Alberto Bressan, Yuxi Zheng
We establish the existence of a conservative weak solution to the Cauchy problem for the nonlinear variational wave equation $u_{tt} - c(u)(c(u)u_x)_x=0$, for initial data of finite energy. Here $c(\cdot)$ is any smooth function with uniformly positive bounded values.
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Uniqueness of conservative solutions to a one-dimensional general quasilinear wave equation through variational principle
本文利用特征分析证明一般拟线性波动方程能量守恒 Hölder 连续弱解的唯一性，且结论适用于任意大小数据。
Relying on the analysis of characteristics, we prove the uniqueness of conservative solutions to the variational wave equation $u_{tt}-c(u) (c(u)u_x)_x=0$. Given a solution $u(t,x)$, even if the wave speed $c(u)$ is only Hölder continuous in the $t$-$x$ plane, one can still define forward and backward characteristics in a unique way. Using a new set of independent variables $X,Y$, constant along characteristics, we prove that $t,x,u$, together with other variables, satisfy a semilinear system with smooth coefficients. From the uniqueness of the solution to this semilinear system, one obtains t…
We prove the existence of a global semigroup for conservative solutions of the nonlinear variational wave equation $u_{tt}-c(u)(c(u)u_x)_x=0$. We allow for initial data $u|_{t=0}$ and $u_t|_{t=0}$ that contain measures. We assume that $0<κ^{-1}\le c(u) \le κ$. Solutions of this equation may experience concentration of the energy density $(u_t^2+c(u)^2u_x^2)dx$ into sets of measure zero. The solution is constructed by introducing new variables related to the characteristics, whereby singularities in the energy density become manageable. Furthermore, we prove that the energy may only focus on a …
Wave breaking for the nonlinear variational wave equation
本文针对非线性变分波动方程，通过特征线方法确定了波浪破碎的判据，并证明非所有行波解为保守解，且保守解可局部类似线性波解，能量集中可能不立即扩散。
In this paper, we study the global conservative weak solutions for a class of nonlinear dispersive wave equations after wave breaking. We first transform the equations into an equivalent semi-linear system by introducing new variables. We then establish the global existence of solutions for the semi-linear system by using the standard theory of ordinary differential equations. Finally, returning to the original variables, we obtain the global conservative weak solutions for the original equations.
We define a kinetic and a potential energy such that the principle of stationary action from Lagrangian mechanics yields a Camassa--Holm system (2CH) as the governing equations. After discretizing these energies, we use the same variational principle to derive a semi-discrete system of equations as an approximation of the 2CH system. The discretizaton is only available in Lagrangian coordinates and requires the inversion of a discrete Sturm--Liouville operator with time-varying coefficients. We show the existence of fundamental solutions for this operator at initial time with appropriate decay…
Existence and uniqueness of the global conservative solutions for the generalized Camassa-Holm equation with dual-power nonlinearities
本文研究广义Camassa-Holm方程的双重非线性情况下全局守恒解的存在性与唯一性，通过引入新变量将原方程转化为半线性系统，证明了解的存在性，并通过构造辅助变量证明了解的唯一性。
The paper is concerned with conservative solutions to the nonlinear wave equation $u_{tt} - c(u)\big(c(u) u_x\big)_x = 0$. For an open dense set of $C^3$ initial data, we prove that the solution is piecewise smooth in the $t$-$x$ plane, while the gradient $u_x$ can blow up along finitely many characteristic curves. The analysis is based on a variable transformation introduced in [7], which reduces the equation to a semilinear system with smooth coefficients, followed by an application of Thom's transversality theorem.
In this paper, we study the blow-up of solutions to the semilinear Moore-Gibson-Thompson (MGT) equation with nonlinearity of derivative type $|u_t|^p$ in the conservative case. We apply an iteration method in order to study both the subcritical case and the critical case. Hence, we obtain a blow-up result for the semilinear MGT equation (under suitable assumptions for initial data) when the exponent $p$ for the nonlinear term satisfies $1<p\leqslant (n+1)/(n-1)$ for $n\geqslant2$ and $p>1$ for $n=1$. In particular, we find the same blow-up range for $p$ as in the corresponding semilinear wave …
We investigate the equation $(u_t + (f(u))_x)_x = f''(u) (u_x)^2/2$ where $f(u)$ is a given smooth function. Typically $f(u)= u^2/2$ or $u^3/3$. This equation models unidirectional and weakly nonlinear waves for the variational wave equation $u_{tt} - c(u) (c(u)u_x)_x =0$ which models some liquid crystals with a natural sinusoidal $c$. The equation itself is also the Euler-Lagrange equation of a variational problem. Two natural classes of solutions can be associated with this equation. A conservative solution will preserve its energy in time, while a dissipative weak solution loses energy at t…
We prove the pathwise well-posedness of stochastic porous media and fast diffusion equations driven by nonlinear, conservative noise. As a consequence, the generation of a random dynamical system is obtained. This extends results of the second author and Souganidis, who considered analogous spatially homogeneous and first-order equations, and earlier works of Lions, Perthame, and Souganidis.
Wave propagation of a generic non--conservative compressible two--fluid model
研究非保守压缩两流体模型的初值问题，通过建立广义惠更斯原理，解决非保守结构带来的分析挑战，发展新的非线性估计方法。
We prove global existence in time of solutions to relaxed conservative cross diffusion systems governed by nonlinear operators of the form $u_i\to \partial_tu_i-Δ(a_i(\tilde{u})u_i)$ where the $u_i, i=1,...,I$ represent $I$ density-functions, $\tilde{u}$ is a spatially regularized form of $(u_1,...,u_I)$ and the nonlinearities $a_i$ are merely assumed to be continuous and bounded from below. Existence of global weak solutions is obtained in any space dimension. Solutions are proved to be regular and unique when the $a_i$ are locally Lipschitz continuous.
We prove several existence and non existence results of solitary waves for a class of nonlinear pseudo-relativistic Hartree equations with general nonlinearities. We use variational methods and some new variational identities involving the half Laplacian.
Entropy solutions to the Dirichlet problem for nonlinear diffusion equations with conservative noise
本文研究带保守噪声的非线性退化扩散方程，证明齐次Dirichlet边界下非负熵解的存在性、唯一性与$L_1$-稳定性，方法结合Kruzhkov双变量技巧与修正强熵条件。
Global conservative weak solutions and global strong solutions for a class of weakly dissipative nonlinear dispersive wave equations
本文研究了一类弱耗散非线性色散波方程的初值问题的全局解的存在性，建立了三种全局存在性结果，包括能量保守弱解和强解，涵盖了小初值和符号变化初值的情况，扩展了已知结果。
A Regularized System for the Nonlinear Variational Wave Equation
本文提出非线性变分波方程的一个新正则化推广系统，证明其局部光滑解的存在性，并在极限情形下恢复原方程。
A central limit theorem for nonlinear conservative SPDEs
本文证明了适用于非线性退化扩散及非正则噪声系数的保守型随机偏微分方程的中心极限定理，并揭示其涨落与相互作用粒子系统流体力学极限涨落的一阶一致性。
The integrable Novikov equation can be regarded as one of the Camassa-Holm-type equations with cubic nonlinearity. In this paper, we prove the global existence and uniqueness of the Hölder continuous energy conservative solutions for the Cauchy problem of the Novikov equation.
We derive a novel two-component generalization of the nonlinear variational wave equation as a model for the director field of a nematic liquid crystal with a variable order parameter. The two-component nonlinear variational wave equation admits solutions locally in time. We show that a particular long time asymptotic expansion around a constant state in a moving frame satisfy the two-component Hunter--Saxton system.
By means of a recent variational technique, we prove the existence of radially monotone solutions to a class of nonlinear problems involving the $p$-Laplace operator. No subcriticality condition (in the sense of Sobolev spaces) is required.
Variational properties of space-periodic standing waves of nonlinear Schr{ö}dinger equations with general nonlinearities
本文研究一般非线性薛定谔方程空间周期驻波的变分刻画，通过固定质量与动量最小化能量及在Nehari流形上最小化作用量，并讨论周期、反周期及聚焦、散焦情形。
Some higher-order quasilinear parabolic, hyperbolic, and nonlinear dispersion equations are shown to admit various blow-up, extinction, and travelling wave solutions, which reduce to variational problems admitting countable families of compactly supported solutions.
We provide a precise description of the set of residual boundary conditions generated by the self-similar viscous approximation introduced by Dafermos et al. We then apply our results, valid for both conservative and non conservative systems, to the analysis of the boundary Riemann problem and we show that, under appropriate assumptions, the limits of the self-similar and the classical vanishing viscosity approximation coincide. We require neither genuinely nonlinearity nor linear degeneracy of the characteristic fields.
Dual variational methods for static Nonlinear Maxwell's Equations
本文提出双变分方法用于研究静态非线性麦克斯韦方程，证明基态和无限多个几何不同解的存在性，提供了一种不同于直接变分方法的替代方法。
We prove new existence results for a Nonlinear Helmholtz equation with sign-changing nonlinearity of the form $$ - Δu - k^{2}u = Q(x)|u|^{p-2}u, \quad u \in W^{2,p}(\mathbb{R}^{N}) $$ with $k>0,$ $N \geq 3$, $p \in \left[\left.\frac{2(N+1)}{N-1},\frac{2N}{N-2}\right)\right.$ and $Q \in L^{\infty}(\mathbb{R}^{N})$. Due to the sign-changes of $Q$, our solutions have infinite Morse-Index in the corresponding dual variational formulation.
Using our results in [15], we provided existence theorems for the general classes of nonlinear evolutions. Finally, we give examples of applications of our results to parabolic, hyperbolic, Shrödinger, Navier-Stokes and other time-dependent systems of equations.
The paper introduces a new way to construct dissipative solutions to a second order variational wave equation. By a variable transformation, from the nonlinear PDE one obtains a semilinear hyperbolic system with sources. In contrast with the conservative case, here the source terms are discontinuous and the discontinuities are not always crossed transversally. Solutions to the semilinear system are obtained by an approximation argument, relying on Kolmogorov's compactness theorem. Reverting to the original variables, one recovers a solution to the nonlinear wave equation where the total energy…
We develop a variational technique for some wide classes of nonlinear evolutions. The novelty here is that we derive the main information directly from the corresponding Euler-Lagrange equations. In particular, we prove that not only the minimizer of the appropriate energy functional but also any critical point must be a solution of the corresponding evolutional system.
Three classes of higher-order nonlinear parabolic hyperbolic, and nonlinear dispersion equations are shown to admit exact blow-up or compacton solutions, which are induced by elliptic equations with non-Lipschitz nonlinearities. Variational techniques give countable families of various compactly supported solutions with oscllatory behaviour cloase to the interfaces. Overall, the whole set of such compact patterns reveals certain chaotic properties and contains solutions of arbitrary complexity.
We develop the complete free boundary analysis for solutions to classical obstacle problems related to nondegenerate nonlinear variational energies. The key tools are optimal $C^{1,1}$ regularity, which we review more generally for solutions to variational inequalities driven by nonlinear coercive smooth vector fields, and the results in \cite{FocGelSp15} concerning the obstacle problem for quadratic energies with Lipschitz coefficients. Furthermore, we highlight similar conclusions for locally coercive vector fields having in mind applications to the area functional, or more generally to area…
In this work, the Cauchy problem for the semilinear Moore-Gibson-Thompson (MGT) equation with power nonlinearity $|u|^p$ on the right-hand side is studied. Applying $L^2-L^2$ estimates and a fixed point theorem, we obtain local (in time) existence of solutions to the semilinear MGT equation. Then, the blow-up of local in time solutions is proved by using an iteration method, under certain sign assumption for initial data, and providing that the exponent of the power of the nonlinearity fulfills $1 < p \leqslant p_{\mathrm{Str}}(n)$ for $n \geqslant2$ and $p>1$ for $n=1$. Here the Strauss expon…
We are concerned with the study of the well-posedness of a nonlinear diffusion equation with a monotonically increasing multivalued time-dependent nonlinearity derived from a convex continuous potential having a superlinear growth to infinity. The results in this paper state that the solution of the nonlinear equation can be retrieved as the null minimizer of an appropriate minimization problem for a convex functional involving the potential and its conjugate. This approach, inspired by the Brezis-Ekeland variational principle, provides new existence results under minimal growth and coercivity…
We construct infinitely many real-valued, time-periodic breather solutions of power-type nonlinear wave equations. These solutions are obtained from critical points of a dual functional and they are weakly localized in space. Our abstract framework allows to find similar existence results for the Klein-Gordon equation or biharmonic wave equations.
Variational Nonlinear and Nonlocal Curvature Flows
本文在一般变分曲率框架下证明Almgren-Taylor-Wang最小化运动格式收敛到带时间依赖强迫项的非线性非局部曲率流水平集解，并在曲率附加假设下建立解的唯一性。
Dual variational methods for time-harmonic nonlinear Maxwell's equations
本文利用对偶变分方法和无需 Palais-Smale 条件的新版对称山路定理，证明了有界域及 $\mathbb{R}^3$ 上时谐非线性 Maxwell 方程存在无穷多个非平凡解。
Globally conservative weak solutions for a class of two-component nonlinear dispersive wave equations beyond wave breaking
本文研究了一类超越波浪破碎的双分量非线性色散波方程，通过引入新变量将其转化为等价半线性系统，并利用常微分方程理论和逆变换方法，证明了原系统全局保守弱解的存在性。
In this paper we study semiclassical states for the problem $$ -\eps^2 Δu + V(x) u = f(u) \qquad \hbox{in} \RN,$$ where $f(u)$ is a superlinear nonlinear term. Under our hypotheses on $f$ a Lyapunov-Schmidt reduction is not possible. We use variational methods to prove the existence of spikes around saddle points of the potential $V(x)$.
We study traveling wave solutions of the nonlinear variational wave equation. In particular, we show how to obtain global, bounded, weak traveling wave solutions from local, classical ones. The resulting waves consist of monotone and constant segments, glued together at points where at least one one-sided derivative is unbounded. Applying the method of proof to the Camassa--Holm equation, we recover some well-known results on its traveling wave solutions.
The primary objective of this research is to investigate an inverse problem of parameter identification in nonlinear mixed quasi-variational inequalities posed in a Banach space setting. By using a fixed point theorem, we explore properties of the solution set of the considered quasi-variational inequality. We develop a general regularization framework to give an existence result for the inverse problem. Finally, we apply the abstract framework to a concrete inverse problem of identifying the material parameter in an implicit obstacle problem given by an operator of $p$-Laplacian type.
A conservative stochastic Dirac-Klein-Gordon system
本文研究由 Yukawa 相互作用耦合的 Dirac-Klein-Gordon 随机系统，在齐次乘性噪声下，于 Bourgain 空间中证明了 Cauchy 问题的存在唯一性，并利用电荷守恒得到了全局存在性。
In this paper, we develop a natural operator-splitting variational scheme for a general class of non-local, degenerate conservative-dissipative evolutionary equations. The splitting-scheme consists of two phases: a conservative (transport) phase and a dissipative (diffusion) phase. The first phase is solved exactly using the method of characteristic and DiPerna-Lions theory while the second phase is solved approximately using a JKO-type variational scheme that minimizes an energy functional with respect to a certain Kantorovich optimal transport cost functional. In addition, we also introduce …
We consider the Cauchy problem for the integrable nonlocal nonlinear Schrödinger (NNLS) equation $ \I\partial_t q(x,t)+\partial_{x}^2q(x,t)+2σq^{2}(x,t)\overline{q(-x,t)}=0 $ with initial data $q(x,0)\in H^{1,1}(\mathbb{R})$. It is known that the NNLS equation is integrable and it has soliton solutions, which can have isolated finite time blow-up points. The main aim of this work is to propose a suitable concept for continuation of weak $H^{1,1}$ local solutions of the general Cauchy problem (particularly, those admitting long-time soliton resolution) beyond possible singularities. Our main to…
Unique continuation for nonlinear variational problems
本文研究非线性变分问题解的唯一延拓性，利用Almgren频率公式的几乎单调性证明弱与强唯一延拓，并给出解与梯度同时消失点集的维数估计。
Finite speed of propagation and waiting time phenomena for stochastic porous media equations with nonlinear conservative noise
本文研究带非线性保守噪声的随机多孔介质方程，证明其动力学解的有限传播速度，提出新颖迭代技术推导Stampacchia型不等式，确定初值平坦性条件以保证等待时间现象发生，结果与预期行为一致。
We propose three new discrete variational schemes that capture the conservative-dissipative structure of a generalized Kramers equation. The first two schemes are single-step minimization schemes while the third one combines a streaming and a minimization step. The cost functionals in the schemes are inspired by the rate functional in the Freidlin-Wentzell theory of large deviations for the underlying stochastic system. We prove that all three schemes converge to the solution of the generalized Kramers equation.
In this paper, we consider the error analysis of a conservative Fourier pseudo-spectral method that conserves mass and energy for the space fractional nonlinear Schrödinger equation. We give a new fractional Sobolev norm that can construct the discrete fractional Sobolev space, and we also can prove some important lemmas for the new fractional Sobolev norm. Based on these lemmas and energy method, a priori error estimate for the method can be established. Then, we are able to prove that the Fourier pseudo-spectral method is unconditionally convergent with order $O(τ^{2}+N^{α/2-r})$ in the disc…
Conservative Numerical Methods for Nonlinear String Dynamics: Non-planar Vibration
本文研究非平面振动的非线性弦动力学，提出多种数值方案，重点在于离散能量和角动量的守恒，通过这些守恒性质在强非线性条件下获得简单的数值稳定性条件，并提供了完整实现细节和多个数值示例，附录中讨论了数值损耗模型和谱方法。
We study the minimizers of $L^2$-subcritical inhomogeneous variational problems with spatially decaying nonlinear terms, which contain $x = 0$ as a singular point. The limit concentration behavior of minimizers is proved as $M\to\infty$ by establishing the refined analysis of the spatially decaying nonlinear term.
We derive an asymptotic solution of the vacuum Einstein equations that describes the propagation and diffraction of a localized, large-amplitude, rapidly-varying gravitational wave. We compare and contrast the resulting theory of strongly nonlinear geometrical optics for the Einstein equations with nonlinear geometrical optics theories for variational wave equations.
Higher Order Fluctuation Expansions for Nonlinear Stochastic Heat Equations in Singular Limits
本文研究了具有非线性噪声的随机热方程（SHE），在奇异极限下推导出描述解渐近行为的高阶Edgeworth型涨落展开，并涵盖了正则与非正则扩散系数及多种相关SPDE情形。
A new partial differential nonlinear system containing quasivariational and parabolic variational inequalities and its application
本文提出了一种包含拟变分不等式和抛物型变分不等式的非线性偏微分系统，用于研究具有长记忆效应、磨损和损伤的粘弹性摩擦接触问题。
A formal methodology for developing variational principles corresponding to a given nonlinear PDE system is discussed. The scheme is demonstrated in the context of the incompressible Navier-Stokes equations, systems of first-order conservation laws, and systems of Hamilton- Jacobi equations.
We present two semidiscretizations of the Camassa-Holm equation in periodic domains based on variational formulations and energy conservation. The first is a periodic version of an existing conservative multipeakon method on the real line, for which we propose efficient computation algorithms inspired by works of Camassa and collaborators. The second method, and of primary interest, is the periodic counterpart of a novel discretization of a two-component Camassa-Holm system based on variational principles in Lagrangian variables. Applying explicit ODE solvers to integrate in time, we compare t…
We establish a variational framework for nonlinear instabilities in a setting of the ideal magnetohydrodynamic (MHD) equations. We apply a variational method to various kind of smooth steady states which are shown to be nonlinearly unstable for both incompressible and compressible ideal MHD equations. Destabilizing effect of compressibility is justified as well as stabilizing effect of magnetic field lines arising in MHD dynamics, which distinguishes from the Rayleigh-Taylor instability in the absence of magnetic field lines.
A dual variational principle for nonlinear dislocation dynamics
该研究定义了描述弹性固体位错动力学的非线性偏微分方程系统的对偶变分原理，并拓展构建了适用于一类通用偏微分方程、满足给定初边值条件的对偶变分原理。
We analyze the asymptotic behavior for a system of fully nonlinear parabolic and elliptic quasi variational inequalities. These equations are related to robust switching control problems introduced in [3]. We prove that, as time horizon goes to infinity (resp. discount factor goes to zero) the long run average solution to the parabolic system (resp. the limiting discounted solution to the elliptic system) is characterized by a solution of a nonlinear system of ergodic variational inequalities. Our results hold under a dissipativity condition and without any non degeneracy assumption on the dif…
Variational problems for the system of nonlinear Schrödinger equations with derivative nonlinearities
本文利用变分方法研究Colin-Colin提出的带导数非线性薛定谔方程组的基态解存在性、全局适定性及行波稳定性，并证明了一维小速度行波集合的稳定性。
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