Using the improved lower bound on the sum of the eigenvalues of the Dirichlet Laplacian proved by A. D. Melas (Proc. Amer. Math. Soc. \textbf{131} (2003) 631-636), we report a new and sharp estimate for the dimension of the global attractor associated to the complex Ginzburg-Landau equation supplemented with Dirichlet boundary conditions.
We study a coupled nonlinear evolution system arising from the Ginzburg-Landau theory for atomic Fermi gases near the BCS-BEC crossover. First, we prove that the initial boundary value problem generates a strongly continuous semigroup on a suitable phase-space which possesses the global attractor. Then we establish the existence of an exponential attractor. As a consequence, we show that the global attractor is of finite fractal dimension.
This paper consists of three results on pattern formation of Ginzburg-Landau $m$-armed vortex solutions and spiral waves in circular and spherical geometries. First, we completely describe the global bifurcation diagram of vortex equilibria. Second, we prove persistence of all bifurcation curves under perturbations of parameters, which yields the existence of spiral waves for the complex Ginzburg-Landau equation. Third, we explicitly construct the global attractor of $m$-armed vortex solutions. Our main tool is a new shooting method that allows us to prove hyperbolicity of vortex equilibria in…
We study the long time behavior of solutions to a nonlinear partial differential equation arising in the description of trapped rotating Bose-Einstein condensates. The equation can be seen as a hybrid between the well-known nonlinear Schrödinger/Gross-Pitaevskii equation and the Ginzburg-Landau equation. We prove existence and uniqueness of global in-time solutions in the physical energy space and establish the existence of a global attractor within the associated dynamics. We also obtain basic structural properties of the attractor and an estimate on its Hausdorff and fractal dimensions.
We give a comprehensive study of the 3D Navier-Stokes-Brinkman-Forchheimer equations in a bounded domain endowed with the Dirichlet boundary conditions and non-autonomous external forces. This study includes the questions related with the regularity of weak solutions, their dissipativity in higher energy spaces and the existence of the corresponding uniform attractors
Discrete Ginzburg-Landau (DGL) equations with non-local nonlinearities have been established as significant inherently discrete models in numerous physical contexts, similar to their counterparts with local nonlinear terms. We study two prototypical examples of non-local and local DGLs on the one-dimensional infinite lattice. For the non-local DGL, we identify distinct scenarios for the asymptotic behavior of the globally existing in time solutions depending on certain parametric regimes. One of these scenarios is associated with a restricted compact attractor according to J. K. Hale's definit…
We study Ginzburg-Landau equations for a complex vector order parameter. We consider the Dirichlet problem in the disk in the plane with a symmetric, degree-one boundary condition, and study its stability, in the sense of the spectrum of the second variation of the energy. We find that the stability of the degree-one equivariant solution depends on both the Ginzburg-Landau parameter as well as the sign of the interaction term in the energy.
We perform the classical Ginzburg-Landau analysis originated from the celebrated paper by Bethuel, Brezis, Hélein for optimal boundary data. More precisely, we give optimal regularity assumptions on the boundary curve of planar domains and Dirichlet boundary data on them. When the Dirichlet boundary data is the tangent vector field of the boundary curve, our framework allows us to define a natural energy minimizing frame for simply connected domains enclosing Weil-Petersson curves.
In this short note, we consider the elliptic problem $$ λϕ+ Δϕ= η|ϕ|^σϕ,\quad ϕ\big|_{\partial Ω}=0,\quad λ, η\in \mathbb{C}, $$ on a smooth domain $Ω\subset \mathbb{R}^N$, $N\ge 1$. The presence of complex coefficients, motivated by the study of complex Ginzburg-Landau equations, breaks down the variational structure of the equation. We study the existence of nontrivial solutions as bifurcations from the trivial solution. More precisely, we characterize the bifurcation branches starting from eigenvalues of the Dirichlet-Laplacian of arbitrary multiplicity. This allows us to discuss the nature…
We establish vortex dynamics for the time-dependent Ginzburg-Landau equation for asymptotically large numbers of vortices for the problem without a gauge field and either Dirichlet or Neumann boundary conditions. As our main tool, we establish quantitative bounds on several fundamental quantities, including the kinetic energy, that lead to explicit convergence rates. For dilute vortex liquids we prove that sequences of solutions converge to the hydrodynamic limit.
We get a new expression of the microscopic renormalized energy for a pinned Ginzburg-Landau type energy modeling small impurities. This is done by obtaining a sharp decomposition for the minimal energy of a Dirichlet type functional with an L $\infty$-weight. In particular we get an explicit expression of the microscopic renormalized energy for a circular impurity. We proceed also to the minimization of this renormalized energy in some cases.
We consider the complex Ginzburg-Landau equation with two pure-power nonlinearities and a damping term. After proving a general global existence result, we focus on the existence and stability of several periodic orbits, namely the trivial equilibrium, bound-states and solutions independent of the spatial variable. In particular, we construct bound-states either explicitly in the real line or through a bifurcation argument for a double eigenvalue of the Dirichlet-Laplace operator on bounded domains.
We prove the existence of non-trivial standing wave solutions of the complex Ginzburg-Landau equation $ϕ_t - e^{iθ}(ρI- Δ) φ- e^{iγ} |ϕ|^αφ=0 $ in $\Rn$, where $(N-2)α<4$, $θ,γ\in (-π/2,π/2)$ and $ρ>0$. Analogous result is obtained in a ball $Ω\in\Rn$ for $ρ>-λ_1$, where $λ_1$ is the first eigenvalue of the Laplace operator with Dirichlet boundary conditions.
Let $Ω$ be a smooth bounded domain in $\mathbb{R}^2$. For $ε>0$ small, we construct non-constant solutions to the Ginzburg-Landau equations $-Δu=\frac{1}{ε^2}(1-|u|^2)u$ in $Ω$ such that on $\partial Ω$ u satisfies $|u|=1$ and $u\wedge \partial_νu=0$. These boundary conditions are called semi-stiff and are intermediate between the Dirichlet and the homogeneous Neumann boundary conditions. In order to construct such solutions we use a variational reduction method very similar to the one used by del Pino-Kowalczyk-Musso to find solutions of the Ginzburg-Landau equations with Dirichlet and homoge…
We study solutions of the 2D Ginzburg-Landau equation -Δu+\frac{1}{\ve^2}u(|u|^2-1)=0 subject to "semi-stiff" boundary conditions: the Dirichlet condition for the modulus, |u|=1, and the homogeneous Neumann condition for the phase. The principal result of this work shows there are stable solutions of this problem with zeros (vortices), which are located near the boundary and have bounded energy in the limit of small epsilon. For the Dirichlet bondary condition ("stiff" problem), the existence of stable solutions with vortices, whose energy blows up as epsilon goes to 0, is well known. By contr…
This paper is concerned with the stochastic generalized Ginzburg-Landau equation driven by a multiplicative noise of jump type. By a prior estimate, weak convergence and monotonicity technique, we prove the existence and uniqueness of the solution of an initial-boundary value problem with homogeneous Dirichlet boundary condition. However, for the generalized Ginzburg-Landau equation, such a locally monotonic condition of the nonlinear term can not be satisfied in a straight way. For this, we utilize the characteristic structure of nonlinear term and refined analysis to overcome this gap.
We prove existence of global attractors for parabolic equations of the form $$u_t+β(x)u-\sum_{ij}\partial_i(a_{ij}(x)\partial_j u)=f(x,u)$$ with Dirichlet boundary condition on an arbitrary unbounded domain $Ω$ in $\R^3$, without smoothness assumptions on $a_{ij}(\cdot)$ and $\partialΩ$.
Slightly compressible Brinkman-Forchheimer equations in a bounded 3D domain with Dirichlet boundary conditions are considered. These equations model fluids motion in porous media. The dissipativity of these equations in higher order energy spaces is obtained and regularity and smoothing properties of the solutions are studied. In addition, the existence of a global and an exponential attractors for these equations in a natural phase space is verified.
The Penrose-Fife system for phase transitions is addressed. Dirichlet boundary conditions for the temperature are assumed. Existence of global and exponential attractors is proved. Differently from preceding contributions, here the energy balance equation is both singular at 0 and degenerate at infinity. For this reason, the dissipativity of the associated dynamical process is not trivial and has to be proved rather carefully.
We study the structure of vortex solutions in a Ginzburg-Landau system for two complex valued order parameters. We consider the Dirichlet problem in the disk in R^2 with symmetric, degree-one boundary condition, as well as the associated degree-one entire solutions in all of R^2. Each problem has degree-one equivariant solutions with radially symmetric profile vanishing at the origin, of the same form as the unique (complex scalar) Ginzburg-Landau minimizer. We find that there is a range of parameters for which these equivariant solutions are the unique locally energy minimizing solutions for …
We study the Dirichlet boundary value problem for viscoelastic diffusion in polymers. We show that its weak solutions generate a dissipative semiflow. We construct the minimal trajectory attractor and the global attractor for this problem.
The effect of rapid oscillations, related to large dispersion terms, on the dynamics of dissipative evolution equations is studied for the model examples of the 1D complex Ginzburg-Landau and the Kuramoto-Sivashinsky equations. Three different scenarios of this effect are demonstrated. According to the first scenario, the dissipation mechanism is not affected and the diameter of the global attractor remains uniformly bounded with respect to the very large dispersion coefficient. However, the limit equation, as the dispersion parameter tends to infinity, becomes a gradient system. Therefore, ad…
The dependence of the fractal dimension of global attractors for the damped 3D Euler--Bardina equations on the regularization parameter $α>0$ and Ekman damping coefficient $γ>0$ is studied. We present explicit upper bounds for this dimension for the case of the whole space, periodic boundary conditions, and the case of bounded domain with Dirichlet boundary conditions. The sharpness of these estimates when $α\to0$ and $γ\to0$ (which corresponds in the limit to the classical Euler equations) is demonstrated on the 3D Kolmogorov flows on a torus.
We exhibit a large class of Lyapunov functionals for nonlinear drift-diffusion equations with non-homogeneous Dirichlet boundary conditions. These are generalizations of large deviation functionals for underlying stochastic many-particle systems, the zero range process and the Ginzburg-Landau dynamics, which we describe briefly. As an application, we prove linear inequalities between such an entropy-like functional and its entropy production functional for the boundary-driven porous medium equation in a bounded domain with positive Dirichlet conditions: this implies exponential rates of relaxa…
This paper is devoted to the asymptotic analysis of a fractional version of the Ginzburg-Landau equation in bounded domains, where the Laplacian is replaced by an integro-differential operator related to the square root Laplacian as defined in Fourier space. In the singular Ginzburg-Landau limit, we show that solutions with uniformly bounded energy converge weakly to sphere valued 1/2-harmonic maps, i.e., the fractional analogues of the usual harmonic maps. In addition, the convergence holds in smooth functions spaces away from a (n-1)-rectifiable closed set of finite (n-1)-Hausdorff measure. …
We prove a $Γ$-convergence result for a class of Ginzburg-Landau type functionals with $\mathcal{N}$-well potentials, where $\mathcal{N}$ is a closed and $(k-2)$-connected submanifold of $\mathbb{R}^m$, in arbitrary dimension. This class includes, for instance, the Landau-de Gennes free energy for nematic liquid crystals. The energy density of minimisers, subject to Dirichlet boundary conditions, converges to a generalised surface (more precisely, a flat chain with coefficients in $π_{k-1}(\mathcal{N})$) which solves the Plateau problem in codimension $k$. The analysis relies crucially on the …
The work is devoted to Dirichlet problem for sub-quintic semi-linear wave equation with damping damping term of the form $(-Δ)^α\partial_t u$, $α\in(0,\frac{1}{2})$, in bounded smooth domains of $\Bbb R^3$. It appears that to prove well-posedness and develop smooth attractor theory for the problem we need additional regularity of the solutions, which does not follow from the energy estimate. Considering the original problem as perturbation of the linear one the task is reduced to derivation of Strichartz type estimate for the linear wave equation with fractional damping, which is the main feat…
We estimate the magnetic Laplacian energy norm in appropriate planar domains under a weak regularity hypothesis on the magnetic field. Our main contribution is an averaging estimate, valid in small cells, allowing us to pass from non-uniform to uniform magnetic fields. As a matter of application, we derive new upper and lower bounds of the lowest eigenvalue of the Dirichlet Laplacian which match in the regime of large magnetic field intensity. Furthermore, our averaging technique allows us to estimate the non-linear Ginzburg-Landau energy, and as a byproduct, yields a non-Gaussian trial state …
We consider minimizers of a Ginzburg-Landau energy with a discontinuous and rapidly oscillating pinning term, subject to a Dirichlet boundary condition of degree $d > 0$. The pinning term models an unbounded number of small impurities in the domain. We prove that for strongly type II superconductor with impurities, minimizers have exactly d isolated zeros (vortices). These vortices are of degree 1 and pinned by the impurities. As in the standard case studied by Bethuel, Brezis and Hélein, the macroscopic location of vortices is governed by vortex/vortex and vortex/ boundary repelling effects. …
We study the asymptotic behaviour, as a small parameter $\varepsilon$ tends to zero, of minimisers of a Ginzburg-Landau type energy with a nonlinear penalisation potential vanishing on a compact submanifold $\mathcal{N}$ and with a given $\mathcal{N}$-valued Dirichlet boundary data. We show that minimisers converge up to a subsequence to a singular $\mathcal{N}$-valued harmonic map, which is smooth outside a finite number of points around which the energy concentrates and whose singularities' location minimises a renormalised energy, generalising known results by Bethuel, Brezis and Hélein for…
We study the Kolmogorov's entropy of uniform attractors for non-autonomous dissipative PDEs. The main attention is payed to the case where the external forces are not translation-compact. We present a new general scheme which allows us to give the upper bounds of this entropy for various classes of external forces through the entropy of proper projections of their hulls to the space of translation-compact functions. This result generalizes well known estimates of Vishik and Chepyzhov for the translation-compact case. The obtained results are applied to three model problems: sub-quintic 3D damp…
In this paper we prove the existence of a trajectory attractor (in the sense of V.V. Chepyzhov and M.I. Vishik) for a nonlinear PDE system coming from a 3D liquid crystal model accounting for stretching effects. The system couples a nonlinear evolution equation for the director d (introduced in order to describe the preferred orientation of the molecules) with an incompressible Navier-Stokes equation for the evolution of the velocity field u. The technique is based on the introduction of a suitable trajectory space and of a metric accounting for the double-well type nonlinearity contained in t…
We consider a Ginzburg-Landau type energy with a piecewise constant pinning term $a$ in the potential $(a^2 - |u|^2)^2$. The function $a$ is different from 1 only on finitely many disjoint domains, called the {\it pinning domains}. These pinning domains model small impurities in a homogeneous superconductor and shrink to single points in the limit $\v\to0$; here, $\v$ is the inverse of the Ginzburg-Landau parameter. We study the energy minimization in a smooth simply connected domain $Ω\subset \mathbb{C}$ with Dirichlet boundary condition $g$ on $\d Ø$, with topological degree ${\rm deg}_{\d Ø…
In this paper, we derive optimal upper and lower bounds on the dimension of the attractor AW for scalar reaction-diffusion equations with a Wentzell (dynamic) boundary condition. We are also interested in obtaining explicit bounds about the constants involved in our asymptotic estimates, and to compare these bounds to previously known estimates for the dimension of the global attractor AK; K \in {D;N; P}, of reactiondiffusion equations subject to Dirichlet, Neumann and periodic boundary conditions. The explicit estimates we obtain show that the dimension of the global attractor AW is of differ…
Let $\mathcal{D} =Ω\setminus\barω \subset \mathbb{R}^2$ be a smooth annular type domain. We consider the simplified Ginzburg-Landau energy $E_ε(u)=\frac{1}{2}\int_{\mathcal{D}} |\nabla u|^2 +\frac{1}{4ε^2}\int_{\mathcal{D}} (1-|u|^2)^2$, where $u: \mathcal{D} \rightarrow \mathbb{C}$, and look for minimizers of $E_ε$ with prescribed degrees $deg(u,\partial Ω)=p$, $deg(u,\partial ω)=q$ on the boundaries of the domain. For large $ε$ and for balanced degrees, i.e., $p=q$, we obtain existence of minimizers for {\it thin} domain. We also prove non-existence of minimizers of $E_ε$, for large $ε$, in …
A doubly nonlinear parabolic equation of the form $α(u_t)-Δu+W'(u)= f$, complemented with initial and either Dirichlet or Neumann homogeneous boundary conditions, is addressed. The two nonlinearities are given by the maximal monotone function $α$ and by the derivative $W'$ of a smooth but possibly nonconvex potential $W$; $f$ is a known source. After defining a proper notion of solution and recalling a related existence result, we show that from any initial datum emanates at least one solution which gains further regularity for $t>0$. Such "regularizing solutions" constitute a semiflow $S$ for…
We consider a Large Eddy Simulation model for a homogeneous incompressible Newtonian fluid in a box space domain with periodic boundary conditions on the lateral boundaries and homogeneous Dirichlet conditions on the top and bottom boundaries, thus simulating a horizontal channel. The model is obtained through the application of an anisotropic horizontal filter, which is known to be less memory consuming from a numerical point of view, but provides less regularity with respect to the standard isotropic one defined as the inverse of the Helmholtz operator. It is known that there exists a unique…
We carry out an asymptotic analysis of a thin nematic liquid crystal in which one elastic constant dominates over the others, namely \begin{align} \label{energyab} \inf E_\varepsilon(u)\quad\mbox{where}\quad E_\varepsilon(u) := \frac{1}{2}\int_Ω\left\{\varepsilon\,|\nabla u|^2 + \frac{1}{\varepsilon} \,(|u|^2 - 1)^2 + L \,(\mathrm{div}\,u)^2\right\} \,dx. \end{align} Here $u: Ω\to \mathbb R^2$ is a vector field, $0 < \varepsilon \ll 1 $ is a small parameter, and $L > 0$ is a fixed constant, independent of $\varepsilon$. We derive the $Γ$-limit $E_0$, which is a sum of a bulk term penalizing di…
In a convex domain $Ø\subset\R^3$, we consider the minimization of a 3D-Ginzburg-Landau type energy $E_\v(u)=1/2\int_Ø|\n u|^2+\frac{1}{2\v^2}(a^2-|u|^2)^2$ with a discontinuous pinning term $a$ among $H^1(Ø,\C)$-maps subject to a Dirichlet boundary condition $g\in H^{1/2}(\pØ,§^1)$. The pinning term $a:\R^3\to\R^*_+$ takes a constant value $b\in(0,1)$ in $ø$, an inner strictly convex subdomain of $Ø$, and 1 outside $ø$. We prove energy estimates with various error terms depending on assumptions on $Ø,ø$ and $g$. In some special cases, we identify the vorticity defects via the concentration of…
We prove that the attractor of the 1D quintic complex Ginzburg-Landau equation with a broken phase symmetry has strictly positive space-time entropy for an open set of parameter values. The result is obtained by studying chaotic oscillations in grids of weakly interacting solitons in a class of Ginzburg-Landau type equations. We provide an analytic proof for the existence of two-soliton configurations with chaotic temporal behavior, and construct solutions which are closed to a grid of such chaotic soliton pairs, with every pair in the grid well spatially separated from the neighboring ones fo…
We construct pullback attractors to the weak solutions of the three-dimensional Dirichlet problem for the incompressible Navier-Stokes equations in the case when the external force may become unbounded as time goes to plus or minus infinity.
We study the well-posedness of the vector-field Peierls-Nabarro model for curved dislocations with a double well potential and a bi-states limit at far field. Using the Dirichlet to Neumann map, the 3D Peierls-Nabarro model is reduced to a nonlocal scalar Ginzburg-Landau equation. We derive an integral formulation of the nonlocal operator, whose kernel is anisotropic and positive when Poisson's ratio $ν\in(-\frac12, \frac13)$. We then prove that any bounded stable solutions to this nonlocal scalar Ginzburg-Landau equation has a 1D profile, which corresponds to the PDE version of flatness resul…
We prove, under generic assumptions, that the special variational traveling wave that minimizes the exponentially weighted Ginzburg-Landau functional associated with scalar reaction-diffusion equations in infinite cylinders is the long-time attractor for the solutions of the initial value problems with front-like initial data. The convergence to this traveling wave is exponentially fast. The obtained result is mainly a consequence of the gradient flow structure of the considered equation in the exponentially weighted spaces and does not depend on the precise details of the problem. It strength…
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）
We study nematic liquid crystalline films within the framework of the Landau-de Gennes theory in the limit when both the thickness of the film and the nematic correlation length are vanishingly small compared to the lateral extent of the film. We prove $Γ$-convergence for a sequence of singularly perturbed functionals with a potential vanishing on a high-dimensional set and a Dirichlet condition imposed on admissible functions. This then allows us to prove the existence of local minimizers of the Landau-de Gennes energy in the spirit of a theorem due to Kohn and Sternberg despite the lack of c…
In this paper we consider the initial-boundary value problem for the heat, damped wave, complex-Ginzburg-Landau and Schr"odinger equations with the power type nonlinearity $|u|^p$ with $p in (1,2]$ in a two-dimensional exterior domain. Remark that $2=1+2/N$ is well-known as the Fujita exponent. If $p>2$, then there exists a small global-in-time solution of the damped wave equation for some initial data small enough (see Ikehata'05), and if $p<2$, then global-in-time solutions cannot exist for any positive initial data (see Ogawa-Takeda'09 and Lai-Yin'17). The result is that for given initial d…
Consider weakly nonlinear complex Ginzburg--Landau (CGL) equation of the form: $$ u_t+i(-Δu+V(x)u)=εμΔu+ε\mathcal{P}( u),\quad x\in {R^d}\,, \quad(*) $$ under the periodic boundary conditions, where $μ\geqslant0$ and $\mathcal{P}$ is a smooth function. Let $\{ζ_1(x),ζ_2(x),\dots\}$ be the $L_2$-basis formed by eigenfunctions of the operator $-Δ+V(x)$. For a complex function $u(x)$, write it as $u(x)=\sum_{k\geqslant1}v_kζ_k(x)$ and set $I_k(u)=\frac{1}{2}|v_k|^2$. Then for any solution $u(t,x)$ of the linear equation $(*)_{ε=0}$ we have $I(u(t,\cdot))=const$. In this work it is proved that if …
