pinecone/core-2020-05-10-deduplication huggingface dataset:
C1Approximations of Inertial Manifolds for Dissipative Nonlinear Equations

AbstractIn this paper we study a class of nonlinear dissipative partial differential equations that have inertial manifolds. This means that the long-time behavior is equivalent to a certain finite system of ordinary differential equations. We investigate ways in which these finite systems can be approximated in theC1sense. Geometrically this may be interpreted as constructing manifolds in phase space that areC1close to the inertial manifold of the partial differential equation. Under such approximations the invariant hyperbolic sets of the global attractor persist
On a Class of Weakly Regular Singular Two-Point Boundary Value Problems, II

AbstractExistence of a unique solution of a class of weakly regular singular two point boundary value problems −(p(x)y′)′=p(x)f(x, y), 0<x⩽b, limx→0+y′(x)=0,y(b)=B, has been established, wherep(x) satisfies: (i) p(x)>0 on (0, b); (ii) p(x)∈C1(0, r), and for somer>b; (iii) xp′(x)/p(x) is analytic in {z:|z|<r} with Taylor expansionxp′(x)/p(x)=b0+b1x+…b0∈[0, 1) and with quite general conditions onf(x, y). These conditions onf(x, y) are sharp, which is seen through one example. Regions for multiple solutions have also been determined
Nonlinear Stability of Travelling Wave Solutions for Viscoelastic Materials with Fading Memory

AbstractIn this paper, we shall discuss the stability of smooth monotone travelling wave solutions for viscoelastic materials with memory. It is known that a smooth monotone travelling wave solution exists for (1.1) if the end states are close and satisfy the Rankine–Hugoniot condition. For such a travelling wave, we shall show that if the initial data are close to a travelling wave solution, the solutions to (1.1) will approach the travelling wave solution in sup norm as the time goes to infinity. For the constitutive relations, we shall discuss two cases: convex and nonconvex
Existence and Asymptotic Behavior of Measure-Valued Solutions for Degenerate Conservation Laws

AbstractWe consider two classes of typical degenerate hyperbolic systems of conservation laws to provide a general approach for solving the existence and large-time asymptotic behavior of measure-valued solutions for initial-boundary value problems. Some existence theorems of the measure-valued solutions are established. The convergence of large time-averages of the measure-valued solutions to a Dirac mass, concentrated at the input state on the boundary, is proved for almost each fixed space variable. Although the measure-valued solutions of the initial-boundary problems may not be unique in general, our results indicate that the asymptotic equilibrium of these measure-valued solutions is unique
Elliptic Eigenvalue Problems and Unbounded Continua of Positive Solutions of a Semilinear Elliptic Equation

AbstractWe derive a result on the limit of certain sequences of principal eigenvalues associated with some elliptic eigenvalue problems. This result is then used to give a complete description of the global structure of the curves of positive steady states of a parameter dependent diffusive version of the classical logistic equation. In particular, we characterize the bifurcation values from infinity to positive steady states. The stability of the positive steady states as well as the asymptotic behaviour of positive solutions is also discussed
Large-Eigenvalue Global Existence and Regularity Results for the Navier–Stokes Equation

AbstractRecently Raugel and Sell obtained global existence results for the Navier–Stokes equation requiring that certain products involving the size of the data and the thinness of the domain be small. Thus the initial and forcing data could actually be quite large if the domain was thin enough. These results were obtained for periodic, and a case of homogeneous mixed periodic-Dirichlet, boundary conditions. We develop integral-equation techniques that allow us to obtain similar results in the case of purely homogeneous-Dirichlet boundary conditions. Our results are fairly simple to state and hold in a general setting, whereby we replace the role of the thinness of the domain by the reciprocal of the first eigenvalue of the Laplacian. We show further utulity of the integral-equation techniques by bootstrapping globalH1-bounds, whenever available in 2-dor 3-d, into higher-order global bounds with slightly smoother forcing functions than those assumed by Guillope, but otherwise more general in thatLp-integrable singularities in time are allowed
Large Solutions and Smoothing Properties for Nonlinear Thermoelastic Systems

AbstractWe study the existence of global solutions for large data for a class of systems of thermoelastic type with non-local nonlinearities. Moreover, we investigate smoothing properties of such coupled systems in an abstract setting, including some models for thermoelastic plates. In this setting, classical second-order thermoelasticity and viscoleasticity of integral type are recognized as limiting cases. Decay rates of Sobolev norms of the solutions as time tends to infinity are also investigated
A Finite-Dimensional Attractor for Three-Dimensional Flow of Incompressible Fluids

AbstractThe asymptotic behavior of the flow for a system of the Navier–Stokes type is investigated. In the considered model, the viscous part of the stress tensor is generally a nonlinear function of the symmetric part of the velocity gradient. Provided that the function describing this dependence satisfies the polynomial (p−1) growth condition, a unique weak solution exists if eitherp⩾(2+n)/2 andu0∈Horp⩾1+2n/(n+2) andu0∈W1, 2(C)n∩H. In the first case, the existence of a global attractor inHis proved. In order to indicate the finite-dimensional behavior of the flow at infinity, the fractal dimension of the new invariant set, composed from all shortδ-trajectories with initial value in the attractor, is estimated in theL2(0, δ; H) topology. Having uniqueness only for more regular data in the second case, many trajectories can start from the initial valueu0∈H. This does not allow one to define a semigroup on the spaceH. Therefore, the set of short trajectoriesXsδ, closed inL2(0, δ; H), is introduced along with a semigroup working on this set. The existence of a global attractor with a finite fractal dimension is then demonstrated
Existence Theorems for Periodic Non-relativistic Maxwell–Chern–Simons Solitons

AbstractWe study the system of elliptic equations[formula]defined over a doubly-periodic domain inR2, where the coefficients are specifically given by the physical model. This system arises in a self-dual non-relativistic Maxwell–Chern–Simons theory coupled with a neutral scalar field in (2+1)-dimensional spacetime and the solutions represent multivortices known as condensates. Our existence results reveal that the number of vortices confined in a periodic cell domain can be arbitrary and that the Chern–Simons coupling parameter imposes no restriction to the existence of solutions
Nilpotent Singularities in Generic 4-Parameter Families of 3-Dimensional Vector Fields

AbstractThis paper deals with singularities of vector fields in R3having a 1-jet linear conjugate toy(∂/∂x)+z(∂/∂y). They first occur in generic 3-parameter families. In codimension?3 all such singularities are mutuallyC0equivalent. We give a proof of this, provide a good normal form for 3-parameter unfoldings, and show that all non-wandering behaviour in such an unfolding is of small amplitude. We also show that for codimension?4 there are exactly 5 types of singularities forC0equivalence. The proof relies on normal form theory, blowing-up, and estimation of Abelian integrals
Near-Collision Dynamics for Particle Systems with Quasihomogeneous Potentials

AbstractWe study collision and ejection orbits of 3-particle systems having the potentialW=U+V, whereUandVare homogeneous functions of degree −aand −b, respectively, with 1⩽a<b. We show that forb≠2, collision and ejection orbits tend to form asymptotically a central configuration. For the caseb=2, which corresponds to Maneff's gravitational law, we find a set of collision and ejection orbits reaching the triple collision manifold without asymptotic phase. This set contains an uncountable union of manifolds and has positive measure within the set of all rectilinear solutions
Global Solution and Regularizing Properties on a Class of Nonlinear Evolution Equation

AbstractIn this paper we will consider the equation[formula]where[formula]The initial value problem is proved to be locally well posed for initial data taken inD(A2)×D(A3/2) and globally well posed for small data, in this case we also show the exponential decay of the solution as time goes to infinity. The main result of this paper is to prove that the solution has the smooting effect property on the initial data. This means that, if the initial data belongs toD(A2)×D(A3/2) then the solutionubelongs toC∞(]0, +∞[; D(Ak)) ∀k∈N, providedM,N, andRareC∞-function
On the Relation between Global Properties of Linear Difference and Differential Equations with Polynomial Coefficients, II

AbstractThis paper is the second one in a series of three articles dealing with applications of the Mellin transformation to the theory of linear differential and difference equations with polynomial coefficients. In the previous part, we studied the case of a differential equation having at most regular singularities atOand ∞ and arbitrary singularities in the rest of the complex plane. This second part is concerned with differential equations having a regular singularity at ∞ and an irregular one at the origin of the complex plane. Using particular types of Mellin (or Pincherle) transforms of appropriate solutions of the differential equation, we construct two fundamental systems of solutions of an associated difference equation. Both fundamental systems admit the same asymptotic representation as the variable tends to ∞, one in the right half plane and the other in the left half plane. We discuss the corresponding connection problem and its relation to the Stokes phenomenon of the differential equation
Commutation Methods for Jacobi Operators

AbstractWe offer two methods of inserting eigenvalues into spectral gaps of a given background Jacobi operator: The single commutation method which introduces eigenvalues into the lowest spectral gap of a given semi-bounded background Jacobi operator and the double commutation method which inserts eigenvalues into arbitrary spectral gaps. Moreover, we prove unitary equivalence of the commuted operators, restricted to the orthogonal complement of the eigenspace corresponding to the newly inserted eigenvalues, with the original background operator. In addition we compute the (matrix-valued) Weylm-functions of the commuted operator in terms of the background Weylm-functions. Finally we show how to iterate the above methods and give explicit formulas for various quantities (such as eigenfunctions and spectra) of the iterated operators in terms of the corresponding background quantities and scattering matrix. Concrete applications include an explicit realization of the isospectral torus for algebro-geometric finite-gap Jacobi operators and theN-soliton solutions of the Toda and Kac–van Moerbeke lattice equations with respect to arbitrary background solutions
Global Attractivity and Singular Perturbation for a Class of Nonlinear Cooperative Systems

AbstractIn the first part of this paper it is proved a general principle for reaction-diffusion cooperative systems with concave nonlinearities. It is proved that there exists a unique coexistence state (u, v) if, and only if, the trivial solution is linearly unstable. In this case, (u, v) is globally asymptotically stable with respect to positive initial data. Moreover, if the trivial solution is linearly stable, it is globally asymptotically stable with respect to non-negative initial data. In the second part it is investigated the singular perturbation problem, it is shown that the positive solution of the diffusion model tends to the positive equilibrium of the purely kinetic model as the diffusion coefficients tend to zero, uniformly in compact subsets of the domain
Viscous Cahn–Hilliard Equation II. Analysis

AbstractThe viscous Cahn–Hilliard equation may be viewed as a singular limit of the phase-field equations for phase transitions. It contains both the Allen–Cahn and Cahn–Hilliard models of phase separation as particular cases; by specific choices of parameters it may be formulated as a one-parameter (sayα) homotopy connecting the Cahn–Hilliard (α=0) and Allen–Cahn (α=1) models. The limitα=0 is singular in the sense that the smoothing property of the analytic semigroup changes from being of the type associated with second order operators to the type associated with fourth order operators. The properties of the gradient dynamical system generated by the viscous Cahn–Hilliard equation are studied asαvaries in [0, 1]. Continuity of the phase portraits near equilibria is established independently ofα∈[0, 1] and, using this, a piecewise, uniform in time, perturbation result is proved for trajectories. Finally the continuity of the attractor is established and, in one dimension, the existence and continuity of inertial manifolds shown and the flow on the attractor detailed
Effective Stability and KAM Theory

AbstractThe two main stability results for nearly-integrable Hamiltonian systems are revisited: Nekhoroshev theorem, concerning exponential lower bounds for the stability time (effective stability), and KAM theorem, concerning the preservation of a majority of the nonresonant invariant tori (perpetual stability). To stress the relationship between both theorems, a common approach is given to their proof, consisting of bringing the system to a normal form constructed through the Lie series method. The estimates obtained for the size of the remainder rely on bounds of the associated vectorfields, allowing one to get the “optimal” stability exponent in Nekhoroshev theorem for quasiconvex systems. On the other hand, a direct and complete proof of the isoenergetic KAM theorem is obtained. Moreover, a modification of the proof leads to the notion of nearly-invariant torus, which constitutes a bridge between KAM and Nekhoroshev theorems
A Dynamical Approach to Convex Minimization Coupling Approximation with the Steepest Descent Method

AbstractWe study the asymptotic behavior of the solutions to evolution equations of the form 0∈u(t)+∂f(u(t), ε(t));  u(0)=u0, where {f(·, ε):ε>0} is a family of strictly convex functions whose minimum is attained at a unique pointx(ε). Assuming thatx(ε) converges to a pointx* as ε tends to 0, and depending on the behavior of the optimal trajectoryx(ε), we derive sufficient conditions on the parametrization ε(t) which ensure that the solutionu(t) of the evolution equation also converges tox* whent→+∞. The results are illustrated on three different penalty and viscosity-approximation methods for convex minimization
Stable Solutions with Zeros to the Ginzburg–Landau Equation with Neumann Boundary Condition

AbstractThis paper is devoted to the Ginzburg–Landau equationΔΦ+λ(1−|Φ|2)Φ=0,Φ=u1+iu2in a bounded domainΩ⊂Rnwith the homogeneous Neumann boundary condition. The previous works [12–14] showed that for largeλthere exist stable non-constant solutions with no zeros in domains, which are topologically non-trivial in a certain sense. In this aritcle it is proved that for a domainΩcontaining a non-trivial domainDas a subset, there exist stable solutions with zeros provided that the volume ofΩ\Dis sufficiently small
Asymptotic Behavior and Symmetry of Internal Waves in Two-Layer Fluids of Great Depth

AbstractThis paper studies certain asymptotic and geometric properties of internal waves at the interface of a two-layer fluid flow of infinite depth bounded below by a rigid bottom under influence of gravity. It is shown that if the governing equations of the flow have a nontrivial solution which approaches to a supercritical equilibrium state at infinity, then the solution decays to the equilibrium exactly with an orderO(1/x2) for largexwherexis the horizontal variable. Furthermore, the solution is symmetric. The interface is always above the equilibrium state and monotonically decreasing for positivexand increasing for negativex. The exact decay estimates are obtained using the properties of Green's function for an integro-differential equation and some tools from harmonic analysis. The proof of symmetry is similar to the one given by Craig and Sternberg for a two-fluid flow of finite depth using the Alexandrov method of moving planes
Singularities of a Variational Wave Equation

AbstractWe analyze several aspects of the singular behavior of solutions of a variational nonlinear wave equation which models orientation waves in a massive nematic liquid crystal director field. We prove that smooth solutions develop singularities in finite time. We construct exact travelling wave solutions with cusp singularities, and use them to illustrate a phenomena of accumulation and annihilation of oscillations in sequences of solutions with bounded energy. We also prove that constant solutions of the equation are nonlinearly unstable
Finite Dimensionality and Upper Semicontinuity of the Global Attractor of Singularly Perturbed Hodgkin–Huxley Systems

AbstractIn an effort to show that the standard Hodgkin–Huxley system approximates the hyperbolic Hodgkin–Huxley system we view the hyperbolic system as a singular perturbation of the standard system. We establish the existence of global attractors for the singularly perturbed hyperbolic system and show that they have finite fractal and Hausdorff dimensions. We then show that the global attractors of the singularly perturbed system converge to the global attractor for the standard system as the small perturbation parameter decreases to zero
Domain Perturbation for Linear and Nonlinear Parabolic Equations

AbstractWe shall develop a general theory for domain perturbation for linear and nonlinear parabolic equations with measurable coefficients subject to Dirichlet boundary conditions. We show how solutions of linear and nonlinear parabolic equations behave as a sequence of domains approaches an open set. Convergence of domains is understood in a very general sense which allows that certain parts of the domains degenerate and are deleted in the limit, or that small sets are removed. We also consider the periodic problem and establish existence and uniqueness results for periodic solutions for the perturbed problem in a neighborhood of a periodic solution of the original equation. The theory can be used for instance to construct domains where a given nonlinear periodic–parabolic equation has many periodic solutions
Shadowing Matching Errors for Wave-Front-Like Solutions

AbstractConsider a singularly perturbed system[formula]Assume that the system has a sequence of regular and internal layers occurring alternatively along thexdirection. These “multiple wave” solutions can formally be constructed by matched asymptotic expansions. To obtain a genuine solution, we derive aSpatial Shadowing Lemmawhich assures the existence of an exact solution that is near the formal asymptotic series provided (1) the residual errors are small in all the layers, and (2) the matching errors are small along the lateral boundaries of the adjacent layers. The method should work on some other systems likeεut=−(−ε2Dxx)mu+…
Oscillation Theory and Renormalized Oscillation Theory for Jacobi Operators

AbstractWe provide a comprehensive treatment of oscillation theory for Jacobi operators with separated boundary conditions. Our main results are as follows: Ifusolves the Jacobi equation (Hu)(n)=a(n)u(n+1)+a(n−1)u(n−1)−b(n)u(n)=λu(n),λ∈R (in the weak sense) on an arbitrary interval and satisfies the boundary condition on the left or right, then the dimension of the spectral projectionP(−∞, λ)(H) ofHequals the number of nodes (i.e., sign flips ifa(n)<0) ofu. Moreover, we present a reformulation of oscillation theory in terms of Wronskians of solutions, thereby extending the range of applicability for this theory; ifλ1, 2∈R and ifu1, 2solve the Jacobi equationHuj=λjuj,j=1, 2 and respectively satisfy the boundary condition on the left/right, then the dimension of the spectral projectionP(λ1, λ2)(H) equals the number of nodes of the Wronskian ofu1andu2. Furthermore, these results are applied to establish the finiteness of the number of eigenvalues in essential spectral gaps of perturbed periodic Jacobi operators
Global Large Time Self-Similarity of a Thermal-Diffusive Combustion System with Critical Nonlinearity

AbstractWe study the initial value problem of the thermal-diffusive combustion systemu1, t=u1, x, x−u1u22,u2, t=du2, xx+u1u22,x∈R1, for non-negative spatially decaying initial data of arbitrary size and for any positive constantd. We show that if the initial data decay to zero sufficiently fast at infinity, then the solution (u1, u2) converges to a self-similar solution of the reduced systemu1, t=u1, xx−u1u22,u2, t=du2, xx, in the large time limit. In particular,u1decays to zero like O(t−1/2−δ), whereδ>0 is an anomalous exponent depending on the initial data, andu2decays to zero with normal rate O(t−1/2). The idea of the proof is to combine the a priori estimates for the decay of global solutions with the renormalization group method for establishing the self-similarity of the solutions in the large time limit
The Initial Boundary Value Problems for Hyperbolic Conservation Laws with Relaxation

AbstractThe hyperbolic conservation laws with relaxation appear in many physical models such as those for gas dynamics with thermo-non-equilibrium, elasticity with memory, flood flow with friction, traffic flow, etc.. The main concern of this article is the long-time effect of the relaxations on the boundary layer behaviors. In this article, we investigate this problem for a simple model of 2×2 systems. Conditions relating the boundary data and far field states are found for the existence of the boundary layers. Also, it is proven that the boundary layers thus obtained are nonlinearly stable
A Functional Integral Approach to Shock Wave Solutions of the Euler Equations with Spherical Symmetry (II)

AbstractThis is a continuation of [15]. As is well known, one dimensional conservation laws without source term have been extensively investigated after the foundamental paper of J. Glimm [3]. And those with integrable source terms were solved by Liu [6, 7], etc. For higher dimensional case with spherical symmetry, Makino [8] first proved a linear growth rate for solutions whenP(ρ)=σ2ργ, whereγ=1. But in order to get global existence forγ≠1 and the decay property, we need to find a uniform bound for the approximate solutions. In [15], we introduced a new norm and a functional integral approach to prove a uniform bound for a model problem of Euler equation inR3with spherical symmetry. In order to overcome the geometric effects of spherical symmetry which leads to a non-integrable source term, we considered an infinite reflection problem and solved it by considering the cancellations between reflections of different orders. In this paper, we consider a system which describes the isentropic and spherically symmetric motion of gas flow surrounding a solid star with radius 1 and massM. It is interesting to note that the wave curves for this problem are no longer continuous and there is an extra term in the wave interaction estimates. By introducing a new norm, we prove a similar result as [15]. In the Appendix, we present a local existence theorem forγ≠1 which was also obtained by Makino [9] by different method. And we extend the results to the cases with different boundary conditions