pinecone/core-2020-05-10-deduplication huggingface dataset:
Higher Energy Solutions in the Theory of Phase Transitions: A Variational Approach

AbstractWe establish the existence of a higher-energy solution to the vector Ginzburg–Landau equation with a triple-well potential on a bounded and smooth domain on the plane. This solution is obtained by a linking argument. In implementing this variational approach we make several considerations on the dynamics of the negative gradient flow. In particular, we use the Conley index to contruct a suitable one-dimensional invariant set. This solution has Morse index two in the nondegenerate case. We discuss its structure in connection with the so-called triple-junction configurations
Some Results on the Navier–Stokes Equations in Thin 3D Domains

AbstractWe consider the Navier–Stokes equations on thin 3D domains Qε=Ω×(0, ε), supplemented mainly with purely periodic boundary conditions or with periodic boundary conditions in the thin direction and homogeneous Dirichlet conditions on the lateral boundary. We prove global existence and uniqueness of solutions for initial data and forcing terms, which are larger and less regular than in previous works on thin domains. An important tool in the proofs are some Sobolev embeddings into anisotropic Lp-type spaces. Better results are proved in the purely periodic case, where the conservation of enstrophy property is used. For example, when the forcing term vanishes, we prove global existence and uniqueness of solutions if ‖(I−M)u0‖H1/2(Qε)exp(C−1ε−1/s‖Mu0 ‖2/sL2(Qε))⩽C for both boundary conditions or ‖Mu0‖H1(Qε)⩽Cε−β, ‖(Mu0)3‖L2(Qε)⩽Cεβ, ‖(I−M)u0‖H1/2(Qε)⩽Cε1/4−β/2 for purely periodic boundary conditions, where 1/2<s<1 and 0⩽β⩽1/2 are arbitrary, C is a prescribed positive constant independent of ε, and M denotes the average operator in the thin direction. We also give a new uniqueness criterium for weak Leray solutions
Singularly Perturbed Elliptic Problems in the Case of Exchange of Stabilities

AbstractWe consider the singularly perturbed boundary value problem (Eε)ε2Δu=f(u, x, ε) for x∈D, ∂u∂n−λ(x)u=0 for x∈Γ where D⊂R2 is an open bounded simply connected region with smooth boundary Γ, ε is a small positive parameter and ∂/∂n is the derivative along the inner normal of Γ. We assume that the degenerate problem (E0)f(u, x, 0)=0 has two solutions ϕ1(x) and ϕ2(x) intersecting in an smooth Jordan curve C located in D such that fu(ϕi(x), x, 0) changes its sign on C for i=1, 2 (exchange of stabilities). By means of the method of asymptotic lower and upper solutions we prove that for sufficiently small ε, problem (Eε) has at least one solution u(x, ε) satisfying α(x, ε)⩽u(x, ε)⩽β(x, ε) where the upper and lower solutions β(x, ε) and α(x, ε) respectively fulfil β(x, ε)−α(x, ε)=O(ε) for x in a δ-neighborhood of C where δ is any fixed positive number sufficiently small, while β(x, ε)−α(x, ε)=O(ε) for x∈D\Dδ. In case that f does not depend on ε these estimates can be improved. Applying this result to a special reaction system in a nonhomogeneous medium we prove that the reaction rate exhibits a spatial jumping behavior
Perturbations of Normally Hyperbolic Manifolds with Applications to the Navier–Stokes Equations

AbstractThere are two objectives in this paper. First we develop a theory which is valid in the infinite dimensional setting and which shows that, under reasonable conditions, if M is a normally hyperbolic, compact, invariant manifold for a semiflow S0(t) generated by a given evolutionary equation on a Banach space W, then for every small perturbation G of the given evolutionary equation, there is a homeomorphism hG:M→W such that MG=hG(M) is a normally hyperbolic, compact, invariant manifold for the perturbed semiflow SG(t), and that hG converges to the identity mapping (on M), as G converges to 0. The second objective is to develop a methodology which is rich enough to show that this theory can be easily applied to a wide range of applications, including the Navier–Stokes equations. It is noteworthy in this regard that, in order to be able to apply this theory in the analysis of numerical schemes used to study discretizations of partial differential equations, one needs to use a new measure or norm of the perturbation term G that arises in these schemes
Dynamical Systems and Traveling Waves in Almost Periodic Structures

AbstractThe paper is devoted to the study of traveling waves in almost periodic structures by applying dynamical system theory. Motivated by the fact that almost automorphic dynamics often exists (though almost periodic dynamics may not) in scalar almost periodic ODEs and parabolic PDEs, we introduce a definition of almost automorphic traveling waves and investigate the existence of such traveling waves in general almost periodic structures. In the author's earlier work, a notion of almost periodic traveling wave solution is introduced. Roughly, a solution is an almost automorphic (almost periodic) traveling wave solution if its propagating profile and speed are almost automorphic (almost periodic) functions. Our basic point of view is that traveling wave solutions are the limits of certain wave-like solutions. We therefore study the existence of almost automorphic traveling wave solutions through the long time behavior of wave-like solutions. We first investigate the existence of wave-like solutions. Then by utilizing dynamical system theory, we explore the “convergence” of wave-like solutions to alsmot automorphic (almost periodic) traveling wave solutions. Besides, based on the arguments established in the author's earlier work on bistable equations, we show the stability, uniqueness, and almost periodicity of almost automorphic traveling wave solutions in almost periodic equations of multi-stable type. Applying the general theory developed in the paper to almost periodic equations of KPP type, we show the existence of a family of almost automorphic traveling waves under some conditions
Constitutive Relations for Rapid Granular Flow of Smooth Spheres: Generalized Rational Approximation to the Sum of the Chapman–Enskog Expansion

AbstractN. Sela and I. Goldhirsch (1998, J. Fluid Mech.361, 41–74) have used the Chapman–Enskog expansion to derive constitutive relations for the pressure deviator P, heat flux q, and rate of energy loss Γ for rapid flows of smooth inelastic spheres. Unfortunately as in the classical Chapman–Enskog expansion for elastic spheres any truncation of the expansion beyond Navier–Stokes order (n=1) will possess unphysical instabilities. This paper uses the method of approximate summation of the Chapman–Enskog expansion presented by the author (in press, Arch. Rational Mech. Anal.) to eliminate these instabilities
Period Function for a Class of Hamiltonian Systems

AbstractThis paper studies the period function of the class of Hamiltonian systems x=−Hy, y=Hx where H(x, y) has the special form H(x, y)=F(x)+G(y) and the origin is a non-degenerate center. More concretely, if T(h) denotes the period of the periodic orbit contained in H(x, y)=h we solve the inverse problem of characterizing all systems with a given function T(h). We also characterize the limiting behaviour of T at infinity when the origin is a global center and apply this result to prove, among other results, that there are no nonlinear polynomial isochronous centers in this family
Multiplicity Result for a Degenerate Elliptic Equation with Logistic Reaction

AbstractA degenerate elliptic equation λΔpu+uq−1(1−ur)=0 with zero Dirichlet boundary condition, where λ is a positive parameter, 2<p<q, and r>0, is studied. It has been known that there exists a positive number Λ such that if λ>Λ, then the problem has no positive solution; if λ⩽Λ, then there exists a maximal solution; and for sufficiently small λ (<Λ), the problem has at least two solutions. In this article, it is shown that if λ<Λ, then there exist at least two positive solutions, via the variational method
The Effect of Domain Squeezing upon the Dynamics of Reaction-Diffusion Equations

AbstractLet Ω be an arbitrary smooth bounded domain in RM×RN and ε>0 be arbitrary. Write (x, y) for a generic point of RM×RN. Squeeze Ω by the factor ε in the y-direction to obtain the squeezed domain Ωε={(x, εy)∣(x, y)∈Ω}. Consider the following reaction-diffusion equation on Ωε:[formula] Here, νε is the exterior normal vector field on ∂Ωε and f:R→R is a nonlinearity satisfying some growth and dissipativeness conditions assuring such that (Eε) generates a semiflow on H1(Ωε) with a global attractor Aε. We prove that, in some strong sense, the equations (Eε) have a limiting equationu+Au=f(u) (E0) as ε→0. This limiting equation is an abstract semilinear parabolic equation which defines a semiflow π on a closed linear subspace of H1(Ω). We show that π has a global attractor A0 and the family of attractors (Aε)ε⩾0 is upper-semicontinuous at ε=0. If M=N=1 and Ω satisfies some natural additional assumptions, then the limiting equation (E0) is equivalent to a parabolic boundary value problem defined on a finite graph. The results of this paper extend previous results obtained by Hale and Raugel for domains which are ordinate sets of a positive function
Diffusion Semigroups in Spaces of Continuous Functions with Mixed Topology

AbstractWe study transition semigroups and Kolmogorov equations corresponding to stochastic semilinear equations on a Hilbert space H. It is shown that the transition semigroup is strongly continuous and locally equicontinuous in the space of polynomially increasing continuous functions on H when endowed with the so-called mixed topology. As a result we characterize cores of certain second order differential operators in such spaces and show that they have unique extensions to generators of strongly continuous semigroups
Stability of the Inverse Conductivity Problem in the Plane for Less Regular Conductivities

AbstractThe Dirichlet to Neumann map Λγ, or voltage to current map, takes Dirichlet data u=f∈∂Ω to the conormal derivatives γ∂u∂η at the boundary ∂Ω, where u is a solution to the elliptic equation div(γ∇u)=0. On a plane Lipchitz domain Ω⊂R2, we consider the inverse conductivity problem that consists of the recovery of γ in the interior of Ω from the knowledge of Λγ. We prove stability for this inverse map, assuming an isotropic conductivity γ∈C1+ε(Ω). We reduce the equation div(γ∇u)=0 to an equivalent system and use the ∂∂-scattering transform. This approach relaxes the number of derivatives of γ required in former works
Marchenko–Ostrovki Mapping for Periodic Zakharov–Shabat Systems

AbstractConsider the Zakharov–Shabat (or Dirac) operator T in the Hilbert space L2(R)⊕L2(R), with real periodic vector potential q=(q1, q2)∈H=L2(T)⊕L2(T). The spectrum of T is absolutely continuous and consists of intervals separated by gaps gn=(z−n, z+n), n∈Z. From the Dirichlet eigenvalue mn, n∈Z, of the Zakharov–Shabat equation with Dirichlet boundary conditions at 0, 1, the square of the height of vertical slits on the quasimomentum domain, and the points on these slits, we construct the Marchenko–Ostrovki (vertical slits) mapping for the periodic Zakharov–Shabat systems h:H→ℓ2⊕ℓ2. Using nonlinear functional analysis in Hilbert spaces, we show that this mapping is a real analytic isomorphism. In the second part of our paper we prove a new identity for the effective masses
Multipulses of Nonlinearly Coupled Schrödinger Equations

AbstractThe capacity of coupled nonlinear Schrödinger (NLS) equations to support multipulse solutions (multibump solitary-waves) is investigated. A detailed analysis is undertaken for a system of quadratically coupled equations that describe the phenomena of second harmonic generation and parametric wave interaction in non-centrosymmetric optical materials. Utilising the framework of homoclinic bifurcation theory, and employing a Lyapunov–Schmidt reduction method developed by Hale, Lin, and Sandstede, a novel mechanism for the generation of multipulses is identified, which arises from a resonant semi-simple eigenvalue configuration of the linearised steady-state equations. Conditions for the existence of multipulses, as well as a description of their geometry, are derived from the analysis
On a Reverse Estimate for Hodge Decompositions of p-Laplacian Type Operators

AbstractLet σ(x, ξ)≈|ξ|p−2ξ be a p-Laplacian type operator and consider the Hodge decomposition σ(x, Du)=Dϕ+H, divH=0. A standard elliptic theory asserts that ‖Dϕ‖q/(p−1)⩽C‖Du‖p−1q for all q>p−1. There has been considerable recent interest in the validity of the reverse estimate ‖Du‖p−1q⩽C‖Dϕ‖q/(p−1) for q>p−1 in the regularity study of certain geometrical mappings. In this paper, we give a relatively new proof of a well-known theorem that this reverse estimate holds for all q sufficiently close to the natural power p and also prove that the estimate holds for all q⩾p−1 for certain special weak solutions u
On the Domain of Analyticity for Solutions of Second Order Analytic Nonlinear Differential Equations

AbstractThe radius of analyticity of periodic analytic functions can be characterized by the decay of their Fourier coefficients. This observation has led to the use of so-called Gevrey norms as a simple way of estimating the time evolution of the spatial radius of analyticity of solutions to parabolic as well as non-parabolic partial differential equations. In this paper we demonstrate, using a simple, explicitly solvable model equation, that estimates on the radius of analyticity obtained by the usual Gevrey class approach do not scale optimally across a family of solutions, nor do they scale optimally as a function of the physical parameters of the equation. We attribute the observed lack of sharpness to a specific embedding inequality, and give a modified definition of the Gevrey norms which is shown to finally yield a sharp estimate on the radius of analyticity
Relaxation Oscillation and Canard Explosion

AbstractWe give a geometric analysis of relaxation oscillations and canard cycles in singularly perturbed planar vector fields. The transition from small Hopf-type cycles to large relaxation cycles, which occurs in an exponentially thin parameter interval, is described as a perturbation of a family of singular cycles. The results are obtained by means of two blow-up transformations combined with standard tools of dynamical systems theory. The efficient use of various charts is emphasized. The results are applied to the van der Pol equation
Self-Adjoint Domains of Products of Differential Expressions

AbstractUnder the assumption that the product l2 of the formally symmetric differential expression l of order n on [a, ∞) is partially separated in L2[a, ∞), we present a new characterization of self-adjoint boundary conditions for l2. For two differential operators T1(l) and T2(l) associated with l, we show that the product T2(l)T1(l) is self-adjoint if and only if T2(l)=T*1(l). It extends the previous result in [1], where both T1(l) and T2(l) are self-adjoint, singular limit-circle Sturm–Liouville operators. Furthermore, we also characterize the boundary conditions of the Friedrichs extension of the minimal operator generated by l2
Time-Averaging under Fast Periodic Forcing of Parabolic Partial Differential Equations: Exponential Estimates

AbstractThe phases of a large class of parabolic partial differential equations with rapid time-periodic forcing can be separated up to exponential small errors. The originally nonautonomous equation is transformed such that the nonautonomous terms are exponentially small in the period h of the forcing. This is a counterpart for partial differential equations of the theorem by A. Neishtadt (1984, J. Appl. Math. Mech.48, 134–139) for ordinary differential equations. In our case the exponential rate depends on time t and the estimates have the form hexp(−c(t)h−1/3)
Convergence to Strong Nonlinear Diffusion Waves for Solutions of p-System with Damping

AbstractThis paper is concerned with the p-system with frictional damping and our main purpose is two-fold: First, we show that for a certain class of given large initial data (v0(x), u0(x)), the Cauchy problem (1.1), (1.2) admits a unique global smooth solution (v(t, x), u(t, x)) and such a solution tends time-asymptotically, at the optimal Lp(2⩽p⩽∞) decay rates, to the corresponding nonlinear diffusion wave (v(t, x), u(t, x)) governed by the classical Darcy's law provided that the corresponding prescribed initial error function (V0(x), U0(x)) lies in (H3×H2)(R)∩(L1×L1)(R). Compared with former results in this direction obtained by L. Hsiao and T.-P. Liu (1992, Comm. Math. Phys.143, 599–605), K. Nishihara (1996, J. Differential Equations131, 171–188), and K. Nishihara, W.-K. Wang, and T. Yang (2000, J. Differential Equations161, 191–218), our main novelty lies in the facts that the nonlinear diffusion wave (v(t, x), u(t, x)) need not to be weak and (V0(x), U0(x)) can be chosen arbitrarily large. Secondly, we show that the nonlinear diffusion waves (v(t, x), u(t, x)) are nonlinear stable provided that the strength of the nonlinear diffusion waves is weak and that the initial disturbance (V0(x), U0(x)) satisfies the assumption that ‖V0xx(x)‖L∞+‖U0x(x)‖L∞ is sufficiently small. We also show that the smallness assumption imposed on the strength of the diffusion waves is a necessary condition to guarantee the nonlinear stability result and compared with the corresponding results obtained by L. Hsiao and T.-P. Liu (1992, Comm. Math. Phys.143, 599–605), the smallness conditions we imposed on the initial disturbance are much more weaker
Well-posedness of a Structural Acoustics Control Model with Point Observation of the Pressure

AbstractWe consider a controlled and observed partial differential equation (PDE) which describes a structural acoustics interaction. Physically, this PDE describes an acoustic chamber with a flexible chamber wall. The control is applied to this flexible wall, and the class of controls under consideration includes those generated by piezoceramic patches. The observation we consider is point measurements of acoustic pressure inside the cavity. Mathematically, the model consists of a wave equation coupled, through boundary trace terms, to a structurally damped plate (or beam) equation, and the point controls and observations for this system are modeled by highly unbounded operators. We analyze the map from the control to the observation, since the properties of this map are central to any control design which is based upon this observation. We also show there exists an appropriate state space X, so that if the initial state is in X and the control is in L2, then the state evolves continuously in X and the observation is in L2. The analysis of this system entails a microlocal analysis of the wave component of the system, and the use of pseudodifferential machinery
Pointwise Estimates for Oblique Derivative Problems in Nonsmooth Domains

AbstractThere is a long history of studying boundary value problems for elliptic and parabolic differential equations when the equation is posed in a domain with sufficiently smooth boundary. In this paper, we prove some pointwise estimates and regularity results for solutions when a directional derivative is prescribed on the boundary, which satisfies an interior cone condition. We consider very weak hypotheses on the data of the problem. In particular, we consider an optimal relation between the interior cone and the direction of the prescribed directional derivative, and we assume very little smoothness of the coefficients in the equation and the boundary condition
Slow Motion Manifolds Far from the Attractor in Multistable Reaction–Diffusion Equations

AbstractWe consider a scalar reaction–diffusion equation with multistable nonlinearity with a particular symmetry. By reduction to a family of transmission problems in R, and by contraction arguments, a manifold close to an invariant manifold formed by functions exhibiting a pattern of transition layers is constructed. An approximation for the associated vector field is also provided. This shows that the motion on those manifolds is exponentially slow, as in the well-known case of the bistable equation. However, in opposition to the bistable case, some of these manifolds are far from the attractor. Since these manifolds correspond to metastable patterns, this shows the importance of the transient motion toward the attractor and the importance of these manifolds in organizing that motion. It is also shown that by a suitable perturbation we can obtain new equilibria on those manifolds
On the Defect of Compactness for the Strichartz Estimates of the Schrödinger Equations

AbstractIn this paper, we prove that every sequence of solutions to the linear Schrödinger equation, with bounded data in H1(Rd), d⩾3, can be written, up to a subsequence, as an almost orthogonal sum of sequences of the type h−(d−2)/2nV((t−tn)/h2n, (x−xn)/ hn ), where V is a solution of the linear Schrödinger equation, with a small remainder term in Strichartz norms. Using this decomposition, we prove a similar one for the defocusing H1-critical nonlinear Schrödinger equation, assuming that the initial data belong to a ball in the energy space where the equation is solvable. This implies, in particular, the existence of an a priori estimate of the Strichartz norms in terms of the energy
The Neumann Problem in a 2-D Exterior Domain with Cuts and Singularities at the Tips

AbstractThe Neumann problem for the harmonic functions in an exterior connected plane region with cuts is studied. The problem is considered with different conditions at infinity, which lead to different theorems on uniqueness and solvability. The existence of a classical solution is proved by potential theory. The problem is reduced to a Fredholm equation of the second kind, which is uniquely solvable. Explicit formulas for singularities of a gradient of the solution at the tips of the cuts are obtained. The results of the paper can be used to model the flow of an ideal fluid over several obstacles, including wings
On Lower Dimensional Invariant Tori in Reversible Systems

AbstractIn this paper, a result on the persistence of lower dimensional invariant tori in reversible systems is obtained under some weak non-degenerate conditions. Such a result can be applied to the case where normal frequencies are not simple and some of them may be zero. We also give an answer of a conjecture of M. B. Sevryuk (1986, “Reversible Systems,” Lecture Notes in Math., Vol. 1211, Springer-Verlag, New York/Berlin) about the existence of lower-dimensional invariant tori in resonant zones
On the Piecewise Smooth Solutions to Non-homogeneous Scalar Conservation Laws

AbstractWe study the structure and smoothness of non-homogeneous convex conservation laws. The question regarding the number of smoothness pieces is addressed. It is shown that under certain conditions on the initial data the entropy solution has only a finite number of discontinuous curves. We also obtain some global estimates on derivatives of the piecewise smooth entropy solution along the generalized characteristics. These estimates play important roles in obtaining the optimal rate of convergence for various approximation methods to conservation laws
The Spectrum of Differential Operators of Order 2n with Almost Constant Coefficients

AbstractWe discuss the spectral properties of higher order ordinary differential operators. If the coefficients differ from constants by small perturbations, then the spectral properties are preserved. In this context, “small perturbations” are either short range (i.e., integrable) or long range, but slowly varying. This generalizes classical results on second order operators. Our approach relies on an analysis of the associated differential equations with the help of uniform asymptotic integration techniques