Alberto Bressan, Adrian Constantin
We construct a continuous semigroup of weak, dissipative solutions to a nonlinear partial differential equations modeling nematic liquid crystals. A new distance functional, determined by a problem of optimal transportation, yields sharp estimates on the continuity of solutions with respect to the initial data.
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This paper is concerned with global existence of weak solution for a periodic two-component $μ$-Hunter-Saxton system. We first derive global existence for strong solutions to the system with smooth approximate initial data. Then, we show that the limit of approximate solutions is a global weak solution of the two-component $μ$-Hunter-Saxton system.
The Hunter-Saxton equation serves as a mathematical model for orientation waves in a nematic liquid crystal. The present paper discusses a modified variant of this equation, coming up in the study of critical points for the speed of orientation waves, as well as a two-component extension. We establish well-posedness and blow-up results for some initial boundary value problems for the modified Hunter-Saxton equation and the two-component Hunter-Saxton system.
We establish the existence of conservative solutions of the initial value problem of the two-component Hunter--Saxton system on the line. Furthermore we investigate the stability of these solutions by constructing a Lipschitz metric.
We prove that the periodic initial value problem for the modified Hunter-Saxton equation is locally well-posed for continuously differentiable initial data. We also study the analytic regularity of this problem and prove a Cauchy-Kowalevski type theorem.
In this paper, we study the Cauchy problem of a periodic 2-component $μ$-Hunter-Saxton system. We first establish the local well-posedness for the periodic 2-component $μ$-Hunter-Saxton system by Kato's semigroup theory. Then, we derive precise blow-up scenarios for strong solutions to the system. Moreover, we present a blow-up result for strong solutions to the system. Finally, we give a global existence result to the system.
This paper is concerned with the local well-posedness and the precise blow-up scenario for a periodic 2-component μ-Hunter-Saxton system in Besov spaces. Moreover, we state a new global existence result to the system. Our obtained results for the system improve considerably earlier results.
We show that the Hunter-Saxton equation $u_t+uu_x=\frac14\big(\int_{-\infty}^x dμ(t,z)- \int^{\infty}_x dμ(t,z)\big)$ and $μ_t+(uμ)_x=0$ has a unique, global, weak, and conservative solution $(u,μ)$ of the Cauchy problem on the line.
In this paper, we consider the Cauchy problem for the Hunter-Saxton (HS) equation on the line. Firstly, we establish the local well-posedness for the integral form of the (HS) equation by constructing some special spaces $E^s_{p,r}$, which mix Lebesgue spaces and homogeneous Besov spaces. Then we present a global existence result and provide a sufficient condition for strong solutions to blow up in finite time for the equation. Finally, we give the ill-posedness and the unique continuation of the Hunter-Saxton equation.
We study the Lipschitz stability in time for $α$-dissipative solutions to the Hunter-Saxton equation, where $α\in [0,1]$ is a constant. We define metrics in both Lagrangian and Eulerian coordinates, and establish Lipschitz stability for those metrics.
This paper is concerned with blow-up phenomena and global existence for a periodic two-component Hunter-Saxton system. We first derive the precise blow-up scenario for strong solutions to the system. Then, we present several new blow-up results of strong solutions and a new global existence result to the system. Our obtained results for the system are sharp and improve considerably earlier results.
A Lipschitz metric for $α$-dissipative solutions to the Hunter-Saxton equation
该研究探究Hunter-Saxton方程解关于初值的Lipschitz稳定性，聚焦用广义特征线法构造的、由函数α决定空间各位置能量损失的α耗散解的稳定性问题。
We study stability of solutions of the Cauchy problem for the Hunter-Saxton equation $u_t+uu_x=\frac14(\int_{-\infty}^xu_x^2 dx-\int_{x}^\infty u_x^2 dx)$ with initial data $u_0$. In particular, we derive a new Lipschitz metric $d_\D$ with the property that for two solutions $u$ and $v$ of the equation we have $d_\D(u(t),v(t))\le e^{Ct} d_\D(u_0,v_0)$.
We study the global existence of solutions to a two-component generalized Hunter-Saxton system in the periodic setting. We first prove a persistence result of the solutions. Then for some particular choices of parameters $(α, κ)$, we show the precise blow-up scenarios and the existence of global solutions to the generalized Hunter-Saxton system under proper assumptions on the initial data. This significantly improves recent results obtained in [M. Wunsch, DCDS Ser. B 12 (2009), 647-656] and [M. Wunsch, SIAM J. Math. Anal. 42 (2010), 1286-1304].
We give explicit solutions to the two-component Hunter-Saxton system on the unit circle. Moreover, we show how global weak solutions can be naturally constructed using the geometric interpretation of this system as a re-expression of the geodesic flow on the semi-direct product of a suitable subgroup of the diffeomorphism group of the circle with the space of smooth functions on the circle. These spatially and temporally periodic solutions turn out to be conservative.
In this paper we develop an existence theory for the Cauchy problem to the stochastic Hunter-Saxton equatio, and prove several properties of the blow-up of its solutions. An important part of the paper is the continuation of solutions to the stochastic equations beyond blow-up (wave-breaking). In the linear noise case, using the method of (stochastic) characteristics, we also study random wave-breaking and stochastic effects unobserved in the deterministic problem. Notably, we derive an explicit law for the random wave-breaking time.
We establish the concept of $α$-dissipative solutions for the two-component Hunter-Saxton system under the assumption that either $α(x)=1$ or $0\leq α(x)<1$ for all $x\in \mathbb{R}$. Furthermore, we investigate the Lipschitz stability of solutions with respect to time by introducing a suitable parametrized family of metrics in Lagrangian coordinates. This is necessary due to the fact that the solution space is not invariant with respect to time.
This article represents a first step towards understanding the well-posedness for the dispersive Hunter-Saxton equation. This problem arises in the study of nematic liquid crystals, and although the equation has formal similarities with the KdV equation, the lack of $L^2$ control gives it a quasilinear character, with only continuous dependence on initial data. Here, we prove the local and global well-posedness of the Cauchy problem using a normal form approach to construct modified energies, and frequency envelopes in order to prove the continuous dependence with respect to the initial data.
In [Zhang and Zheng, 2005] it was conjectured by Zhang and Zheng that dissipative solutions of the Hunter-Saxton equation, which are known to be unique in the class of weak solutions, dissipate the energy at the highest possible rate. The conjecture of Zhang and Zheng was proven in [Dafermos, 2012] by Dafermos for monotone increasing initial data with bounded energy. In this note we prove the conjecture in [Zhang and Zheng, 2005] in full generality. To this end we examine the evolution of the energy of any weak solution of the Hunter-Saxton equation. Our proof shows in fact that for every time…
The generalized, two-component Hunter-Saxton system comprises several well-known models of fluid dynamics and serves as a tool for the study of one-dimensional fluid convection and stretching. In this article a general representation formula for periodic solutions to the system, which is valid for arbitrary values of parameters $(λ,κ)\in\mathbb{R}\times\mathbb{R}$, is derived. This allows us to examine in great detail qualitative properties of blow-up as well as the asymptotic behaviour of solutions, including convergence to steady states in finite or infinite time.
In this work we introduce the r-Hunter-Saxton equation, a generalisation of the Hunter-Saxton equation arising as extremals of an action principle posed in L_r. We characterise solutions to the Cauchy problem, quantifying the blow-up time and studying various symmetry reductions. We construct piecewise linear functions and show that they are weak solutions to the r-Hunter-Saxton equation.
In this paper we characterize the regularity structure, as well as show the global-in-time existence and uniqueness, of (energy) conservative solutions to the Hunter-Saxton equation by using the method of characteristics. The major difference between the current work and previous results is that we are able to characterize the singularities of energy measure and their nature in a very precise manner. In particular, we show that singularities, whose temporal and spatial locations are also explicitly given in this work, may only appear at at most countably many times, and are completely determin…
We analyze stability of conservative solutions of the Cauchy problem on the line for the (integrated) Hunter-Saxton (HS) equation. Generically, the solutions of the HS equation develop singularities with steep gradients while preserving continuity of the solution itself. In order to obtain uniqueness, one is required to augment the equation itself by a measure that represents the associated energy, and the breakdown of the solution is associated with a complicated interplay where the measure becomes singular. The main result in this paper is the construction of a Lipschitz metric that compares…
We show that the two-component Hunter-Saxton system with negative coupling constant describes the geodesic flow on an infinite-dimensional pseudosphere. This approach yields explicit solution formulae for the Hunter-Saxton system. Using this geometric intuition, we conclude by constructing global weak solutions. The main novelty compared with similar previous studies is that the metric is indefinite.
In this paper we study the large time asymptotic behavior of (energy) conservative solutions to the Hunter-Saxton equation in a generalized framework that consists of the evolutions of solution and its energy measure. We describe the large time asymptotic expansions of the conservative solutions, and rigorously verify the validity of the leading order term in $L^{\infty}(\mathbb{R})$ and ${\dot{H}}^1(\mathbb{R})$ spaces respectively. The leading order term is given by a kink-wave that is determined by the total energy of the system only. As a corollary, we also show that the singular part of t…
In the article a convergent numerical method for conservative solutions of the Hunter--Saxton equation is derived. The method is based on piecewise linear projections, followed by evolution along characteristics where the time step is chosen in order to prevent wave breaking. Convergence is obtained when the time step is proportional to the square root of the spatial step size, which is a milder restriction than the common CFL condition for conservation laws.
Existence, uniqueness and long-time behavior of the $λ$-dissipative solutions to the two-component Hunter-Saxton system
本文研究双组分Hunter-Saxton系统的λ耗散解，构造其显式特征，得出全耗散与部分耗散情形下解的唯一性结果，还揭示了解的长时间渐近行为，即密度与能量奇异部分衰减，能量集中于u分量，扭结波可由初始数据和λ显式计算。
Many important equations of mathematical physics arise geometrically as geodesic equations on Lie groups. In this paper, we study an example of a geodesic equation, the two-component Hunter-Saxton (2HS) system, that displays a number of unique geometric features. We show that 2HS describes the geodesic flow on a manifold which is isometric to a subset of a sphere. Since the geodesics on a sphere are simply the great circles, this immediately yields explicit formulas for the solutions of 2HS. We also show that when restricted to functions of zero mean, 2HS reduces to the geodesic equation on an…
Strong Ill-Posedness in critical and subcritical regimes for the Hunter-Saxton equation
该研究针对实直线上的Hunter-Saxton方程，在临界和次临界区域构造初值，证明其全局耗散解在Lipschitz阈值及以下会瞬时丧失Sobolev正则性，得到强不适定性结果。
A geometric view on the generalized Proudman-Johnson and $r$-Hunter-Saxton equations
本研究证明广义无粘Proudman-Johnson方程与$r$-Hunter-Saxton方程在特定参数范围等价，给出其几何解释并通过等距映射分析两类情形，得到非周期情形显式解，解答了相关极限行为问题。
In this paper, we study two-component versions of the periodic Hunter-Saxton equation and its $μ$-variant. Considering both equations as a geodesic flow on the semidirect product of the circle diffeomorphism group $\Diff(§)$ with a space of scalar functions on $§$ we show that both equations are locally well-posed. The main result of the paper is that the sectional curvature associated with the 2HS is constant and positive and that 2$μ$HS allows for a large subspace of positive sectional curvature. The issues of this paper are related to some of the results for 2CH and 2DP presented in [J. Esc…
The generalized Hunter-Saxton system comprises several well-known models from fluid dynamics and serves as a tool for the study of fluid convection and stretching in one-dimensional evolution equations. In this work, we examine the global regularity of periodic smooth solutions of this system in $L^p$, $p \in [1,\infty)$, spaces for nonzero real parameters $(λ,κ)$. Our results significantly improve/extend those by Wunsch et al. [27-29] and Sarria [21]. Furthermore, we study the effects that different boundary conditions have on the global regularity of solutions by replacing periodicity with a…
A numerical view on α-dissipative solutions of the Hunter-Saxton equation
本文提出一种结合投影算子、广义特征线法和最小时间步长迭代格式的数值方法，用于求解Hunter-Saxton方程的α-耗散解，显著提升效率并证明波形在C([0,T],L^∞)中的收敛性。
On Global Weak Solutions for the Magnetic Two-Component Hunter-Saxton System
本文从偏微分方程角度研究磁双组分猎人-萨克斯顿系统的全局弱解，通过黎卡提约简推导显式解公式并构建保守弱解。
We introduce a natural class of multicomponent local Poisson structures $\mathcal P_k + \mathcal P_1$, where $\mathcal P_1$ is a local Poisson bracket of order one and $\mathcal P_k$ is a homogeneous Poisson bracket of odd order $k$ under assumption that is has Darboux coordinates (Darboux-Poisson bracket) and non-degenerate. For such brackets we obtain the general formulas in arbitrary coordinates, find normal forms (related to Frobenius triples) and provide the description of the Casimirs, using purely algebraic procedure. In two-component case we completely classify such brackets up to the …
We study geodesic equations for a family of right-invariant Riemannian metrics on the group of diffeomorphisms of a compact manifold. The metrics descend to Fisher's information metric on the space of smooth probability densities. The right reduced geodesic equations are higher-dimensional generalisations of the $μ$--Hunter--Saxton equation, used to model liquid crystals under influence of magnetic fields. Local existence and uniqueness results are established by proving smoothness of the geodesic spray. The descending property of the metrics is used to obtain a novel factorisation of diffeomo…
A Convergent Numerical Algorithm for $α$-Dissipative Solutions of the Hunter-Saxton Equation
本文提出一种求解Hunter-Saxton方程α-耗散解的收敛数值算法，通过专门设计的投影算子处理初始数据并结合广义特征线法精确求解，证明了波剖面在L∞中收敛且能量密度子序列几乎处处弱收敛。
Exact Analytic Solution for the Time-Fractional Hunter-Saxton Equation with Caputo derivative
该研究将Hunter-Saxton方程推广至分数阶时间导数，利用分离变量法得到首个精确闭式解，为数值方法提供基准并揭示分数阶记忆对非线性波传播的影响。
Rate of convergence for numerical $α$-dissipative solutions of the Hunter-Saxton equation
本文证明 Hunter-Saxton 方程 $\alpha$-耗散解在初始导数满足 Hölder 条件时，数值收敛阶为 $\mathcal{O}(\Delta x^{1/8}+\Delta x^{\beta/4})$，并通过数值实验验证。
On the convergence rate of a numerical method for the Hunter-Saxton equation
该研究针对Hunter-Saxton方程的α耗散解数值方法推导误差估计，给出不同条件下的波形及能量测度收敛阶，并通过算例验证，为该方程数值求解提供理论支撑。
Long-time asymptotic behavior of the Hunter-Saxton equation
本研究利用推广的Deift-Zhou最速下降法，通过Riemann-Hilbert问题变形，得到Hunter-Saxton方程柯西问题解在两个时空区域的长时间渐近近似，明确了不同区域的衰减速率与误差阶。
On Mañé's critical value for the two-component Hunter-Saxton system and a infnite dimensional magnetic Hopf-Rinow theorem
本文研究了两组分磁Hunter-Saxton系统的Mañé临界值及其无限维磁Hopf-Rinow定理，通过几何框架分析解的破裂行为并扩展解至全局保守弱解。
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.
Global weak solutions of a Hamiltonian regularised Burgers equation
研究无粘 Burgers 方程的非色散守恒正则化，构造连接其与 Hunter–Saxton 方程的 Galilean 不变模型，并证明两类全局弱解及参数极限。
We study the propagation of orientation waves in a director field with rotational inertia and potential energy given by the Oseen-Frank energy functional from the continuum theory of nematic liquid crystals. There are two types of waves, which we call splay and twist waves. Weakly nonlinear splay waves are described by the quadratically nonlinear Hunter-Saxton equation. Here, we show that weakly nonlinear twist waves are described by a new cubically nonlinear, completely integrable asymptotic equation. This equation provides a surprising representation of the Hunter-Saxton equation as an advec…
We show that the weakly dissipative Camassa-Holm, Degasperis-Procesi, Hunter-Saxton, and Novikov equations can be reduced to their non-dissipative versions by means of an exponentially time-dependent scaling. Hence, up to a simple change of variables, the non-dissipative and dissipative versions of these equations are equivalent. Similar results hold also for the equations in the so-called b-family of equations as well as for the two-component and μ-versions of the above equations.
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.
In this paper we construct global weak conservative solutions of the Camassa-Holm equation on the periodic domain. We first express the equation in Lagrangian flow variable $η$ and then transform it using a change of variable $ρ=\sqrt{η_x}$. The new variable removes the singularity of the CH equation, and we obtain both the global weak conservative solution and global spatial smoothness of the Lagrangian trajectories of the CH equation. This work is motivated by J. Lenells who proved similar results for the Hunter-Saxton equation using the geometric interpretation.
Considered herein is the initial-value problem for the generalized periodic Camassa-Holm equation which is related to the Camassa-Holm equation and the Hunter-Saxton equation. Sufficient conditions guaranteeing the development of breaking waves in finite time are demonstrated. On the other hand, the existence of strong permanent waves is established with certain initial profiles depending on the linear dispersive parameter in a range of the Sobolev spaces. Moreover, the admissible global weak solution in the energy space is obtained.
The Hunter-Saxton equation and the Gurevich-Zybin system are considered as two mutually non-equivalent representations of one and the same Whitham-type equation, and all their common solutions are obtained exactly.
Stable $C^{7/9}$ cusp formation for the Novikov equation
该研究针对Novikov方程确定光滑初值开集，通过引入Galilean型变量变换克服困难，证明其会形成稳定的$C^{7/9}$尖点，揭示了非局部项与非线性结构耦合决定尖点正则性。
Among hyperbolic Initial Boundary Value Problems (IBVP), those coming from a variational principle 'generically' admit linear surface waves, as was shown by Serre [J. Funct. Anal. 2006]. At the weakly nonlinear level, the behavior of surface waves is expected to be governed by an amplitude equation that can be derived by means of a formal asymptotic expansion. Amplitude equations for weakly nonlinear surface waves were introduced by Lardner [Int. J. Engng Sci. 1983], Parker and co-workers [J. Elasticity 1985] in the framework of elasticity, and by Hunter [Contemp. Math. 1989] for abstract hype…
We describe a way of solving a partial differential equation using the differential invariants of its point symmetries. By first solving its quotient PDE, which is given by the differential syzygies in the algebra of differential invariants, we obtain new differential constraints which are compatible with the PDE under consideration. Adding these constraints to our system makes it overdetermined, and thus easier to solve. We focus on second-order scalar PDEs whose quotients are first-order scalar PDEs. This situation occurs only when the Lie algebra of symmetries of the second-order PDE is inf…
Large data global well-posedness for a one-dimensional quasilinear wave equation
本文研究了一维准线性波动方程在大初始数据下的全局存在性问题，通过新的比较原理建立了黎曼变量的上界和下界估计，解决了Glassey、Hunter和Zheng提出的开放问题的一部分。
Instability of the peaked traveling wave in a local model for shallow water waves
该研究基于与Hunter–Saxton方程相关的浅水波局域模型，结合理论推导与数值近似，证明尖峰行波存在谱不稳定性与非线性不稳定性，且该不稳定性无法通过光滑行波族取极限得到。
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