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Xiangqian Yan

Publications and source records attributed to Xiangqian Yan.

10 recordsLinked to original sources
arXiv2026

Almost sure spatial decay and almost sure nonlinear smoothing of some stochastic dispersive equations

In this paper, we consider the almost sure nonlinear smoothing, the almost sure spatial decay and the almost sure uniform convergence of the stochastic mKdV equation and the stochastic cubic KdV-Benjamin-Ono equation. Firstly, for initial data $g\in H^{s}(\mathbb{R})(s\geq\frac{1}{4})$ and $Φ_{2}\in L_{2}^{0,s}$, we prove the local well-posedness for the stochastic cubic KdV-Benjamin-Ono equation. Secondly, we establish the almost sure nonlinear smoothing of the stochastic mKdV equation and the stochastic cubic KdV-Benjamin-Ono equation. Finally, by using the almost sure nonlinear smoothing, we obtain the almost sure spatial decay and the almost sure uniform convergence of the integral term in the pathwise solutions to the stochastic mKdV equation and the stochastic cubic KdV-Benjamin-Ono equation. More precisely, we have the following results: for the stochastic mKdV equation, let $s>\frac{1}{3}$, $f\in H^{s}(\mathbb{R})$ and $Φ_{1}\in L_{2}^{0,s}$. Then, the local pathwise solution $u$ satisfies \begin{eqnarray*} &&\mathbb{P}\Big(\Big\{ω: \lim_{t\rightarrow0}\Big\|u-U(t)f-\int_{0}^{t}U(t-s)Φ_{1}dW(s)\Big\|_{L_{x}^{\infty}}=0\Big\}\Big)=1,\\ &&\mathbb{P}\Big(\Big\{ω: \forall t\in[0,T_ω], \lim_{|x|\rightarrow\infty}\Big(u-U(t)f-\int_{0}^{t}U(t-s)Φ_{1}dW(s)\Big)=0\Big\}\Big)=1. \end{eqnarray*} For the stochastic cubic KdV-Benjamin-Ono equation, let $s>\frac{1}{3}$, $g\in H^{s}(\mathbb{R})$ and $Φ_{2}\in L_{2}^{0,s}$. Then, the local pathwise solution $v$ satisfies \begin{eqnarray*} &&\mathbb{P}\Big(\Big\{ω: \lim_{t\rightarrow0}\Big\|v-V(t)g-\int_{0}^{t}V(t-s)Φ_{2}dW(s)\Big\|_{L_{x}^{\infty}}=0\Big\}\Big)=1,\\ &&\mathbb{P}\Big(\Big\{ω: \forall t\in[0,T_ω],\lim_{|x|\rightarrow\infty}\Big(v-V(t)g-\int_{0}^{t}V(t-s)Φ_{2}dW(s)\Big)=0\Big\}\Big)=1. \end{eqnarray*}

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arXiv2026

The probabilistic convergence problem of density functions related to $\partial_{x}^{3}+\partial_{x}^{-1}$

In this article, by using full randomization introduced by Hadama and Yamamoto (Probabilistic Strichartz estimates in Schatten classes and their applications to Hartree equation, J. Math. Phys. 67(2026), 35pp) and high-low frequency technique as well as the property of $\mathfrak{S}^{2}$, we establish the probabilistic convergence of the density function related to $\partial_{x}^{3}+\partial_{x}^{-1}$ on $\R$, which extends the Theorem 1.3 of Yan et al. (Convergence problem of Ostrovsky equation with rough data and random data, Indiana Univ. Math. J. 71(2022), 1897-1921.).

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arXiv2026

Spatial decay and nonlinear smoothing of the sixth-order Boussinesq equation

In this paper, we study the initial value problem of the sixth-order Boussinesq equation with quadratic and cubic nonlinearities in arbitrary spatial dimensions. First, by using the Fourier restriction norm method and a high-low frequency decomposition, we establish the nonlinear smoothing for this equation, namely, the integral form of the solution to the Duhamel formulation enjoys higher regularity than its linear counterpart. Finally, by using the nonlinear smoothing, we establish the uniform convergence of the integral term and its spatial decay for each fixed $t$.

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arXiv2026

Spatial decay and nonlinear smoothing of the generalized Ostrovsky equation

This paper is devoted to studying the generalized Ostrovsky equation \begin{eqnarray*} u_{t}-β\partial_{x}^{3}u-γ\partial_{x}^{-1}u+\frac{1}{k+1}(u^{k+1})_{x}=0,k\geq5 \end{eqnarray*} with $β<0,γ>0$. Firstly, by using the density theorem in the mixed Lebesgue spaces, we prove that $X_{s,b}\hookrightarrow C(\mathbb{R};H^{s}(\mathbb{R})) \hookrightarrow C(\mathbb{R};L_{x}^{\infty})$ with $s>1/2,b>1/2.$ Secondly, we present a new proof of the convergence problem of linear Ostrovsky equation, which is slightly different from the proof of Theorem 1.1 (Convergence problem of Ostrovsky equation with rough data and random data, Indiana Univ. Math. J. 71(2022), 1897-1921.) Thirdly, we investigate the pointwise convergence problem of the generalized Ostrovsky equation. Fourthly, for the solution $u$ to the Cauchy problem for the generalized Ostrovsky equation, we prove that $u=u_{1}+u_{2},t\in[-δ,δ]$, and $u_{2}$ possesses better regularity than $u$, where $u_{1}$ is the linear part of $u$ and $u_{2}$ is the nonlinear integral part. Fifthly, we investigate the nonlinear smoothing and the uniform convergence problem of the generalized Ostrovsky equation. Finally, when data $f$ belongs to $H^{s}(\mathbb{R})(s>\frac{1}{2}-\frac{2}{k+1},k\geq6)$ and $\lim\limits_{|x|\rightarrow{\infty}}f=0$ and $\mathscr{F}_{x}(U(t)f)\in L^{1}(\mathbb{R}),$ for $t\in [-δ,δ],$ we prove that $\lim\limits_{|x|\rightarrow{\infty}}u=0$. The key ingredients are high-low frequency technique, maximal function estimates related to low frequency and some Strichartz estimates which can be proved with the aid of the Stein complex interpolation Theorem.

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arXiv2026

The Cauchy problem for the generalized KdV equation in the Sobolev space $H^{s}(\mathbf{R})$

In this paper, we are concerned with the Cauchy problem for the generalized KdV equation with random data and rough data. Firstly, when $s\in\mathbf{R}$, by using the initial value randomization technique introduced by Shen et al. (arXiv:2111.11935) and the construction of appropriate auxiliary spaces, we establish the almost sure local well-posedness of the generalized KdV equation in $H^{s}(\mathbf{R})$, which improves Theorem 1.3 of Hwang and Kwak (Proc. Amer. Math. Soc. 146(2018), 267-280.) and Theorem 1.5 of Yan et al.(arXiv:2011.07128.). Secondly, by using the well-posedness results proved in Theorem 1.1, for $f\in H^{s}(\mathbf{R}),\, s\in\mathbf{R}$, we obtain \begin{eqnarray*} &&\mathbb{P}\left(\left\{ω:\lim_{t\rightarrow0}\|u(t,x)-U(t)f^ω(x)\|_{L_{x}^{\infty}}=0\right\}\right)=1, \end{eqnarray*} which improves Theorem 1.6 of Yan et al.(arXiv:2011.07128.). Thirdly, by using the dyadic decomposition and constructing appropriate function spaces, we establish nonlinear smoothing for the generalized KdV equation with rough data. Furthermore, by using this estimate, when data $f\in H^{s}(\mathbf{R})\cap\hat{L}^{\infty}(\mathbf{R}),\, s>\frac{1}{2}-\frac{2}{k+1},\, k\geq4$, we obtain \begin{eqnarray*} &&\lim_{|x|\rightarrow \infty}u(t,x)=0,\quad t\in[0, T]. \end{eqnarray*} In particular, for $f(x)\in H^{s}(\mathbf{R}),\,s>\frac{1}{2}-\frac{2}{k+1},\,k\geq4$, we prove \begin{eqnarray*} &&\lim_{|x|\rightarrow \infty}(u(t,x)-U(t)f(x))=0. \end{eqnarray*} Finally, by using Theorem 1.1, when $f\in H^{s}(\mathbf{R}),\, s\in\mathbf{R}$, we obtain \begin{eqnarray*} &&\mathbb{P}\left(\left\{ω: \forall t\in I_ω, \lim_{|x|\rightarrow \infty}\left(u(t,x)-U(t)f^ω(x)\right)=0\right\}\right)=1. \end{eqnarray*}

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arXiv2024

Strichartz estimates for orthonormal functions and convergence problem of density functions of Boussinesq operator on manifolds

This paper is devoted to studying the maximal-in-time estimates and Strichartz estimates for orthonormal functions and convergence problem of density functions related to Boussinesq operator on manifolds. Firstly, we present the pointwise convergence of density function related to Boussinesq operator with $γ_{0}\in\mathfrak{S}^β(\dot{H}^{\frac{1}{4}}(\mathbf{R}))(β<2)$ with the aid of the maximal-in-time estimate related to Boussinesq operator with orthonormal function on $\R$. Secondly, we present the pointwise convergence of density function related to Boussinesq operator with $γ_{0}\in\mathfrak{S}^β(\dot{H}^{s})(\frac{d}{4}\leq s<\frac{d}{2},\, 0<α\leq d, 1\leqβ<\fracα{d-2s})$ with the aid of the maximal-in-time estimates related to Boussinesq operator with orthonormal function on the unit ball $\mathbf{B}^{d}(d\geq1)$ established in this paper; we also present the Hausdorff dimension of the divergence set of density function related to Boussinesq operator $dim_{H}D(γ_{0})\leq (d-2s)β$. Thirdly, we show the Strichartz estimates for orthonormal functions and Schatten bound with space-time norms related to Boussinesq operator on $\mathbf{T}$ with the aid of the noncommutative-commutative interpolation theorems established in this paper, which are just Lemmas 3.1-3.4 in this paper; we also prove that Theorems 1.5, 1.6 are optimal. Finally, by using full randomization, we present the probabilistic convergence of density function related to Boussinesq operator on $\R$, $\mathbf{T}$ and $Θ=\{x\in\R^{3}:|x|<1\}$ with $γ_{0}\in\mathfrak{S}^{2}$.

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arXiv2021

Convergence problem of the Kawahara equation on the real line

In this paper, we consider the convergence problem of the Kawahara equation \begin{eqnarray*} &&u_{t}+α\partial_{x}^{5}u+β\partial_{x}^{3}u+\partial_{x}(u^{2})=0 \end{eqnarray*} on the real line with rough data. Firstly, by using Strichartz estimates as well as high-low frequency idea, we establish two crucial bilinear estimates, which are just Lemmas 3.1-3.2 in this paper; we also present the proof of Lemma 3.3 which shows that $s>-\frac{1}{2}$ is necessary for Lemma 3.2. Secondly, by using frequency truncated technique and high-low frequency technique, we show the pointwise convergence of the Kawahara equation with rough data in $H^{s}(\R)(s\geq\frac{1}{4})$; more precisely, we prove \begin{eqnarray*} &&\lim\limits_{t\rightarrow0}u(x,t)=u(x,0), \qquad a.e. x\in\R, \end{eqnarray*} where $u(x,t)$ is the solution to the Kawahara equation with initial data $u(x,0).$ Lastly, we show \begin{eqnarray*} &&\lim\limits_{t\rightarrow0}\sup\limits_{x\in\SR}|u(x,t)-U(t)u_{0}|=0 \end{eqnarray*} with rough data in $H^{s}(\R)(s>-\frac{1}{2})$.

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arXiv2021

The Cauchy problem for the generalized KdV equation with rough data and random data

In this paper, we consider the Cauchy problem for the generalized KdV equation with rough data and random data. Firstly, we prove that $u(x,t)\longrightarrow u(x,0)$ as $t\longrightarrow0$ for a.e. $x\in \mathbb{R}$ with $u(x,0)\in H^{s}(\mathbb{R})(s>\frac{1}{2}-\frac{2}{k},k\geq8).$ Secondly, we prove that $u(x,t)\longrightarrow e^{-t\partial_{x}^{3}}u(x,0)$ as $t\longrightarrow0$ for a.e. $x\in \mathbb{R}$ with $u(x,0)\in H^{s}(\mathbb{R})(s>\frac{1}{2}-\frac{2}{k},k\geq8).$ Thirdly, we prove that $\lim\limits_{t\longrightarrow 0}\left\|u(x,t)-e^{-t\partial_{x}^{3}}u(x,0)\right\|_{L_{x}^{\infty}}=0$ with $u(x,0)\in H^{s}(\mathbb{R})(s>\frac{1}{2}-\frac{2}{k+1},k\geq5)$. Fourthly, by using Strichartz estimates, probabilistic Strichartz estimates, we establish the probabilistic well-posedness in $H^{s}(\mathbb{R})\left(s>{\rm max} \left\{\frac{1}{k+1}\left(\frac{1}{2}-\frac{2}{k}\right), \frac{1}{6}-\frac{2}{k}\right\}\right)$ with random data. Our result improves the result of Hwang, Kwak (Proc. Amer. Math. Soc. 146(2018), 267-280.). Fifthly, we prove that $\forall ε>0,$ $\forall ω\in Ω_{T},$ $\lim\limits_{t\longrightarrow0}\left\|u(x,t)-e^{-t\partial_{x}^{3}}u^ω(x,0)\right\|_{L_{x}^{\infty}}=0$ with $u(x,0)\in H^{s}(\mathbb{R})(s>\frac{1}{6},k\geq6),$ where ${\rm P}(Ω_{T})\geq 1- C_{1}{\rm exp} \left(-\frac{C}{T^{\fracε{48k}}\|u(x,0)\|_{H^{s}}^{2}}\right)$ and $u^ω(x,0)$ is the randomization of $u(x,0)$. Finally, we prove that $\forall ε>0,$ $\forall ω\in Ω_{T}, \lim\limits_{t\longrightarrow0}\left\|u(x,t)-u^ω(x,0)\right\|_{L_{x}^{\infty}}=0$ with ${\rm P}(Ω_{T})\geq 1- C_{1}{\rm exp} \left(-\frac{C}{T^{\fracε{48k}}\|u(x,0)\|_{H^{s}}^{2}}\right)$ and $u(x,0)\in H^{s}(\mathbb{R})(s>\frac{1}{6},k\geq6)$.

math.AP↗
arXiv2021

The Cauchy problem for the generalized Ostrovsky equation with negative dispersion

This paper is devoted to studying the Cauchy problem for the generalized Ostrovsky equation \begin{eqnarray*} u_{t}-β\partial_{x}^{3}u-γ\partial_{x}^{-1}u+\frac{1}{k+1}(u^{k+1})_{x}=0,k\geq5 \end{eqnarray*} with $βγ<0,γ>0$. Firstly, we prove that the Cauchy problem for the generalized Ostrovsky equation is locally well-posed in $H^{s}(\mathbb{R})\left(s>\frac{1}{2}-\frac{2}{k}\right)$. Then, we prove that the Cauchy problem for the generalized Ostrovsky equation is locally well-posed in $X_{s}(\mathbb{R}): =\|f\|_{H^{s}}+\left\|\mathscr{F}_{x}^{-1}\left(\frac{\mathscr{F}_{x} f(ξ)}ξ\right)\right\|_{H^{s}}\left(s>\frac{1}{2}-\frac{2}{k}\right).$ Finally, we show that the solution to the Cauchy problem for generalized Ostrovsky equation converges to the solution to the generalized KdV equation as the rotation parameter $γ$ tends to zero for data belonging to $X_{s}(\mathbb{R})(s>\frac{3}{2})$. The main difficulty is that the phase function of Ostrosvky equation with negative dispersive $βξ^{3}+\fracγξ$ possesses the zero singular point.

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arXiv2021

Convergence problem of Schrödinger equation in Fourier-Lebesgue spaces with rough data and random data

In this paper, we consider the convergence problem of Schrödinger equation. Firstly, we show the almost everywhere pointwise convergence of Schrödinger equation in Fourier-Lebesgue spaces $\hat{H}^{\frac{1}{p},\frac{p}{2}}(\mathbb{R})(4\leq p<\infty),$ $\hat{H}^{\frac{3 s_{1}}{p},\frac{2p}{3}}(\mathbb{R}^2)(s_{1}>\frac{1}{3},3\leq p<\infty),$ $\hat{H}^{\frac{2 s_{1}}{p},p}(\mathbb{R}^n)(s_{1}>\frac{n}{2(n+1)},2\leq p<\infty,n\geq3)$ with rough data. Secondly, we show that the maximal function estimate related to one Schrödinger equation can fail with data in $\hat{H}^{s,\frac{p}{2}}(\mathbb{R})(s<\frac{1}{p})$. Finally, we show the stochastic continuity of Schrödinger equation with random data in $\hat{L}^{r}(\mathbb{R}^n)(2\leq r<\infty)$ almost surely. The main ingredients are Lemmas 2.4, 2.5, 3.2-3.4.

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