( \widetilde{\eta }\right) \chi _{3}\left( \frac{\eta }{\xi }\right) ,\) with \(G_{1}\left( \xi ,\eta \right) =S\left( \xi ,\eta \right) -2i\left( \frac{\eta }{\left| \eta \right| }\Lambda \left( \eta \right) -\left| \eta \right| \Lambda ^{\prime }\left( \eta \right) \right) \) and
\(G_{2}\left( \xi ,\eta \right) =S\left( \xi ,\eta \right) +4i\left( \frac{\eta }{\left| \eta \right| }\Lambda \left( \eta \right) -\left| \eta \right| \Lambda ^{\prime }\left( \eta \right) \right) ,\) also we denote
Note that
in the domain \(\left| \eta \right| \ge 2\left| \xi \right| .\) Similarly \(G_{1}\left( \xi ,\eta \right) \ge 2\left| \eta \right| ^{\alpha }\) in the domain \(\left| \eta \right| \ge 2\left| \xi \right| .\)Therefore \(h_{5}\left( t,\xi ,\eta \right) =O\left( 1\right) \) and \(h_{6}\left( t,\xi ,\eta \right) =O\left( 1\right) .\) Then as in Lemma 2.10 we get
for \(k=5,6.\) Thus
since by Corollary 2.1 we have \(\left| v\right| \le C\varepsilon t^{\frac{\alpha -4}{2\alpha }}\left\langle \widetilde{\eta }\right\rangle ^{-\frac{\alpha -4}{2}}\) in the domain \(\left| \eta \right| \le 2t^{-\frac{1}{\alpha }}.\) We have
Also by Corollary 2.1 we find \(\left| v\right| \le C\varepsilon t^{\frac{\alpha -4}{2\alpha }}\left\langle \widetilde{\eta }\right\rangle ^{-\frac{\alpha -4}{2}}\left\langle \eta \right\rangle ^{-\frac{1}{2}}\) in the domain \(\left| \eta \right| \ge t^{-\frac{1}{\alpha }}.\) Hence
Therefore we get
Denoting \(y_{1}=\) \(\partial _{\xi }\widehat{\varphi }_{1}-{\mathcal {V}}_{h_{5} }^{*}M^{2}\frac{1}{\eta }v^{3}-{\mathcal {V}}_{h_{6}}^{*}{\overline{M}} ^{4}\frac{1}{\eta }{\overline{v}}^{3},\) we find
To estimate \(\partial _{t}v\) we write \(\alpha t\partial _{t}v=\widehat{{\mathcal {P}}}v+\eta \partial _{\eta }v.\) Then we obtain
Hence \(\alpha t\partial _{t}v=\eta \partial _{\eta }v+\frac{\alpha -4}{2}v+i{\mathcal {A}}_{1}^{2}{\mathcal {Q}}\partial _{\xi }\widehat{\varphi } +\alpha {\mathcal {A}}_{1}{\overline{M}}\left( Mv+{\overline{M}}{\overline{v}}\right) ^{3}\) and also \({\mathcal {A}}_{1}{\overline{M}}\left( Mv+{\overline{M}}\overline{v}\right) ^{3}=3\left( Mv+{\overline{M}}{\overline{v}}\right) ^{2}\left( {\mathcal {A}}_{1}v+{\overline{M}}^{2}\overline{{\mathcal {A}}_{1}v}\right) .\) Then
The last term can be represented as
Therefore we get
Hence it follows that \(\left\| \partial _{t}y_{1}\right\| _{{\textbf{L}} _{\xi }^{2}}\le C\varepsilon ^{3}t^{\frac{1}{2\alpha }-1}.\) Integrating we find \(\left\| y_{1}\right\| _{{\textbf{L}}_{\xi }^{2}}\le C\varepsilon ^{3}t^{\frac{1}{2\alpha }}.\) Thus
Lemma 3.2 is proved. \(\square \)
Next we estimate \(\partial _{\xi }\widehat{\varphi }_{2}\left( t,\xi \right) \).
Lemma 3.3
Suppose that \(\left\| u\right\| _{{\textbf{X}}_{T}}\le C\varepsilon .\) Then the estimate \(\left\| \partial _{\xi }\widehat{\varphi }_{2}\right\| _{{\textbf{L}}^{2}}\le C\varepsilon t^{\frac{1}{2\alpha }}\) is true for all \(t\in \left[ 1,T\right] .\)
Proof
Let us prove the following estimate \(\left\| \frac{1}{\xi }\widehat{{\mathcal {P}}}\widehat{\varphi }_{2}\right\| _{{\textbf{L}}^{2}}\le C\varepsilon t^{\frac{1}{2\alpha }},\) where \(\widehat{{\mathcal {P}}}=\alpha t\partial _{t}-\xi \partial _{\xi }\). Then it follows that
since by Lemma 2.10 and by the estimate of Corollary 2.1\(\left| v\right| \le C\varepsilon t^{\frac{\alpha -4}{2\alpha }}\left\langle \widetilde{\eta }\right\rangle ^{-\frac{\alpha -4}{2}}\left\langle \eta \right\rangle ^{-\frac{1}{2}}\) we have
Thus we apply the operator \(\frac{1}{\xi }\widehat{{\mathcal {P}}}\), with \(\widehat{{\mathcal {P}}}=\alpha t\partial _{t}-\xi \partial _{\xi },\) to Eq. (3.2). Then we get
Note that \(\left( \alpha t\partial _{t}-\xi \partial _{\xi }-\eta \partial _{\eta }\right) e^{itS\left( \xi ,\eta \right) }=0,\) therefore we have the commutator\(\left[ \widehat{{\mathcal {P}}},{\mathcal {Q}}_{6}^{*}\right] =\frac{2-\alpha }{2}{\mathcal {Q}}_{6}^{*}.\) Similarly, we find \(\left[ \widehat{{\mathcal {P}}},{\mathcal {Q}}\right] =\frac{\alpha -2}{2}{\mathcal {Q}}.\) Then we obtain
where \(R_{5}=\frac{2-3\alpha }{2}\frac{1}{\xi t}{\mathcal {Q}}_{6}^{*} {\overline{M}}\left( Mv+{\overline{M}}{\overline{v}}\right) ^{3}\) has the estimate similarly to the above
Denote \(y=\frac{1}{\xi }\widehat{{\mathcal {P}}}\widehat{\varphi }_{2}-y_{1},\) \(y_{1}=\) \(\partial _{\xi }\widehat{\varphi }_{1}-{\mathcal {V}}_{h_{5}}^{*} M^{2}\frac{1}{\eta }v^{3}-{\mathcal {V}}_{h_{6}}^{*}{\overline{M}}^{4}\frac{1}{\eta }{\overline{v}}^{3}\), then since \(\widehat{{\mathcal {P}}}=\alpha t\partial _{t}-\xi \partial _{\xi }\), we get
where \(Y=\frac{\alpha -4}{2}v+\alpha {\mathcal {A}}_{1}{\mathcal {Q}}\left( {\mathcal {Q}}_{5}^{*}{\overline{M}}\left( Mv+{\overline{M}}{\overline{v}}\right) ^{3}\right) \) \(-{\mathcal {A}}_{1}{\mathcal {Q}}\left( \xi {\mathcal {V}}_{h_{5} }^{*}M^{2}\frac{1}{\eta }v^{3}+\xi {\mathcal {V}}_{h_{6}}^{*}{\overline{M}} ^{4}\frac{1}{\eta }{\overline{v}}^{3}\right) .\) Then we obtain
where \(R_{6}=\frac{24}{\xi t}{\mathcal {Q}}_{6}^{*}\left( {\overline{M}}\left( {\text {Re}}Mv\right) ^{2}\left( {\text {Re}}MY\right) \right) .\) We represent
where we denote
As above we find \(\left\| \frac{1}{\xi t}{\mathcal {Q}}_{6}^{*}\left( {\overline{M}}\left( {\text {Re}}Mv\right) ^{3}\right) \right\| _{{\textbf{L}}^{2}}\le C\varepsilon ^{3}t^{\frac{1}{2\alpha }-1}.\) By Lemma 2.10 we have the estimate
and by Corollary 2.1\(\left| v\right| \le C\varepsilon t^{\frac{\alpha -4}{2\alpha }}\left\langle \widetilde{\eta }\right\rangle ^{-\frac{\alpha -4}{2}}\left\langle \eta \right\rangle ^{-\frac{1}{2}},\) then we get
Also using Lemma 2.11 we find
Then we obtain
Hence \(\left\| R_{6}\right\| _{{\textbf{L}}^{2}}\le C\varepsilon ^{3}t^{\frac{1}{2\alpha }-1}.\) Finally, we transform
where
Using estimate of Corollary 2.1\(\left\langle \widetilde{\eta }\right\rangle ^{-1}\left| v\right| \left| {\mathcal {A}}_{1} ^{2}v\right| \le C\varepsilon ^{2},\) as above we have the estimates \(\sum _{j=7}^{10}\left\| R_{j}\right\| _{{\textbf{L}}^{2}}\le C\varepsilon ^{3}t^{\frac{1}{2\alpha }-1}.\) Thus we get the equation
where the remainder term has the estimate \(\left\| R\right\| _{{\textbf{L}}^{2}}\le C\varepsilon ^{3}t^{\frac{1}{2\alpha }-1}.\) Multiplying the above equation by \({\overline{y}}\), taking the real part and integrating over \({\mathbb {R}}\) we get
Note that
and
since by Corollary 2.1 we have \(\left\| v{\mathcal {A}} _{1}v\right\| _{{\textbf{L}}^{\infty }}\le C\varepsilon ^{2}.\) Thus we find
Integrating in time we obtain the estimate of the lemma. Lemma 3.3 is proved. \(\square \)
3.3 A-priori estimates of local solutions
We first state the local existence of solutions to the Cauchy problem (1.2) which can be obtained by the classical energy method (see [14, 15]).
Theorem 3.1
Assume that the initial data \(u_{0}\in {\textbf{H}}^{3} \cap {\textbf{H}}^{2,1},\) and the norm \(\varepsilon =\left\| u_{0}\right\| _{{\textbf{H}}^{3}\cap {\textbf{H}}^{2,1}}\) is sufficiently small. Then there exists a time \(T>1\) such that the Cauchy problem (1.2) has a unique solution \(u\in {\textbf{C}}\left( \left[ 0,T\right] ;{\textbf{H}}^{3} \cap {\textbf{H}}^{2,1}\right) \) such that \(\left\| u\right\| _{{\textbf{X}}_{T}}\le C\varepsilon .\)
To prove the global result, we need a priori estimate of the norm \(\left\| u\right\| _{{\textbf{X}}_{T}}\) uniformly with respect to \(T\ge 1.\)
Lemma 3.4
Let the initial data \(u_{0}\in {\textbf{H}}^{3}\cap {\textbf{H}}^{2,1}\) have a small norm \(\left\| u_{0}\right\| _{{\textbf{H}}^{3}\cap {\textbf{H}}^{2,1}}\). Then the estimate \(\left\| u\right\| _{{\textbf{X}}_{T}}<C\varepsilon \) is true for all \(T\ge 1.\)
Proof
Arguing by the contradiction, we can find a time interval \(T\ge 1\) such that \(\left\| u\right\| _{{\textbf{X}}_{T}}=C\varepsilon .\) We need the estimate the norm \(\left\| \left\langle \xi \right\rangle ^{2}\widehat{\varphi }\right\| _{{\textbf{L}}^{\infty }}.\) In the domain \(\left| \xi \right| \ge \left\langle t\right\rangle ^{\nu }\), by the Sobolev embedding inequality we get
if \(\frac{\nu }{2}>\gamma .\) Therefore we need to estimate \(\left\langle \xi \right\rangle ^{2}\widehat{\varphi }\left( t,\xi \right) \) in the domain \(\left| \xi \right| \le \left\langle t\right\rangle ^{\nu }.\) Applying the operator \({{\mathcal {F}}}{{\mathcal {U}}}\left( -t\right) \) to Eq. (1.2), we get (2.1). Then by virtue of Lemma 3.1 since \(\left\langle \xi \right\rangle ^{2}\le C\left\langle t\right\rangle ^{2\nu }\), we obtain
Multiplying the above equation by \(\overline{\left\langle \xi \right\rangle ^{2}\widehat{\varphi }}\), and taking the real part of the result, we find
in the domain \(\left| \xi \right| \le \left\langle t\right\rangle ^{\nu }.\) Define \(t_{1}\) such that \(\left\langle t_{1}\right\rangle ^{\nu }=\left| \xi \right| ,\) then integrating in time from \(t_{1}\) to t we obtain
Integration by parts via the identity \(e^{i\tau \Omega }=\left( 1+i\tau \Omega \right) ^{-1}\partial _{\tau }\left( \tau e^{i\tau \Omega }\right) \) yields the estimate \(\left| \left\langle \xi \right\rangle ^{2} \widehat{\varphi }\left( t\right) \right| <C\varepsilon \). Since the solution u is real, we have \(\overline{\widehat{\varphi }\left( t,\xi \right) }=\widehat{\varphi }\left( t,-\xi \right) .\) Therefore \(\left\| \left\langle \xi \right\rangle ^{2}\widehat{\varphi }\right\| _{{\textbf{L}}^{\infty }}<C\varepsilon .\) By Corollary 2.1 we get
Then applying the energy method to Eq. (1.2), we have
Using the Grönwall inequality we get \(\left\| u\left( t\right) \right\| _{{\textbf{H}}^{3}}<C\varepsilon \left\langle t\right\rangle ^{C\varepsilon }.\) To estimate the norm \(\left\| \xi ^{2}\partial _{\xi }\widehat{\varphi }\right\| _{{\textbf{L}}_{\xi }^{2}}\) we apply the operator \(\partial _{x}{\mathcal {P}}\) to Eq. (1.2). In view of the commutators \(\left[ {\mathcal {L}},{\mathcal {P}}\right] =\alpha {\mathcal {L}},\) \(\left[ {\mathcal {P}},\partial _{x}\right] =-\partial _{x},\) we obtain \({\mathcal {L}} \partial _{x}{\mathcal {P}}u=\partial _{x}\left( {\mathcal {P}}+\alpha \right) {\mathcal {L}}u=\partial _{x}\left( {\mathcal {P}}+\alpha \right) \left| u_{x}\right| ^{2}u_{x}.\) Application of the energy method yields
Hence integrating in time we obtain \(\left\| \partial _{x}{\mathcal {P}} u\right\| _{{\textbf{L}}^{2}}<C\varepsilon t^{\frac{1}{2\alpha }}\). Therefore, using the identity \({\mathcal {J}}=\partial _{x}^{-1}{\mathcal {P}}+i\alpha t\partial _{x}^{-1}{\mathcal {L}}\), we find
Also using Lemma 3.2 and Lemma 3.3 we obtain \(\left\| \partial _{\xi }\widehat{\varphi }\right\| _{{\textbf{L}}_{\xi }^{2} }<C\varepsilon t^{\frac{1}{2\alpha }}.\) Hence we get the estimate \(\left\| \left\langle \xi \right\rangle ^{2}\partial _{\xi }\widehat{\varphi }\right\| _{{\textbf{L}}_{\xi }^{2}}<C\varepsilon t^{\frac{1}{2\alpha }}.\) Thus we find \(\left\| u\right\| _{{\textbf{X}}_{T_{1}}}<C\varepsilon ,\) which yields a desired contradiction. Lemma 3.4 is proved. \(\square \)
Next lemma states the asymptotics of the solution.
Lemma 3.5
Let \(\left\| u\right\| _{{\textbf{X}}_{\infty }}\le C\varepsilon .\) Then the asymptotics
is true for \(t\rightarrow \infty \) uniformly with respect to \(x\in {\mathbb {R}},\) where \(\delta >0.\)
Proof
In the domain \(\left| \xi \right| \ge \left\langle t\right\rangle ^{\nu }\), by the Sobolev embedding inequality we get
with \(\nu >\gamma .\) Therefore we need to compute the asymptotics of the function \(\widehat{\varphi }\) in the domain \(\left| \xi \right| \le \left\langle t\right\rangle ^{\nu }.\) Applying the operator \({{\mathcal {F}}}{{\mathcal {U}}} \left( -t\right) \) to Eq. (1.2), we get Eq. (2.1). Then by virtue of Lemma 3.1, we obtain
Then we change the dependent variable \(\widehat{\varphi }\left( t,\xi \right) =y\left( t,\xi \right) \Psi \left( t,\xi \right) \) with
to get
Integration by parts via the identity \(e^{i\tau \Omega }=\left( 1+i\tau \Omega \right) ^{-1}\partial _{\tau }\left( \tau e^{i\tau \Omega }\right) \) yields the estimate \(\left\| y\left( t\right) -y\left( s\right) \right\| _{{\textbf{L}}^{\infty }}\le C\varepsilon s^{-\delta _{2}}\) for all \(t>s>0,\) with some \(\delta _{2}>0.\) Therefore there exists a unique final state \(y_{+}\in {\textbf{L}}^{\infty }\) such that \(\left\| y\left( t\right) -y_{+}\right\| _{{\textbf{L}}^{\infty }}\le C\varepsilon t^{-\delta _{2}}\) for all \(t>0.\) Denote the remainder term \(\Phi \left( t\right) =\int _{1} ^{t}\left| y\left( \tau \right) \right| ^{2}\frac{d\tau }{\tau }-\left| y_{+}\right| ^{2}\log t,\) then we have
and \(\left\| \Phi \left( t\right) -\Phi \left( s\right) \right\| _{{\textbf{L}}^{\infty }}\le C\varepsilon ^{2}s^{-\delta _{2}}\) for all \(t>s>0.\) Hence there exists a unique real-valued function \(\Phi _{+}\), such that \(\Phi _{+}\in {\textbf{L}}^{\infty }\) and \(\left\| \Phi \left( t\right) -\Phi _{+}\right\| _{{\textbf{L}}^{\infty }}\le C\varepsilon ^{2}t^{-\delta _{2}}.\) Therefore we obtain
for all \(t>0.\) Then we have
for all \(t>0.\) Thus we get the large time asymptotics
where \(\widetilde{W_{+}}\left( \xi \right) =y_{+}\left( \xi \right) \exp \left( \frac{3i\xi ^{3}\theta \left( \xi \right) }{\Lambda ^{\prime \prime }\left( \xi \right) }\Phi _{+}\left( \xi \right) \right) .\) Note that \(\widetilde{W_{+}}\in {\textbf{L}}^{\infty }.\) Finally, by the factorization formula we get \(\left| \partial _{x}\right| ^{\frac{\alpha -2}{2} }u\left( t\right) =2{\text {Re}}{\mathcal {D}}_{t}{\mathcal {B}} M{\mathcal {Q}}\left( \left| \xi \right| ^{\frac{\alpha -2}{2}} \widehat{\varphi }\right) \). Hence applying Lemma 2.1, Lemma 2.2 and Lemma 2.3, we find
where \(W_{+}\left( \xi \right) =\frac{1}{\sqrt{i\Lambda ^{\prime \prime }\left( v\right) }}\widetilde{W_{+}}\left( \xi \right) .\) Note that \(\left| \xi \right| ^{\frac{\alpha -2}{2}}W_{+}\left( \xi \right) \in {\textbf{L}}^{\infty }.\) This completes the proof of the asymptotics. If we substitute the definitions of the operators \({\mathcal {D}}_{t}\) and \({\mathcal {B}}\), then the asymptotics has the form
Lemma 3.5 is proved. \(\square \)
4 Proof of Theorem 1.1
By Lemma 3.4 we see that a priori estimate \(\left\| u\right\| _{{\textbf{X}}_{\infty }}\le C\varepsilon \) is true. Therefore global existence of solutions of the Cauchy problem (1.2) satisfying estimate \(\left\| u\right\| _{{\textbf{X}}_{\infty }}\le C\varepsilon ,\) follows by a standard continuation argument via the local existence Theorem 3.1. Then the asymptotic formula (1.3) for the solutions u of the Cauchy problem (1.2) follows from Lemma 3.5. Theorem 1.1 is proved.
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Acknowledgements
We are grateful to unknown referee for many useful suggestions and comments. The work of P.I.N. is partially supported by CONACYT and PAPIIT project IN103221.
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Carreño-Bolaños, R., Naumkin, P.I. Large time asymptotics for the fractional modified Korteweg-de Vries equation of order \(\alpha \in \left[ 4,5\right) \). J. Pseudo-Differ. Oper. Appl. 14, 42 (2023). https://doi.org/10.1007/s11868-023-00536-4
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DOI: https://doi.org/10.1007/s11868-023-00536-4