We propose a gradient flow perspective to the spatially homogeneous Landau equation for soft potentials. We construct a tailored metric on the space of probability measures based on the entropy dissipation of the Landau equation. Under this metric, the Landau equation can be characterized as the gradient flow of the Boltzmann entropy. In particular, we characterize the dynamics of the PDE through a functional inequality which is usually referred as the Energy Dissipation Inequality. Furthermore, analogous to the optimal transportation setting, we show that this interpretation can be used in a minimizing movement scheme to construct solutions to a regularized Landau equation.
We prove convergence of the gradient flow of the Ginzburg-Landau energy functional on a Riemann surface in the self-dual Bogomolny case, in Coulomb gauge. The proof is direct and makes use of the associated nonlinear first order differential operators (the Bogomolny operators). One aim is to illustrate that the Bogomolny structure, which is known to be of great utility in the static elliptic case, can also be used effectively in evolution problems. We also identify the minimizers and minimum value of the energy when the Bogomolny bound is not achieved (below the Bradlow limit).
We study the spectral flow of Landau-Robin hamiltonians in the exterior of a compact domain with smooth boundary. This provides a method to study the spectrum of the exterior Landau-Robin hamiltonian's dependence on the choice of Robin data, even explaining the heuristics of how the spectrum of the Robin problem asymptotically tends to the spectrum of the Dirichlet problem. The main technical result concerns the continuous dependence of Landau-Robin hamiltonians on the Robin data in the gap topology. The problem can be localized to the compact boundary where the asymptotic behavior of the spec…
We discuss a 3D model describing the time evolution of nematic liquid crystals in the framework of Landau-de Gennes theory, where the natural physical constraints are enforced by a singular free energy bulk potential proposed by J.M. Ball and A. Majumdar. The thermal effects are present through the component of the free energy that accounts for intermolecular interactions. The model is consistent with the general principle of thermodynamics and mathematically tractable. We identify the a priori estimates for the associated system of evolutionary partial differential equations and construct glo…
We prove that the isotropic Landau equation equipped with the Coulomb potential introduced by Krieger-Strain and Gualdani-Guillen can be identified with the gradient flow of the entropy in the probability space with respect to a Riemannian metric tensor with nonlocal mobility. We give characterizations of the corresponding geodesics equations and present a convergence rate result by estimating its Hessian operator.
We revisit the grazing collision limit connecting the Boltzmann equation to the Landau(-Fokker-Planck) equation from their recent reinterpretations as gradient flows. Our results are in the same spirit as the $Γ$-convergence of gradient flows technique introduced by Sandier and Serfaty. In this setting, the grazing collision limit reduces to showing the lower semi-continuous convergence of the Boltzmann entropy-dissipation to the Landau entropy-dissipation.
In this paper we revisit a singular bulk potential in the Landau-de Gennes free energy that describes nematic liquid crystal configurations in the framework of the Q-tensor order parameter. This singular potential, called Ball-Majumdar potential, is introduced in [3], and is considered as a natural enforcement of a physical constraint on the eigenvalues of symmetric, traceless Q-tensors. Specifically, we establish blowup rates of both this singular potential and its gradient as Q approaches its physical boundary.
We follow the idea of Wang \cite{W} to show the existence of global weak solutions to the Cauchy problems of Landau-Lifshtiz type equations and related heat flows from a $n$-dimensional Euclidean domain $\Om$ or a $n$-dimensional closed Riemannian manifold $M$ into a 2-dimensional unit sphere $\U^{2}$. Our conclusions extend a series of related results obtained in the previous literature.
We consider the Ginzburg-Landau heat flow on the two-dimensional flat torus, starting from an initial data with a finite number of nondegenerate zeros -- but possibly very high initial energy. We show that the initial zeros are conserved and the flow rapidly enters a logarithmic energy regime, from which the evolution of vortices can be described by the works of Bethuel, Orlandi and Smets.
We consider the mixed Ginzburg-Landau flow that is supplemented with convective derivatives of the unknown function. We show that the associated vortex motion law is the mixed flow of the renormalized energy with new nonlinear forcing terms. These terms are uniquely determined by the extra terms in the initial PDE. Our proof relies on the assumption that the initial data are close to optimal.
In this paper, we consider the Landau-Lifshitz equation of the ferromagnetic spin chain from $\R^2$ to the unit sphere $S^2$ under the general Oseen-Frank energy. We obtain global existence and uniqueness of weak solutions for large energy data; moreover, the number of singular points is finite.
In this paper, we prove that the solution of the Landau-Lifshitz flow $u(t,x)$ from $\mathbb{H}^2$ to $\mathbb{H}^2$ converges to some harmonic map as $t\to\infty$. The essential observation is that although there exist infinite numbers of harmonic maps from $\Bbb H^2$ to $\Bbb H^2$, the heat flow initiated from $u(t,x)$ for any given $t>0$ converges to the same harmonic map as the heat flow initiated from $u(0,x)$. This observation enables us to construct a variant of Tao's caloric gauge to reduce the convergence to harmonic maps for the Landau-Lifshitz flow to the decay of the corresponding …
We study the convergence of the system of the Allen-Cahn equations to the weak solution for the multi-phase mean curvature flow in the sense of Brakke. The Landau-Lifshitz equation in this paper can be regarded as a system of Allen-Cahn equations with the Lagrange multiplier, which is a phase field model of the multi-phase mean curvature flow. Under an assumption that the limit of the energies of the solutions for the equations matches with the total variation for the singular limit of the solutions, we show that the family of the varifolds derived from the energies is a Brakke flow.
We consider the dissipative heat flow and conservative Gross-Pitaevskii dynamics associated with the Ginzburg-Landau energy posed on a Riemannian 2-manifold M. We show the limiting vortices of the solutions to these two problems evolve according to the gradient flow and Hamiltonian point-vortex flow respectively, associated with the renormalized energy on M. For the heat flow, we then specialize to the case where M is a sphere and study the limiting system of ODE's and establish an annihilation result. Finally, for the Ginzburg-Landau heat flow on a sphere, we derive some weighted energy ident…
In this paper we prove the convergence for all time for a Ginzburg- Landau type approximation of a simplified Ericksen-Leslie model in two dimension. Moreover, we are able to show that the singular set consists in at most finitely many singular points and we give a characterizations of the singularities.
In this paper, we study the asymptotic behaviors of finite energy solutions to the Landau-Lifshitz flows from $\Bbb R^2$ into Kähler manifolds. First, we prove that the solution with initial data below the critical energy converges to a constant map in the energy space as $t\to \infty$ for the compact Riemannian surface targets. In particular, when the target is a two dimensional sphere, we prove that the solution to the Landau-Lifshitz-Gilbert equation with initial data having an energy below $4π$ converges to some constant map in the energy space. Second, for general compact Kähler manifolds…
We introduce a gradient flow formulation of linear Boltzmann equations. Under a diffusive scaling we derive a diffusion equation by using the machinery of gradient flows.
In this paper, we study the energy critical 1-equivariant Landau-Lifschitz flow mapping $\mathbb{R}^2$ to $\mathbb{S}^2$ with arbitrary given coefficients $ρ_1\in \mathbb{R}$, $ρ_2>0$. We prove that there exists a codimension one smooth well-localized set of initial data arbitrarily close to the ground state which generates type-II finite-time blowup solutions, and give a precise description of the corresponding singularity formation. In our proof, both the Schrödinger part and the heat part play important roles in the construction of approximate solutions and the mixed energy/Morawetz functio…
In this paper we study a gradient flow generated by the Landau-de Gennes free energy that describes nematic liquid crystal configurations in the space of $Q$-tensors. This free energy density functional is composed of three quadratic terms as the elastic energy density part, and a singular potential in the bulk part that is considered as a natural enforcement of a physical constraint on the eigenvalues of $Q$. The system is a non-diagonal parabolic system with a singular potential which trends to infinity logarithmically when the eigenvalues of $Q$ approaches the physical boundary. We give a r…
We consider the Landau-Lifshitz equations of ferromagnetism (including the harmonic map heat-flow and Schroedinger flow as special cases) for degree m equivariant maps from R^2 to S^2. If m \geq 3, we prove that near-minimal energy solutions converge to a harmonic map as t goes to infinity (asymptotic stability), extending previous work down to degree m = 3. Due to slow spatial decay of the harmonic map components, a new approach is needed for m=3, involving (among other tools) a "normal form" for the parameter dynamics, and the 2D radial double-endpoint Strichartz estimate for Schroedinger op…
Contact set of solutions to gradient flow of Landau-de Gennes energy with a singular entropy potential
NYU Shanghai（纽约大学上海分校）；NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai（纽约大学上海分校纽约-华东师范大学数学科学研究所）；Department of Mathematics and Statistics, Old Dominion University（老道明大学数学与统计系）
We study a modified Landau-de Gennes model for nematic liquid crystals, where the elastic term is assumed to be of subquadratic growth in the gradient. We analyze the behaviour of global minimizers in two- and three-dimensional domains, subject to uniaxial boundary conditions, in the asymptotic regime where the length scale of the defect cores is small compared to the length scale of the domain. We obtain uniform convergence of the minimizers and of their gradients, away from the singularities of the limiting uniaxial map. We also demonstrate the presence of maximally biaxial cores in minimize…
We prove existence, uniqueness and asymptotics of global smooth solutions for the Landau-Lifshitz-Gilbert equation in dimension $n \ge 3$, valid under a smallness condition of initial gradients in the $L^n$ norm. The argument is based on the method of moving frames that produces a covariant complex Ginzburg-Landau equation, and a priori estimates that we obtain by the method of weighted-in-time norms as introduced by Fujita and Kato.
We establish the global well-posedness of the Landau-Lifshitz-Gilbert equation in $\mathbb R^n$ for any initial data ${\bf m}_0\in H^1_*(\mathbb R^n,\mathbb S^2)$ whose gradient belongs to the Morrey space $M^{2,2}(\mathbb R^n)$ with small norm $\displaystyle\|\nabla {\bf m}_0\|_{M^{2,2}(\mathbb R^n)}$. The method is based on priori estimates of a dissipative Schrödinger equation of Ginzburg-Landau types obtained from the Landau-Lifshitz-Gilbert equation by the moving frame technique.
In this paper we are concerned with the stochastic Landau-Lifshitz-Slonczewski equation (LLS) that describes magnetisation of an inifnite nanowire evolving under current driven spin torque. The current brings into the system a multiplicative gradient noise that appears as a transport term in the equation. We prove the existence, uniqueness and regularity of pathwise solutions to this equation.
We consider the gradient flow of a Ginzburg-Landau functional of the type \[ F_\varepsilon^{\mathrm{extr}}(u):=\frac{1}{2}\int_M \left|D u\right|_g^2 + \left|\mathscr{S} u\right|^2_g +\frac{1}{2\varepsilon^2}\left(\left|u\right|^2_g-1\right)^2\mathrm{vol}_g \] which is defined for tangent vector fields (here $D$ stands for the covariant derivative) on a closed surface $M\subseteq\mathbb{R}^3$ and includes extrinsic effects via the shape operator $\mathscr{S}$ induced by the Euclidean embedding of~$M$. The functional depends on the small parameter $\varepsilon>0$. When $\varepsilon$ is small it…
In this note we prove that, under some minimal regularity assumptions on the initial datum, solutions to the spatially homogenous Boltzmann and Landau equations for hard potentials uniformly propagate the Fisher information. The proof of such a result is based upon some explicit pointwise lower bound on solutions to Boltzmann equation and strong diffusion properties for the Landau equation. We include an application of this result related to emergence and propagation of exponential tails for the solution's gradient. These results complement estimates provided in the literature.
The existence of finite time blowup solutions for the two-dimensional Landau--Lifshitz equation is a long-standing problem, which exists in the literature at least since 2001 (E, Mathematics Unlimited--2001 and Beyond, Springer, Berlin, P.410, 2001). A more refined description in the equivariant class is given in (van den Berg and Williams, European J. Appl. Math., 24(6), 912--948, 2013). In this paper, we consider the blowup dynamics of the Landau--Lifshitz equation $$ \partial_tu=\mathfrak{a}_1u\timesΔu-\mathfrak{a}_2u\times(u\timesΔu),\quad x\in\mathbb{R}^2, $$ where $u\in\mathbb{S}^2$, $\m…
The Ginzburg-Landau system with general potential: maximum principle and gradient estimates
Institut de Mathématiques de Toulouse, UMR 5219, Université de Toulouse, CNRS, UPS IMT（图卢兹数学研究所，UMR 5219，图卢兹大学，CNRS，UPS IMT）；Mathematical Institute and St Edmund Hall, University of Oxford（牛津大学数学研究所与圣埃德蒙堂）；School of Mathematics, University of Bristol（布里斯托尔大学数学学院）；IKERBASQUE, Basque Foundation for Science（伊克尔巴斯克，巴斯克科学基金会）；BCAM, Basque Center for Applied Mathematics（BCAM，巴斯克应用数学中心）；“Simion Stoilow” Institute of the Romanian Academy（罗马尼亚科学院“西米翁·斯托伊洛”研究所）
The energy gradient theory was proposed in our previous studies. The mechanism of flow instability is very different in shear driven flows from pressure driven flows. In present paper, the relationship for the energy variation, work done, and energy dissipation in unit volumetric fluid of incompressible flow is derived. A universal equation for calculating the energy gradient function in shear driven flows is presented. With the calculation of the energy gradient function which is a field variable and is considered as a local Reynolds number, the stability of a basic flow can be analyzed. The …
This paper first proposes a new approximate scheme to construct a harmonic heat flow $u$ between a parabolic cylinder to a sphere. Y.Chen and M.Struwe have proved an existence and discussed a partial regularity of harmonic heat flows by using Ginzburg-Landau heat flow and passing to the limit of a parameter appeared in the equation. To construct a new harmonic heat flow, we propose a Ginzburg-Landau type heat flow with a time-dependent parameter. and we next establish the existence of a harmonic heat flow into spheres with (i) a global energy inequality, (ii) a monotonicity for the scaled ener…
We investigate a variational approach to nonpotential perturbations of gradient flows of nonconvex energies in Hilbert spaces. We prove existence of solutions to elliptic-in-time regularizations of gradient flows by combining the minimization of a parameter-dependent functional over entire trajectories and a fixed-point argument. These regularized solutions converge up to subsequence to solutions of the gradient flow as the regularization parameter goes to zero. Applications of the abstract theory to nonlinear reaction-diffusion systems are presented.
We consider an asymptotic regime for two-dimensional ferromagnetic films that is consistent with the formation of transition layers (Néel walls). We first establish compactness of S2-valued magnetizations in the energetic regime of Néel walls and characterize the set of accumulation points. We then prove that Néel walls are asymptotically the unique energy minimizing configurations. We finally study the corresponding dynamical issues, namely the compactness properties of the magnetizations under the flow of the Landau-Lifshitz-Gilbert equation.
We consider the variant of a stochastic parabolic Ginzburg-Landau equation that allows for the formation of point defects of the solution. The noise in the equation is multiplicative of the gradient type. We show that the family of the Jacobians associated to the solution is tight on a suitable space of measures. Our main result is the characterization of the limit points of this family. They are concentrated on finite sums of delta measures with integer weights. The singular set of the solution coincides with the points at which the delta measures are centered.
It is well-known that the mean curvature flow is a formal gradient flow of the perimeter functional. However, by the work of Michor and Mumford [7,8], the formal Riemannian structure that is compatible with the gradient flow structure induces a degenerate metric on the space of hypersurfaces. It is then natural to ask whether there is a nondegenerate metric space of hypersurfaces, on which the mean curvature flow admits a gradient flow structure. In this paper we study the mean curvature flow on two nondegenerate metric spaces of simple closed plane curves: the uniformness-preserving metric st…
This paper gives a framework to study a continuum limit of a gradient flow on a graph where the number of vertices increases in an appropriate way. As examples we prove the convergence of a discrete total variation flow and a discrete Allen--Cahn flow on discretised tori to their respective continuum limits.
We present new abstract results on the interrelation between the minimizing movement scheme for gradient flows along a sequence of Gamma-converging functionals and the gradient flow motion for the corresponding limit functional, in a general metric space. We are able to allow a relaxed form of minimization in each step of the scheme, and so we present new relaxation results too.
In this paper, we study two kind of L^2 norm preserved non-local heat flows on closed manifolds. We first study the global existence, stability and asymptotic behavior to such non-local heat flows. Next we give the gradient estimates of positive solutions to these heat flows.
Based on the idea of a recent paper by Ambrosio-Gigli-Savaré in Invent. Math. (2013), we show that flow of the $q$-Cheeger energy, called $q$-heat flow, solves the gradient flow problem of the Renyi entropy functional in the $p$-Wasserstein. For that, a further study of the $q$-heat flow is presented including a condition for its mass preservation. Under a convexity assumption on the upper gradient, which holds for all $q\ge2$, one gets uniqueness of the gradient flow and the two flows can be identified. Smooth solution of the $q$-heat flow are solution the parabolic q-Laplace equation, i.e. $…
