Stephen C. Anco, Huijun He, Zhijun Qiao
The Cauchy problem for a unified family of integrable $U(1)$-invariant peakon equations from the NLS hierarchy is studied. As main results, local well-posedness is proved in Besov spaces, and blow-up is established through use of an $L^1$ conservation law.
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This paper is devoted to an integrable two-component Camassa-Holm system with cubic nonlinearity, which includes the cubic Camassa-Holm equation (also called the Fokas-Olver-Rosenau-Qiao equation) as a special case. The one peaked solitons (peakons) and two peakon solutions are described in an explicit formula. Then, the local well-posedness for the Cauchy problem of the system is studied. Moreover, we target at the precise blow-up scenario for strong solutions to the system, and establish a new blow-up result with respect to the initial data.
In this paper we mainly investigate the initial value problem of the periodic Euler-Poincaré equations. We first present a new blow-up result to the system for a special class of smooth initial data by using the rotational invariant properties of the system. Then, we prove that the periodic Euler-Poincaré equations is ill-posed in critical Besov spaces by a contradiction argument. Finally, we verify the system possesses a class of peakon solutions in the sense of distributions.
This paper is contributed to study the Cauchy problem of a new integrable two-component system with peaked soliton (peakon) and weak kink solutions. We first establish the local well-posedness result for the Cauchy problem in Besov spaces, and then present a precise blow-up scenario and a new blow-up result for strong solutions to the system.
It is known from the previous works that the peakon solutions of the Novikov equation are orbitally and asymptotically stable in $H^1$. We prove, via the method of characteristics, that these peakon solutions are unstable under $W^{1,\infty}$-perturbations. Moreover, we show that small initial $W^{1,\infty}$-perturbations of the Novikov peakons can lead to the finite time blow-up of the corresponding solutions.
Consideration here is a generalized $μ$-type integrable equation, which can be regarded as a generalization to both the $μ$-Camassa-Holm and modified $μ$-Camassa-Holm equations. It is shown that the proposed equation is formally integrable with the Lax-pair and the bi-Hamiltonian structure and its scale limit is an integrable model of hydrodynamical systems describing short capillary-gravity waves. Local well-posedness of the Cauchy problem in the suitable Sobolev space is established by the viscosity method. Existence of peaked traveling-wave solutions and formation of singularities of soluti…
This paper investigates the Cauchy problem of a generalized Camassa-Holm equation with quadratic and cubic nonlinearities (alias the mCH-Novikov-CH equation), which is a generalization of some special equations such as the Camassa-Holm (CH) equation, the modified CH (mCH) equation ((alias the Fokas-Olver-Rosenau-Qiao equation), the Novikov equation, the CH-mCH equation, the mCH-Novikov equation, and the CH-Novikov equation. We first show the local well-posedness for the strong solutions of the mCH-Novikov-CH equation in Besov spaces by means of the Littlewood-Paley theory and the transport equ…
The Degasperis-Procesi equation can be derived as a member of a one-parameter family of asymptotic shallow water approximations to the Euler equations with the same asymptotic accuracy as that of the Camassa-Holm equation. In this paper, we study the orbital stability problem of the peaked solitons to the Degasperis-Procesi equation on the line. By constructing a Liapunov function, we prove that the shapes of these peakon solitons are stable under small perturbations.
Using the tri-hamiltonian splitting method, the authors of [Anco and Mobasheramini, Physica D, 355:1--23, 2017] derived two $U(1)$-invariant nonlinear PDEs that arise from the hierarchy of the nonlinear Schrödinger equation and admit peakons ($non-smooth\ solitons$). In the present paper, these two peakon PDEs are generalized to a family of $U(1)$-invariant peakon PDEs parametrized by the real projective line $ \mathbf{RP} ^1$. All equations in this family are shown to posses $conservative\ peakon \ solutions$ (whose Sobolev $H^1(\mathbf{R})$ norm is time invariant). The Hamiltonian structure …
Peakon solutions and analytical properties for the Camassa-Holm type equations with quadratic nonlinearities
本文研究了具有二次非线性的Camassa-Holm型方程的多峰解动力学系统及其初值问题的分析性质。首先在Besov空间中建立了局部well-posed性，并给出了破散准则。随后，通过适当的初始条件保证解要么全局存在要么在有限时间内破散。最后，利用非行波解证明了在Besov空间$B_{2,\infty}^{3/2}$中的ill-posed性。
We give a proof for the uniqueness of dissipative solution for the Camassa-Holm equation with some peakon-antipeakon initial data following Dafermos' earlier resut in [5] on the Hunter-Saxton equation. Our result shows that two existing global existence frameworks, through the vanishing viscosity method by Xin-Zhang in [11] and the transformation of coordinate method for dissipative solutions by Bressan-Constantin in [3], give the same solution, for a special but typical initial data forming finite time gradient blowup.
We prove that the peakons are asymptotically H 1-stable, under the flow of the Degasperis-Procesi equation, in the class of functions with a momentum density that belongs to M + (R). The key argument is a rigidity result for uniformly in time exponentially decaying global solutions that is shared by the Holm-Staley b-family of equations for b $\ge$ 1. This extends previous results obtained for the Camassa-Holm equation (b = 2).
We study the cubic ab-family of equations, which includes both the Fokas-Olver-Rosenau-Qiao (FORQ) and the Novikov (NE) equations. For $a\neq0$, it is proved that there exist initial data in the Sobolev space $H^s$, $s<3/2$, with non-unique solutions. Multiple solutions are constructed by studying the collision of 2-peakon solutions. Furthermore, we prove the novel phenomenon that for some members of the family, collision between 2-peakons can occur even if the "faster" peakon is in front of the "slower" peakon.
On local energy decay for solutions to the b-family of peakon equations
该研究针对b族尖峰方程解的长时间行为，在不同假设条件下证明了整体解在不同区域的局部能量衰减性质，丰富了该类方程的动力学行为理论。
The Lax pair formulation of the two-component Camassa-Holm equation (CH2) is generalized to produce an integrable multi-component family, CH(n,k), of equations with $n$ components and $1\le |k|\le n$ velocities. All of the members of the CH(n,k) family show fluid-dynamics properties with coherent solitons following particle characteristics. We determine their Lie-Poisson Hamiltonian structures and give numerical examples of their soliton solution behaviour. We concentrate on the CH(2,k) family with one or two velocities, including the CH(2,-1) equation in the Dym position of the CH2 hierarchy.…
We study a family of equations defined on the space of tensor densities of weight $λ$ on the circle and introduce two integrable PDE. One of the equations turns out to be closely related to the inviscid Burgers equation while the other has not been identified in any form before. We present their Lax pair formulations and describe their bihamiltonian structures. We prove local wellposedness of the corresponding Cauchy problem and include results on blow-up as well as global existence of solutions. Moreover, we construct "peakon" and "multi-peakon" solutions for all $λ\neq 0,1$, and "shock-peako…
Spectral instability of peakons for the $b$-family of Novikov equations
本文研究三次非线性单参数 $b$-Novikov peakon 方程族，通过延拓线性化算子证明了任意 $b$ 下 peakon 在 $L^2(\mathbb{R})$ 上的谱与线性不稳定性，并证明仅在 $b=3$ 时于 $H^1(\mathbb{R})$ 上稳定。
In this paper we discuss integrable higher order equations {\em of Camassa-Holm (CH) type}. Our higher order CH-type equations are "geometrically integrable", that is, they describe one-parametric families of pseudo-spherical surfaces, in a sense explained in Section 1, and they are integrable in the sense of zero curvature formulation ($\simeq$ Lax pair) with infinitely many local conservation laws. The major focus of the present paper is on a specific fifth order CH-type equation admitting {\em pseudo-peakons} solutions, that is, weak bounded solutions with differentiable first derivative an…
We prove spectral instability of peakons in the $b$-family of Camassa--Holm equations in $L^2(\mathbb{R})$ that includes the integrable cases of $b = 2$ and $b = 3$. We start with a linearized operator defined on functions in $H^1(\mathbb{R}) \cap W^{1,\infty}(\mathbb{R})$ and extend it to a linearized operator defined on weaker functions in $L^2(\mathbb{R})$. For $b \neq \frac{5}{2}$, the spectrum of the linearized operator in $L^2(\mathbb{R})$ is proved to cover a closed vertical strip of the complex plane. For $b = \frac{5}{2}$, the strip shrinks to the imaginary axis, but an additional pai…
Considered herein is a multi-component Novikov equation, which admits bi-Hamiltonian structure, infinitely many conserved quantities and peaked solutions. In this paper, we deduce two blow-up criteria for this system and global existence for some two-component case in $H^s$. Finally we verify that the system possesses peakons and periodic peakons.
We consider a Lax pair found by Xia, Qiao and Zhou for a family of two-component analogues of the Camassa-Holm equation, including an arbitrary function $H$, and show that this apparent freedom can be removed via a combination of a reciprocal transformation and a gauge transformation, which reduces the system to triangular form. The resulting triangular system may or may not be integrable, depending on the choice of $H$. In addition, we apply the formal series approach of Dubrovin and Zhang to show that scalar equations of Camassa-Holm type with homogeneous nonlinear terms of degree greater th…
In this Letter we propose that for Lax integrable nonlinear partial differential equations the natural concept of weak solutions is implied by the compatibility condition for the respective distributional Lax pairs. We illustrate our proposal by comparing two concepts of weak solutions of the modified Camassa-Holm equation pointing out that in the peakon sector (a family of non-smooth solitons) only one of them, namely the one obtained from the distributional compatibility condition, supports the time invariance of the Sobolev $H^1$ norm.
The Camassa-Holm equation with linear dispersion was originally derived as an asymptotic equation in shallow water wave theory. Among its many interesting mathematical properties, which include complete integrability, perhaps the most striking is the fact that in the case where linear dispersion is absent it admits weak multi-soliton solutions - "peakons" - with a peaked shape corresponding to a discontinuous first derivative. There is a one-parameter family of generalized Camassa-Holm equations, most of which are not integrable, but which all admit peakon solutions. Numerical studies reported…
In this paper we mainly study the Cauchy problem for a generalized Camassa-Holm equation. First, by using the Littlewood-Paley decomposition and transport equations theory, we establish the local well-posedness for the Cauchy problem of the equation in Besov spaces. Then we give a simply blow-up criterion for the Cauchy problem of the equation. we present a blow-up result and the exact blow-up rate of strong solutions to the equation by making use of the conservation law and the obtained blow-up criterion.Finally, we verify that the system possesses peakon solutions.
Analytic solutions and Singularity formation for the Peakon b--Family equations
利用抽象柯西-柯瓦列夫斯卡娅定理证明b-族方程局部时间存在唯一解析解，并针对b≥1的情况，在初始数据实解析且动量密度不变号时证明解全局解析；通过伪谱数值方法和奇点追踪方法研究奇点形成。
The local well-posedness, blow-up and global solution of a new integrable system in Besov spaces
该研究在临界与超临界Besov空间中建立一类新可积N-peakon系统柯西问题的局部适定性，推导爆破准则与全局解存在条件，并通过一阶差分法给出相关仿真算例。
In this paper, we study the Cauchy problem for an integrable multi-component (2N-component) peakon system which is involved in an arbitrary polynomial function. Based on a generalized Ovsyannikov type theorem, we first prove the existence and uniqueness of solutions for the system in the Gevrey-Sobolev spaces with the lower bound of the lifespan. Then we show the continuity of the data-to-solution map for the system. Furthermore, by introducing a family of continuous diffeomorphisms of a line and utilizing the fine structure of the system, we demonstrate the system exhibits unique continuation…
Considered in this paper is the modified Camassa-Holm equation with cubic nonlinearity, which is integrable and admits the single peaked solitons and multi-peakon solutions. The short-wave limit of this equation is known as the short-pulse equation. The main investigation is the Cauchy problem of the modified Camassa-Holm equation with qualitative properties of its solutions. It is firstly shown that the equation is locally well-posed in a range of the Besov spaces. The blow-up scenario and the lower bound of the maximal time of existence are then determined. A blow-up mechanism for solutions …
On the well-posedness of the Cauchy problem for the two-component peakon system in $C^k\cap W^{k,1}$
本文研究带三次非线性且推广Fokas-Olver-Rosenau-Qiao方程的双分量peakon系统，在C^k∩W^{k,1}类中利用拉格朗日坐标方法证明了解的局部适定性，并给出爆破准则。
In this paper, we study the Cauchy problem for a generalized integrable Camassa-Holm equation with both quadratic and cubic nonlinearity. By overcoming the difficulties caused by the complicated mixed nonlinear structure, we firstly establish the local well-posedness result in Besov spaces, and then present a precise blow-up scenario for strong solutions. Furthermore, we show the existence of single peakon by the method of analysis.
We consider a family of non-local problems that model the effects of transport and vortex stretching in the incompressible Euler equations. Using modulation techniques, we establish stable self-similar blow-up near a family of known self-similar blow-up solutions.
We consider the Yangs-Mills equations in 4+1 dimensions. This is the energy critical case and we show that it admits a family of solutions which blow up in finite time. They are obtained by the spherically symmetric ansatz in the SO(4) gauge group and result by rescaling of the instanton solution. The rescaling is done via a prescribed rate which in this case is a modification of the self-similar rate by a power of |log t|. The powers themselves take any value exceeding 3/2 and thus form a continuum of distinct rates leading to blow-up. The methods are related to the authors' previous work on …
In this paper, we give two family of explicit blowup solution for $3$D incompressible Navier-Stokes equations in $\RR^3$. Here one family of solutions admit the smooth initial data, and the initial data of another family of solutions are not smooth. The energy of those solutions is infinite.
This paper is concerned with global existence and blow-up phenomena for two-component Degasperis-Procesi system and two-component b-family system. The strategy relies on our observation on new conservative quantities of these systems. Several new global existence results and a new blowup result of strong solutions to the two-component Degasperis- Procesi system and the two-component b-family system are presented by using these new conservative quantities.
In this paper we consider the $b$-family of equations on the torus $u\_t- u\_{txx}+ (b+1) u u\_x=b u\_x u\_{xx} + u u\_{xxx}$, which for appropriate values of $b$ reduces to well-known models, such as the Camassa-Holm equation or the Degasperis-Procesi equation. We establish a local-in-space blow-up criterion
Blow-up phenomena and the local well-posedness of a Two-Component b-Family Equations in Besov spaces
该研究针对双分量b族方程，在非齐次Besov空间中证明了其局部适定性，并给出了新的爆破结果，丰富了该类方程的定性理论。
An inverse problem for the modified Camassa-Holm equation and multi-point Padé approximants
本文研究修正Camassa-Holm方程非光滑孤立子的逆谱构造，提出利用多点Padé近似解决逆问题的方法。
A blow-down mechanism for the Landau-Coulomb equation
本研究针对Landau-Coulomb方程的一类麦克斯韦函数超临界扰动初值，提出显式blow-down机制，通过双尺度线性化方法证明全局适定性与解的指数衰减平衡态收敛性。
This paper concerns with the explicit blowup phenomenon for 3D incompressible MHD equations in R^3. More precisely, we find two family of explicit blowup solutions for 3D incompressible MHD equations in R^3. One family of solutions admit the smooth initial data, and the initial data of another family of solutions are not smooth. The energy of those solutions is infinite. Moreover, our results tell us that the blowup phenomenon of 3D incompressible MHD can only take place in the velocity field of the fluid, but no blowup for the magnetic field.
Existence and stability of discretely self-similar blowup for a wave maps type equation
本文研究了波映射类型方程中离散自相似爆破的存在性与稳定性，通过非线性稳定性分析和谱分析，证明了在所有维度中该类解的稳定性。
We consider the blow-up of solutions to the following parameterized nonlinear wave equation: $ u_{tt} = c(u)^{2} u_{xx} + λc(u)c'(u)( u_x)^2$ with the real parameter $λ$. In previous works, it was reported that there exist finite time blow-up solutions with $λ=1$ and $2$. However, the construction of a blow-up solution depends on the symmetric structure of the equation (e.g., the energy conservation law). In the present paper, we extend the blow-up result with $λ=1$ to the case with $λ\in (0,1]$ by using a new $L^{2/λ}$ estimate. Moreover, some properties for the blow-up solution including the…
In this paper we investigate a non-linear and non-local one dimensional transport equation under random perturbations on the real line. We first establish a local-in-time theory, i.e., existence, uniqueness and blow-up criterion for pathwise solutions in Sobolev spaces $H^{s}$ with $s>3$. Thereafter, we give a complete picture of the long time behavior of the solutions based on the type of noise we consider. On one hand, we identify a family of noises such that blow-up can be prevented with probability $1$, guaranteeing the existence and uniqueness of global solutions almost surely. On the oth…
This paper construct a family of explicit self-similar blowup axisymmetric solutions for the 3D incompressible Euler equations in R^3. Those singular solutions admit infinite energy.
In this paper, we study the Cauchy problem of a two-component b-family equation. We first establish the local well-posedness for a two-component b-family equation by Kato's semigroup theory. Then, we derive precise blow-up scenarios for strong solutions to the equation. Moreover, we present several blow-up results for strong solutions to the equation.
This paper studies the dynamical behavior near a new family of explicit self-similar solutions for the one dimensional Born-Infeld equation. This quasilinear scalar field equation arises from nonlinear electromagnetism, as well as branes in string theory and minimal surfaces in Minkowski spacetimes. We show that both this model and the linear wave equation admit the same family of explicit timelike self-similar blow up solutions, meanwhile, Lyapunov nonlinear stability of those self-similar blow up solutions are given inside a strictly proper subset of the backward light cone.
We consider the 3d cubic focusing nonlinear Schroedinger equation (NLS) i\partial_t u + Δu + |u|^2 u=0, which appears as a model in condensed matter theory and plasma physics. We construct a family of axially symmetric solutions, corresponding to an open set in H^1_{axial}(R^3) of initial data, that blow-up in finite time with singular set a circle in xy plane. Our construction is modeled on Raphaël's construction \cite{R} of a family of solutions to the 2d quintic focusing NLS, i\partial_t u + Δu + |u|^4 u=0, that blow-up on a circle.
A note on the stability of self-similar blow-up solutions for superconformal semilinear wave equations
本文研究超共形半线性波动方程自相似爆破解的稳定性，利用洛伦兹变换下线性化算子的谱等价性证明模式稳定性，并将Merle-Zaag及Ostermann的渐近稳定性结果推广到整个ODE爆破族。
We consider a non-local Liouville equation corresponding to the prescription of the geodesic curvature on the circle. We build a family of solutions which blow up at a critical point of the harmonic extension of the prescribed curvature function, provided some generic assumptions are satisfied.
We extend the slow blow up solutions of Krieger, Schlag, and Tataru to semilinear wave equations on a curved background. In particular, for a class of manifolds $(M,g)$ we show the existence of a family of blow-up solutions with finite energy norm to the equation {equation} \partial_t^2 u - Δ_g u = |u|^4 u, \notag {equation} with a continuous rate of blow up. In contrast to the case where $g$ is the Minkowski metric, the argument used to produce these solutions can only obtain blow up rates that are bounded above.
Blow-up phemomenon for the Geng-Xue system and related models
本文研究了Geng-Xue系统及相关模型的爆破现象，通过低贝塞尔空间中的准则和非守恒方法证明了爆破，并扩展到双组分b-族系统。
Optimal Spectral Lower Bounds and Nonradial Nonlinear Asymptotic Stability of a Family of Three-Dimensional Keller--Segel Self-Similar Blow-Up Solutions
本文通过谱分析和能量方法，证明了三维Keller--Segel系统一族自相似爆破解在去除不稳定模式后的非径向非线性渐近稳定性，并给出了最优谱下界1/4。
We consider generic 2 x 2 singular Liouville systems on a smooth bounded domain in the plane having some symmetry with respect to the origin. We construct a family of solutions to which blow-up at the origin and whose local mass at the origin is a given quantity depending on the parameters of the system. We can get either finitely many possible blow-up values of the local mass or infinitely many. The blow-up values are produced using an explicit formula which involves Chebyshev polynomials.
In this paper, we consider a nonlocal parabolic equation associated with initial and Dirichlet boundary conditions. Firstly, we discuss the vacuum isolating behavior of solutions with the help of a family of potential wells. Then we obtain a threshold of global existence and blow up for solutions with critical initial energy. Furthermore, for those solutions satisfy $J(u_0)\leq d$ and $I(u_0)\neq 0$, we show that global solutions decay to zero exponentially as time tends to infinity and the norm of blow-up solutions increase exponentially.
The Landau equation does not blow up
本文证明了具有一般相互作用势（含Coulomb相互作用）的空齐Landau方程解的Fisher信息随时间单调递减且全局有界，从而推导出该方程的解永不爆破。
Partially invariant solution with an arbitrary surface of blow-up for the gas dynamics equations admitting pressure translation
本文对允许压力平移的气体动力学方程，通过4维子代数对称约化得到部分不变子模型，导出两族精确解，分别描述等容运动和任意爆破面，并分析了粒子轨迹。
We consider a family of non-local evolution equations including the $0-$Holm-Staley equation. We show that the family considered does not posses compactly supported solutions as long as the initial data is non-trivial. Also, we prove different unique continuation results for the solutions of the family studied. In addition, some special solutions, such as peakons and kinks, are studied and their dynamics are analyzed. Persistence properties of the solutions are also investigated as well as we describe the scenario for the global existence of solutions of the $0-$Holm-Staley equation. In partic…
In this article, a non linear family of spaces, based on the energy dissipation, is introduced. This family bridges an energy space (containing weak solutions to Navier-Stokes equation) to a critical space (invariant through the canonical scaling of the Navier-Stokes equation). This family is used to get uniform estimates on higher derivatives to solutions to the 3D Navier-Stokes equations. Those estimates are uniform, up to the possible blowing-up time. The proof uses blow-up techniques. Estimates can be obtained by this means thanks to the galilean invariance of the transport part of the equ…
In this paper, we consider a blow-up solution for the complex-valued semilinear wave equation with power nonlinearity in one space dimension. We first characterize all the solutions of the associated stationary problem as a two-parameter family. Then, we use a dynamical system formulation to show that the solution in self-similar variables approaches some particular stationary one in the energy norm, in the non-characteristic case. This gives the blow-up profile for the original equation in the non-characteristic case. Our analysis is not just a simple adaptation of the already handled real ca…
Stable Self-Similar Blow-Up In Nonlinear Wave Equations With Quadratic Time-Derivative Nonlinearities
本文针对两类含二次时间导数非线性项的一维非线性波动模型，构造分类五参数广义自相似爆破解族，证明相关分支渐近稳定性，统一刻画其爆破机制。
We consider the sinh-Poisson equation $$(P)_λ\quad -Δu=\la\sinh u\ \hbox{in}\ Ω,\ u=0\ \hbox{on}\ \partialΩ,$$ where $Ω$ is a smooth bounded domain in $\rr^2$ and $λ$ is a small positive parameter. If $0\inΩ$ and $Ω$ is symmetric with respect to the origin, for any integer $k$ if $\la$ is small enough, we construct a family of solutions to $(P)_\la$ which blows-up at the origin whose positive mass is $4πk(k-1)$ and negative mass is $4πk(k+1).$ It gives a complete answer to an open problem formulated by Jost-Wang-Ye-Zhou in [Calc. Var. PDE (2008) 31: 263-276].
Self-similar blow up for energy supercritical semilinear wave equation
研究能量超临界半线性波动方程的自相似爆破，证明可数族自相似轮廓的存在性及其非径向有限余维稳定性。
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