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Bing-Yu Zhang

Publications and source records attributed to Bing-Yu Zhang.

At least 19 recordsLinked to original sources
arXiv2023

Effect of lower order terms on the well-posedness of Majda-Biello systems

This paper investigates a noteworthy phenomenon within the framework of Majda-Biello systems, wherein the inclusion of lower-order terms can enhance the well-posedness of the system. Specifically, we investigate the initial value problem (IVP) of the following system: \[ \left\{ \begin{array}{l} u_{t} + u_{xxx} = - v v_x, v_{t} + αv_{xxx} + βv_x = - (uv)_{x}, (u,v)|_{t=0} = (u_0,v_0) \in H^{s}(\mathbb{R}) \times H^{s}(\mathbb{R}), \end{array} \right. \quad x \in \mathbb{R}, \, t \in \mathbb{R}, \] where $α\in \mathbb{R}\setminus \{0\}$ and $β\in \mathbb{R}$. Let $s^{*}(α, β)$ be the smallest value for which the IVP is locally analytically well-posed in $H^{s}(\mathbb{R})\times H^{s}(\mathbb{R}) $ when $s > s^{}(α, β)$. Two interesting facts have already been known in literature: $s^{*}(α, 0) = 0$ for $α\in (0,4)\setminus\{1\}$ and $s^*(4,0) = \frac34$. Our key findings include the following: For $s^{*}(4,β)$, a significant reduction is observed, reaching $\frac12$ for $β> 0$ and $\frac14$ for $β< 0$. Conversely, when $α\neq 4$, we demonstrate that the value of $β$ exerts no influence on $s^*(α, β)$. These results shed light on the intriguing behavior of Majda-Biello systems when lower-order terms are introduced and provide valuable insights into the role of $α$ and $β$ in the well-posedness of the system.

math.AP↗
arXiv2023

Non-homogeneous boundary value problems for coupled KdV-KdV systems posed on the half line

In this article, we study an initial-boundary-value problem of a coupled KdV-KdV system on the half line $ \mathbb{R}^+ $ with non-homogeneous boundary conditions: \begin{equation*} \left\{ \begin{array}{l} u_t+v_x+u u_x+v_{xxx}=0, \quad v_t+u_x+(vu)_x+u_{xxx}=0, \quad u(x,0)=ϕ(x),\quad v(x,0)=ψ(x), \quad u(0,t)=h_1(t),\quad v(0,t)=h_2(t),\quad v_x(0,t)=h_3(t), \end{array} \right. \qquad x,\,t>0. \end{equation*} It is shown that the problem is locally unconditionally well-posed in $H^s(\mathbb{R}^+)\times H^s(\mathbb{R}^+)$ for $s> -\frac34 $ with initial data $(ϕ,ψ)$ in $H^s(\mathbb{R}^+)\times H^{s}(\mathbb{R}^+)$ and boundary data $(h_1,h_2,h_3) $ in $H^{\frac{s+1}{3}}(\mathbb{R}^+)\times H^{\frac{s+1}{3}}(\mathbb{R}^+)\times H^{\frac{s}{3}}(\mathbb{R}^+)$. The approach developed in this paper can also be applied to study more general KdV-KdV systems posed on the half line.

math.AP↗
arXiv2022

Well-posedness and Critical Index Set of the Cauchy Problem for the Coupled KdV-KdV Systems on $\mathbb{T}$

Studied in this paper is the well-posedness of the Cauchy problem for the coupled KdV-KdV systems \[ u_t+a_1u_{xxx} = c_{11}uu_x+c_{12}vv_x+d_{11}u_{x}v+d_{12}uv_{x}, \quad u(x,0)= u_0(x) \] \[ v_t+a_2v_{xxx}= c_{21}uu_x+c_{22}vv_x +d_{21}u_{x}v+d_{22}uv_{x}, \quad v(x,0)=v_0(x)\] posed on the torus $\mathbb{T}$ in the spaces \[ {\cal H}^s_1:=H^s_0 (\mathbb{T})\times H^s_0 (\mathbb{T}), \quad {\cal H}^s_2:=H^s_0 (\mathbb{T})\times H^s(\mathbb{T}), \quad {\cal H}^s_3:=H^s (\mathbb{T})\times H^s_0 (\mathbb{T}), \quad {\cal H}^s_4:=H^s (\mathbb{T})\times H^s (\mathbb{T}).\] For $k=1,2,3,4$, it is shown that for given $a_1$, $a_2$, $(c_{ij})$ and $(d_{ij})$, there exists a unique $s^*_k \in (-\infty, +\infty]$, called the critical index, such that the system is analytically well-posed in $\cal{H}^s_k$ for $s>s^*_k$ while the bilinear estimate, the key for the proof of the analytical well-posedness, fails if $s<s^{*}_k$. Viewing the critical index $s^*_k$ as a function of the coefficients $a_1$, $a_2$, $(c_{ij})$ and $(d_{ij})$, its range $\cal{C}_k$ is called the critical index set for the analytical well-posedness of the system in the space $\cal{H}^s_k$. Invoking some classical results of Diophantine approximation in number theory, we are able to identify that \[ \mbox{$ {\cal C}_1= \left \{ -\frac12, \infty \right\} \bigcup \left \{ α: \frac12\leq α\leq 1 \right \}$ } \quad\text{and}\quad \mbox{${\cal C}_q= \left \{ -\frac12, -\frac14, \infty \right\} \bigcup \left \{ α: \frac12\leq α\leq 1 \right \}$ $\quad$ for $\quad$ $q=2,3,4$.}\] This is in sharp contrast to the $R$ case in which the critical index set ${\cal C}$ for the analytical well-posedness of in the space $H^s (R)\times H^s (R)$ consists of exactly four numbers: $ {\cal C}=\left \{ -\frac{13}{12}, -\frac34, 0, \frac34 \right \}.$

math.AP↗
arXiv2022

Local Well-posedness of the Coupled KdV-KdV Systems on $\mathbb{R}$

Inspired by the recent successful completion of the study of the well-posedness theory for the Cauchy problem of the Korteweg-de Vries (KdV) equation \[ u_t +uu_x +u_{xxx}=0, \quad \left. u \right |_{t=0}=u_{0} \] in the space $H^{s} (\mathbb{R})$ (or $H^{s} (\mathbb{T})$), we study the well-posedness of the Cauchy problem for a class of coupled KdV-KdV (cKdV) systems \[\left\{\begin{array}{rcl} u_t+a_{1}u_{xxx} &=& c_{11}uu_x+c_{12}vv_x+d_{11}u_{x}v+d_{12}uv_{x},\\ v_t+a_{2}v_{xxx}&=& c_{21}uu_x+c_{22}vv_x +d_{21}u_{x}v+d_{22}uv_{x},\\ \left. (u,v)\right |_{t=0} &=& (u_{0},v_{0}) \end{array}\right.\] in the space $\mathcal{H}^s (\mathbb{R}) := H^s (\mathbb{R})\times H^s (\mathbb{R})$. Typical examples include the Gear-Grimshaw system, the Hirota-Satsuma system and the Majda-Biello system, to name a few. In this paper we look for those values of $s\in \mathbb{R}$ for which the cKdV systems are well-posed in $\mathcal{H}^s (\mathbb{R})$. Our findings enable us to provide a complete classification for the cKdV systems in terms of the analytical well-posedness in $\mathcal{H}^s (\mathbb{R})$ based on its coefficients $a_i$, $c_{ij}$ and $d_{ij}$ for $i,j=1,2$. The key ingredients in the proofs are the bilinear estimates under the Fourier restriction space norms. There are four types of the bilinear estimates that need to be investigated. Sharp results are established for all of them. In contrast to the lone critical index $-\frac{3}{4}$ for the single KdV equation, the critical indexes for the cKdV systems are $-\frac{13}{12}$, $-\frac{3}{4}$, $0$ and $\frac{3}{4}$. As a result, the cKdV systems are classified into four classes, each of which corresponds to a unique index $s^{*}\in\{-\frac{13}{12},\,-\frac{3}{4},\,0,\,\frac{3}{4}\}$ such that any system in this class is locally analytically well-posed if $s>s^{*}$ while the bilinear estimate fails if $s<s^{*}$.

math.AP↗
arXiv2021

Lower Regularity Solutions of the Non-homogeneous Boundary-Value Problem for a Higher Order Boussinesq Equation in a Quarter Plane

We continue to study the initial-boundary-value problem of the sixth order Boussinesq equation in a quarter plane with non-homogeneous boundary conditions: \begin{equation*} \begin{cases} u_{tt}-u_{xx}+βu_{xxxx}-u_{xxxxxx}+(u^2)_{xx}=0,\quad x,t\in \mathbb{R}^+,\\ u(x,0)=φ(x), u_t(x,0)=ψ''(x), \\ u(0,t)=h_1(t), u_{xx}(0,t)=h_2(t), u_{xxxx}(0,t)=h_3(t), \end{cases} \end{equation*} where $β=\pm1$. We show that the problem is locally analytically well-posed in the space $H^s(\mathbb{R}^+)$ for any $ s> -\frac34 $ with the initial-value data $$(φ,ψ)\in H^s(\mathbb{R}^+)\times H^{s-1}(\mathbb{R}^+)$$ and the boundary-value data $$(h_1,h_2,h_3) \in H^{\frac{s+1}{3}}(\mathbb{R}^+)\times H^{\frac{s-1}{3}}(\mathbb{R}^+)\times H^{\frac{s-3}{3}}(\mathbb{R}^+).$$

math.AP↗
arXiv2018

Exact controllability and stability of the Sixth Order Boussinesq equation

The article studies the exact controllability and the stability of the sixth order Boussinesq equation \[ u_{tt}-u_{xx}+βu_{xxxx}-u_{xxxxxx}+(u^2)_{xx}=f, \quad β=\pm1, \] on the interval $S:=[0,2π]$ with periodic boundary conditions. It is shown that the system is locally exactly controllable in the classic Sobolev space, $H^{s+3}(S)\times H^s(S)$ for $s\geq 0$, for "small" initial and terminal states. It is also shown that if $f$ is assigned as an internal linear feedback, the solution of the system is uniformly exponential decay to a constant state in $H^{s+3}(S)\times H^s(S)$ for $s\geq 0$ with "small" initial data assumption.

math.AP↗
arXiv2018

Hexagonal standing wave patterns of a two-dimensional Boussinesq system

We prove the existence of a large family of two-dimensional standing waves, that are triple periodic solutions, for a Boussinesq system which describes two-way propagation of water waves in a channel. Our proof uses the Lyapunov-Schmidt method to find the bifurcation standing waves.

math.AP↗
arXiv2017

General Boundary Value Problems of the Korteweg-de Vries Equation on a Bounded Domain

In this paper we consider the initial boundary value problem of the Korteweg-de Vries equation posed on a finite interval \begin{equation} u_t+u_x+u_{xxx}+uu_x=0,\qquad u(x,0)=ϕ(x), \qquad 0 0 \qquad (1) \end{equation} subject to the nonhomogeneous boundary conditions, \begin{equation} B_1u=h_1(t), \qquad B_2 u= h_2 (t), \qquad B_3 u= h_3 (t) \qquad t>0 \qquad (2) \end{equation} where \[ B_i u =\sum _{j=0}^2 \left(a_{ij} \partial ^j_x u(0,t) + b_{ij} \partial ^j_x u(L,t)\right), \qquad i=1,2,3,\] and $a_{ij}, \ b_{ij}$ $ (j,i=0, 1,2,3)$ are real constants. Under some general assumptions imposed on the coefficients $a_{ij}, \ b_{ij}$, $ j,i=0, 1,2,3$, the IBVPs (1)-(2) is shown to be locally well-posed in the space $H^s (0,L)$ for any $s\geq 0$ with $ϕ\in H^s (0,L)$ and boundary values $h_j, j=1,2,3$ belonging to some appropriate spaces with optimal regularity.

math.AP↗
arXiv2017

Nonhomogeneous Boundary Value Problems of Nonlinear Schrödinger Equations in a Half Plane

This paper discusses the initial-boundary-value problems (IBVP) of nonlinear Schrödinger equations posed in a half plane $\mathbb{R} \times \mathbb{R}^+$ with nonhomogeneous Dirichlet boundary conditions. For any given $s \ge 0$, if the initial data $φ(x, y)$ are in Sobolev space $H^s(\mathbb{R}\times \mathbb{R}^+) $ with the boundary data $ h ( x, t) $ in an optimal space ${\cal H}^s(0,T)$ as defined in the introduction, which is slightly weaker than the space $$H^{(2s+1)/4}_{t} ([0, T]; L_x^2(\mathbb{R} ) ) \cap L^2_t ( [ 0, T]; H^{s+ 1/2} _x ( \mathbb{R} ) ),$$ the local well-posedness of the IBVP in $ C ( [0, T] ; H^s ( \mathbb{R}\times \mathbb{R}^+ ) )$ is proved. The global well-posedness is also discussed for $s = 1$. The main idea of the proof is to derive a boundary integral operator for the corresponding nonhomogeneous boundary condition and obtain the Strichartz's estimates for this operator. The results presented in the paper hold for the IBVP posed in a half space $ \mathbb{R}^n\times \mathbb{R}^+$ with any $n>1$.

math.AP↗
arXiv2016

Nonhomogeneous Boundary-Value Problems for One-Dimensional Nonlinear Schrödinger Equations

This paper is concerned with initial-boundary-value problems (IBVPs) for a class of nonlinear Schrödinger equations posed either on a half line $\mathbb{R}^+$ or on a bounded interval $(0, L)$ with nonhomogeneous boundary conditions. For any $s$ with $0\leq s < 5/2$ and $s \not = 3/2$, it is shown that the relevant IBVPs are locally well-posed if the initial data lie in the $L^2$--based Sobolev spaces $H^s(\mathbb{R}^+) $ in the case of the half line and in $H^s (0, L)$ on a bounded interval, provided the boundary data are selected from $H^{(2s+1)/4}_{loc} (\mathbb{R}^+)$ and $H^{(s+ 1) /2}_{loc} (\mathbb{R}^+)$, respectively. (For $s > \frac12$, compatibility between the initial and boundary conditions is also needed.) Global well-posedness is also discussed when $s \ge 1$. From the point of view of the well-posedness theory, the results obtained reveal a significant difference between the IBVP posed on $\mathbb{R}^+$ and the IBVP posed on $(0,L)$. The former is reminiscent of the theory for the pure initial-value problem (IVP) for these Schrödinger equations posed on the whole line $\mathbb{R}$ while the theory on a bounded interval looks more like that othe pure IVP posed on a periodic domain. In particular, the regularity demanded of the boundary data for the IBVP on $\mathbb{R}^+$ is consistent with the temporal trace results that obtain for solutions of the pure IVP on $\mathbb{R}$, while the slightly higher regularity of boundary data for the IBVP on $(0, L)$ resembles what is found for temporal traces of spatially periodic solutions.

math.AP↗
arXiv2016

A nonhomogeneous boundary value problem for the Kuramoto-Sivashinsky equation in a quarter plane

We study the initial boundary value problem for one-dimensional Kuramoto-Sivashinsky equation with nonhomogeneous boundary conditions. Through the analysis of the boundary integral operator, and applying the known results on the Cauchy problem, we obtain both the local well-posedness and the global well-posedness for the nonhomogeneous initial boundary value problem. It is shown that the Kuramoto-Sivashinsky equation is well-posed in Sobolev space $C([0,T]; H^s (R^+)) \bigcap L^2(0,T; H^{s+2}(R^+))$ for $s>-2$.

math.AP↗
arXiv2012

Boundary Controllability of the Korteweg-de Vries Equation on a Bounded Domain

This paper is devoted to study boundary controllability of the Korteweg-de Vries equation posed on a finite interval, in which, because of the third-order character of the equation, three boundary conditions are required to secure the well-posedness of the system. We consider the cases where one, two, or all three of those boundary data are employed as boundary control inputs. The system is first linearized around the origin and the corresponding linear system is shown to be exactly boundary controllable if using two or three boundary control inputs. In the case where only one control input is allowed to be used, the linearized system is known to be only \emph{null} controllable if the single control input acts on the left end of the spatial domain. By contrast, if the single control input acts on the right end of the spatial domain, the linearized system is exactly controllable if and only if the length of the spatial domain does not belong to a set of critical values. Moreover, the nonlinear system is shown to be locally exactly boundary controllable via contraction mapping principle if the associated linearized system is exactly controllable.

math.AP↗
arXiv2012

Unique continuation property and control for the Benjamin-Bona-Mahony equation on the torus

We consider the Benjamin-Bona-Mahony (BBM) equation on the one dimensional torus T = R/(2πZ). We prove a Unique Continuation Property (UCP) for small data in H^1(T) with nonnegative zero means. Next we extend the UCP to certain BBM-like equations, including the equal width wave equation and the KdV-BBM equation. Applications to the stabilization of the above equations are given. In particular, we show that when an internal control acting on a moving interval is applied in BBM equation, then a semiglobal exponential stabilization can be derived in H^s(T) for any s \geq 1. Furthermore, we prove that the BBM equation with a moving control is also locally exactly controllable in H^s(T) for any s \geq 0 and globally exactly controllable in H s (T) for any s \geq 1.

math.AP↗
arXiv2010

Global Well-posedness and Asymptotic Behavior of a Class of Initial-Boundary-Value Problem of the Korteweg-de Vries Equation on a Finite Domain

In this paper, we study a class of initial boundary value problem (IBVP) of the Korteweg- de Vries equation posed on a finite interval with nonhomogeneous boundary conditions. The IBVP is known to be locally well-posed, but its global $L^2 $ a priori estimate is not available and therefore it is not clear whether its solutions exist globally or blow up in finite time. It is shown in this paper that the solutions exist globally as long as their initial value and the associated boundary data are small, and moreover, those solutions decay exponentially if their boundary data decay exponentially

math.AP↗
arXiv2010

Control and Stabilization of the Nonlinear Schroedinger Equation on Rectangles

This paper studies the local exact controllability and the local stabilization of the semilinear Schrödinger equation posed on a product of $n$ intervals ($n\ge 1$). Both internal and boundary controls are considered, and the results are given with periodic (resp. Dirichlet or Neumann) boundary conditions. In the case of internal control, we obtain local controllability results which are sharp as far as the localization of the control region and the smoothness of the state space are concerned. It is also proved that for the linear Schrödinger equation with Dirichlet control, the exact controllability holds in $H^{-1}(Ω)$ whenever the control region contains a neighborhood of a vertex.

math.AP↗
arXiv2009

Control and Stabilization of the Korteweg-de Vries Equation on a Periodic Domain

This paper aims at completing an earlier work of Russell and Zhang to study internal control problems for the distributed parameter system described by the Korteweg-de Vries equation on a periodic domain T^1. In their article, Russell and Zhang showed that the system is locally exactly controllable and locally exponentially stabilizable when the control acts on an arbitrary nonempty subdomain of T^1. In this paper, we show that the system is in fact globally exactly controllable and globally exponentially stabilizable. The global exponential stabilizability corresponding to a natural feedback law is first established with the aid of certain properties of propagation of compactness and propagation of regularity in Bourgain spaces for solutions of the associated linear system. Then, using a different feedback law, the resulting closed-loop system is shown to be locally exponentially stable with an arbitrarily large decay rate. A time-varying feedback law is further designed to ensure a global exponential stability with an arbitrary large decay rate.

math.AP↗