Lax Pairs and Darboux Transformations for Euler Equations

半线性正则Euler-Poisson-Darboux-Tricomi方程的寿命估计

Lifespan estimate for the semilinear regular Euler-Poisson-Darboux-Tricomi equation

本文研究半线性正则Euler-Poisson-Darboux-Tricomi方程，建立局部适定性，并利用基于高斯超几何函数的新检验函数和二阶常微分不等式得到Strauss型寿命估计，进而在δ=1时借助Kato引理证明相应爆破结果。

The Inverse Problem for the Euler-Poisson-Darboux Equation and Shifted $k$-Plane Transforms

The inverse problem for the Euler-Poisson-Darboux equation deals with reconstruction of the Cauchy data for this equation from incomplete information about its solution. In the present article, this problem is studied in connection with the injectivity of the shifted $k$-plane transform, which assigns to functions in $L^p(\mathbb {R}^n)$ their mean values over all k-planes at a fixed distance from the given $k$-planes. Several generalizations, including the Radon transform over strips of fixed width in $\mathbb {R}^2$ and a similar transform over tubes of fixed diameter in $\mathbb {R}^3$, are…

A Lax Pair for 2D Euler Equation

A Lax pair for the 2D Euler equation is found.

三维 Euler方程的Constantin-Lax-Majda模型的多尺度自相似有限时间爆破

Multi-scale self-similar finite-time blowups of the Constantin-Lax-Majda model for the 3D Euler equations

本文为1D Constantin-Lax-Majda模型构造了一类具有两个坍缩空间尺度的新渐近自相似有限时间爆破，以揭示高维方程中多尺度爆破的共同机制。

具有时间导数非线性的两个Euler-Poisson-Darboux-Tricomi方程弱耦合系统的爆破结果

Blow-up result for a weakly coupled system of two Euler-Poisson-Darboux-Tricomi equations with time derivative nonlinearity

本文研究具有尺度不变阻尼、时间导数非线性、质量项和Tricomi项的两个Euler-Poisson-Darboux-Tricomi方程弱耦合系统，在相同传播速度假设下得到新的爆破区域并给出最大存在时间的寿命估计。

弱耦合半线性欧拉-泊松-达布-特里科米方程组的临界曲线

Critical curve for weakly coupled system of semilinear Euler-Poisson-Darboux-Tricomi equations

本文针对阻尼项主导质量项的弱耦合半线性欧拉-泊松-达布-特里科米方程组（EPDTS），给出p-q平面中划分全局存在与爆破阈值的临界曲线Γₘ(n,p,q,β₁,β₂)=0，通过新测试函数解决爆破问题，结合线性方程估计推导全局存在性。

三维半线性欧拉 - 泊松 - 达布方程小数据解的全局存在性

Global existence of small data solutions to 3-D semilinear Euler-Poisson-Darboux equations

研究三维半线性欧拉 - 泊松 - 达布方程小数据弱解全局存在性问题，通过特定条件，证明当\(\mu\geq\frac{14}{5}\)且\(p>\max\{\frac53, 1+\frac{2}{\mu}\}\)时全局存在性。

Asymptotics to all orders of the Euler--Darboux equation in a triangle

In Einstein's theory of relativity, the interaction of two collinearly polarized plane gravitational waves can be described by a Goursat problem for the Euler--Darboux equation in a triangular domain. In this paper, using a representation of the solution in terms of Abel integrals, we give a full asymptotic expansion of the solution near the diagonal of the triangle. The expansion is related to the formation of a curvature singularity of the spacetime. In particular, our framework allows for boundary data with derivatives which are singular at the corners. This level of generality is crucial f…

欧拉-泊松-达布方程的瞬时快照问题

The Snapshot Problem for the Euler-Poisson-Darboux Equation

本文研究广义欧拉-泊松-达布方程在给定两个时刻数据时的解存在性与唯一性条件，并由此发现一类与贝塞尔函数相关的类刘维尔数并分析其性质。

Linear instability of breathers for the focusing nonlinear Schr{ö}dinger equation

Relying upon tools from the theory of integrable systems, we discuss the linear instability of the Kuznetsov-Ma breathers and the Akhmediev breathers of the focusing nonlinear Schr{ö}dinger equation. We use the Darboux transformation to construct simultaneously the breathers and the exact solutions of the Lax system associated with the breathers. We obtain a full description of the Lax spectra for the two breathers, including multiplicities of eigenvalues. Solutions of the linearized NLS equations are then obtained from the eigenfunctions and generalized eigenfunctions of the Lax system. While…

半线性Euler-Poisson-Darboux方程：具有临界耗散的波动情形

The semilinear Euler-Poisson-Darboux equation: a case of wave with critical dissipation

本文研究半线性Euler-Poisson-Darboux方程（具有临界耗散）的Cauchy问题，证明在初始数据小且 $p>\max\{p_0(1+\mu),3\}$ 时存在整体能量解，并给出多维情形在 $\mu$ 足够大时的结果。

On 2D Euler Equations: Part II. Lax Pairs and Homoclinic Structures

In Part I of our study on 2D Euler equation, we established the spectral theorem for a linearized 2D Euler equation. We also computed the point spectrum through continued fractions, and identified the eigenvalues with nonzero real parts. In this Part II of our study, first we discuss the Lax pairs for both 2D and 3D Euler equations. The existence of Lax pairs suggests that the hyperbolic foliations of 2D and 3D Euler equations may be degenerate, i.e., there exist homoclinic structures. Then we investigate the question on the degeneracy v.s. nondegeneracy of the hyperbolic foliations for Galerk…

Chaos in PDEs and Lax Pairs of Euler Equations

Recently, the author and collaborators have developed a systematic program for proving the existence of homoclinic orbits in partial differential equations. Two typical forms of homoclinic orbits thus obtained are: (1). transversal homoclinic orbits, (2). Silnikov homoclinic orbits. Around the transversal homoclinic orbits in infinite dimensional autonomous systems, the author was able to prove the existence of chaos through a shadowing lemma. Around the Silnikov homoclinic orbits, the author was able to prove the existence of chaos through a horseshoe construction. Very recently, there has be…

Unified approach to Miura, Bäcklund and Darboux transformations for nonlinear partial differential equations

This paper is an attempt to present and discuss at some length the Singular Manifold Method. This Method is based upon the Painlevé Property systematically used as a tool for obtaining clear cut answers to almost all the questions related with Nonlinear Partial Differential Equations: Lax pairs, Miura, Bäcklund or Darboux Transformations as well as $τ$-functions, in a unified way. Besides to present the basics of the Method we exemplify this approach by applying it to four equations in $(1+1)$-dimensions. Two of them are related with the other two through Miura transformations that are also de…

Periodic travelling waves of the modified KdV equation and rogue waves on the periodic background

We address the most general periodic travelling wave of the modified Korteweg-de Vries (mKdV) equation written as a rational function of Jacobian elliptic functions. By applying an algebraic method which relates the periodic travelling waves and the squared periodic eigenfunctions of the Lax operators, we characterize explicitly the location of eigenvalues in the periodic spectral problem away from the imaginary axis. We show that Darboux transformations with the periodic eigenfunctions remain in the class of the same periodic travelling waves of the mKdV equation. In a general setting, there …

Formation of singularity and smooth wave propagation for the non-isentropic compressible Euler equations

We define compressive and rarefactive waves and give the differential equations describing smooth wave steepening for the compressible Euler equations with a varying entropy profile and general pressure laws. Using these differential equations, we directly generalize P. Lax's singularity (shock) formation results for hyperbolic systems with two variables to the compressible Euler equations for a polytropic ideal gas. Our results are valid globally without restriction on the size of the variation of initial data.

三维半线性Euler-Poisson-Darboux方程的全局小数据径向对称解

Global small data radial symmetric solutions of 3D semilinear Euler-Poisson-Darboux equations

本文针对三维半线性Euler-Poisson-Darboux方程的临界指数推测，利用超几何Riemann表示与精细逐点时空加权估计，证明了特定参数范围内小数据径向对称解的全局存在性，解决了对应条件下的开放问题。

Isospectral Theory of Euler Equations

Isospectral problem of both 2D and 3D Euler equations of inviscid fluids, is investigated. Connections with the Clay problem are described. Spectral theorem of the Lax pair is studied.

On the blow-up problem for the axisymmetric 3D Euler equations

In this paper we study the finite time blow-up problem for the axisymmetric 3D incompressible Euler equations with swirl. The evolution equations for the deformation tensor and the vorticity are reduced considerably in this case. Under the assumption of local minima for the pressure on the axis of symmetry with respect to the radial variations we show that the solution blows-up in finite time. If we further assume that the second radial derivative vanishes on the axis, then system reduces to the form of Constantin-Lax-Majda equations, and can be integrated explicitly.

Integrable Structures for 2D Euler Equations of Incompressible Inviscid Fluids

In this article, I will report a Lax pair structure, a Backlund-Darboux transformation, and the investigation of homoclinic structures for 2D Euler equations of incompressible inviscid fluids.

A variational representation of weak solutions for the pressureless Euler-Poisson equations

We derive an explicit formula for global weak solutions of the one dimensional system of pressure-less Euler-Poisson equations. Our variational formulation is an extension of the well-known formula for entropy solutions of the scalar inviscid Burgers' equation: since the characteristics of the Euler-Poisson equations are parabolas, the representation of their weak solution takes the form of a "quadratic" version of the celebrated Lax-Oleinik variational formula. Three cases are considered. (i) The variational formula recovers the "sticky particle" solution in the attractive case; (ii) It repre…

On the Constantin-Lax-Majda Model with Convection

The well-known Constantin-Lax-Majda (CLM) equation, an important toy model of the 3D Euler equations without convection, can develop finite time singularities [5]. De Gregorio modified the CLM model by adding a convective term [6], which is known important for fluid dynamics [10,14]. Presented are two results on the De Gregorio model. The first one is the global well-posedness of such a model for general initial data with non-negative (or non-positive) vorticity which is based on a newly discovered conserved quantity. This verifies the numerical observations for such class of initial data. The…

Herglotz' variational principle and Lax-Oleinik evolution

We develop an elementary method to give a Lipschitz estimate for the minimizers in the problem of Herglotz' variational principle proposed in \cite{CCWY2018} in the time-dependent case. We deduce Erdmann's condition and the Euler-Lagrange equation separately under different sets of assumptions, by using a generalized du Bois-Reymond lemma. As an application, we obtain a representation formula for the viscosity solution of the Cauchy problem for the Hamilton-Jacobi equation \begin{align*} D_tu(t,x)+H(t,x,D_xu(t,x),u(t,x))=0 \end{align*} and study the related Lax-Oleinik evolution.

解Euler-Poisson-Darboux方程的临界非线性幂次下的有限时间爆破

Blow-up of solutions to the Euler-Poisson-Darbox equation with critical power nonlinearity

本文研究了Euler-Poisson-Darboux方程在临界非线性幂次下的有限时间爆破问题，通过改进的测试函数方法，得到了非线性项时空积分的下界，从而获得了对应初值问题的爆破结果和寿命上界估计。

On some model equations of Euler and Navier-Stokes equations

We propose a two-dimensional generalization of Constantin-Lax-Majda model [2]. Some results about singular solutions are given. This model might be the first step toward the singular solutions of the Euler equations. Along the same line (vorticity formulation), we present some further model equations. They possibly models various aspects of difficulties related with the singular solutions of the Euler and Navier-Stokes equations. We also make some discussions on the possible connection between turbulence and the singular solutions of the Navier-Stokes equations.