Abstract. Inspired by the fundamental work of Escobedo and Velázquez [ Invent. Math., 200 (2015), pp. 761–847; Mem. Amer. Math. Soc., 238 (2015), 1124], we prove that solutions of 4-wave kinetic equations, under very general forms of the dispersion relations, exhibit energy transfer toward large wavenumbers as time evolves. We also establish a global existence result for mild solutions of the equation under general dispersion relations. To the best of our knowledge, this is the first study on the long-term asymptotic behavior of solutions to 4-wave kinetic equations in such a general setting.
Abstract. This paper demonstrates the unconditional global stability of solutions to the initial-boundary value problem for a parabolic-elliptic coupled system in the two-dimensional or three-dimensional half-space. It is shown that as time approaches infinity, the solution converges to a planar rarefaction wave at a specific decay rate. The concept of unconditional global stability means that these conclusions hold under arbitrarily large initial perturbation and wave strength. The proof relies on transforming the parabolic-elliptic coupled system into a nonlocal scalar equation in the half-space. By capitalizing on this transformation, the maximum principle is employed to deduce the boundedness and monotonicity of the solution, which are crucial for energy estimates under large perturbations and the construction of one-dimensional smooth rarefaction waves. The primary challenge lies in the higher-order energy estimates due to boundary layer. We first estimate the energy of tangential derivatives and then use iterative methods based on the governing equations to derive estimates of all derivatives.
Abstract. We prove that, among all subsets [Formula: see text] having circular symmetry and prescribed measure, the ball is the only maximizer of the sum of the first [Formula: see text] eigenvalues ([Formula: see text]) of the corresponding Toeplitz operator [Formula: see text] on the Fock space [Formula: see text]. As a byproduct, we prove that, again among circularly symmetric sets of prescribed measure, balls maximize any Schatten [Formula: see text]-norm of [Formula: see text] for [Formula: see text] (and minimize the corresponding quasinorm for [Formula: see text]), and that the second eigenvalue is maximized by a particular annulus. Moreover, we extend some of these results to general radial symbols in [Formula: see text] with [Formula: see text], characterizing those that maximize the sum of the first [Formula: see text] eigenvalues. We also show a symmetry breaking phenomenon for the second eigenvalue, when the assumption of circular symmetry is dropped.
Abstract. This paper is concerned with the existence of genuine 3D large amplitude subsonic steady-state for the isentropic Euler–Poisson system when the flow sufficiently approaches the sonic speed, namely, the Mach number [Formula: see text] but sufficiently close to 1. We first choose a subsonic radial solution as a background solution, and then we investigate the existence of a genuine 3D solution as a perturbation of the radial background solution. By utilizing estimates for a quasi-linear elliptic system, we prove that when [Formula: see text] with [Formula: see text] is a positive constant, where [Formula: see text] is the relaxation time and [Formula: see text] is the largest Mach number of the background solution, the perturbation of the background solution exists for the potential flow. Thus, we obtain the unique genuine 3D subsonic steady-state solution [Formula: see text] around the background state with large amplitude in [Formula: see text] with [Formula: see text]. Although the targeted system under consideration is still subsonic—even for the flow with the Mach number sufficiently close to 1—the genuine 3D subsonic solution still exists when the relaxation time is big enough at [Formula: see text]. This result significantly improves and develops the existing studies.
Abstract. We construct a series of classic vorticity solutions for the incompressible Euler equation on [Formula: see text], which constitute the [Formula: see text] type regularization for a general traveling point vortex system. The construction is accomplished by applying a tangent mapping on [Formula: see text] and the Lyapunov–Schmidt reduction argument. By introducing a normalized condition to deal with the rotation invariance on [Formula: see text], we prove that the vortices will be located near a 1-dimensional degenerate normalized critical point of the Kirchhoff–Routh function. Moreover, in the tangent space at each vortex center, the scaled stream function is verified as a perturbation of the ground state for a generalized plasma problem. Some other qualitative and quantitative estimates for the regularization series are also obtained in this paper.
Abstract. We consider the Cauchy problem for an integrable cubic nonlinear Schrödinger equation in one space dimension. We prove global spatial analyticity of solutions via the inverse scattering transform. Consequently we remove the size restriction on the analytic norm when initial data have radially or odd symmetry.
We construct a series of classic vorticity solutions for the incompressible Euler equation on 2, which constitute the C1 type regularization for a general traveling point vortex system. The construction is accomplished by applying a tangent mapping on 2 and the Lyapunov-Schmidt reduction argument. By introducing a normalized condition to deal with the rotation invariance on 2, we prove that the vortices will be located near a 1-dimensional degenerate normalized critical point of the Kirchhoff--Routh function. Moreover, in the tangent space at each vortex center, the scaled stream function is verified as a perturbation of the ground state for a generalized plasma problem. Some other qualitative and quantitative estimates for the regularization series are also obtained in this paper.
We continue our study, initiated in [E. O. Hiltunen et al., SIAM J. Math. Anal., 56 (2024), pp. 3802--3831], of the quantum optics of a single photon interacting with a system of two level atoms. In this work, we investigate the case of a periodic arrangement of atoms. We provide a general structure theorem characterizing the band functions of this problem, which comprise the spectrum of the associated Hamiltonian. Additionally, we study atomic densities arising as periodically arranged scaled inclusions. For this family of examples, we obtain explicit asymptotic formulas for the band functions.
We derive the one-fluid ion model from the partial-compressible two-fluid Euler-- Maxwell system with a collision mechanism in which the ions are considered compressible while electrons are not. Deriving such a model based solely on the zero-electron-mass limit or the quasi-neutral limit is quite open because of the singularities of the electric field and the strong coupling effects of the Lorentz forces. In this work, we apply the combined zero-electron-mass and quasi-neutral limit. We prove the global-in-time existence and the combined limit of smooth solutions. Furthermore, for well-prepared initial data, we establish the error estimates of the convergence between smooth solutions to one-fluid and two-fluid models. It is worth mentioning that the velocities of ions and electrons are different in this study.
We study the local behavior of bending deformations (isometric immersions) which have a fold along a regular curve \Sigma in the reference configuration. We derive a classical formula for the folding angle in terms of the curvature of \Sigma and the normal curvature of its deformed configuration, and we derive consequences of this formula. Then we introduce a natural condition (the ``torsion constraint"") under which we can prove the following intuitive dichotomy: either the folding angle is nonzero everywhere on \Sigma or there is no fold at all. Finally, we construct folded isometric immersions with a prescribed folding angle or with a prescribed deformation on one side of the fold. Most of our results in fact apply to pairs of framed curves with a common base curve---namely the deformed fold---which satisfy appropriate additional conditions on their curvatures.
We investigate the structural stability of spherically symmetric subsonic flows to the steady Euler-Poisson system in a three-dimensional divergent nozzle under axisymmetric perturbations of suitable boundary conditions, and establish the existence and uniqueness of smooth axisymmetric subsonic Euler--Poisson flows with nonzero swirl velocity and vorticity. The solution shares the same regularity for the velocity, the pressure, the entropy, and the electrostatic potential. The main difficulty for the axisymmetric flows with swirls is the singularity near the symmetry axis. With the aid of the deformation-curl-Poisson decomposition, the steady axisymmetric Euler--Poisson system is transformed into a deformation-curl-Poisson system and several transport equations. One of the crucial ingredients of the analysis is to obtain the well-posedness of the boundary value problem for the associated linearized second order singular elliptic system, which is achieved by reformulating the problem in three-dimensional spaces to remove the singularity and discovering a special structure of the system to derive a priori estimates.
This paper is concerned with the inverse scattering problem of determining the unknown potential for the classical Schro"\dinger equation in two and three dimensions. For the first time, increasing stability estimates of the inverse scattering problem are achieved from either multiwavenumber near-field or far-field data. The stability estimate consists of two parts: one part is the Lipschitz stability, and the other part is the logarithmic stability. As the wavenumber increases, the logarithmic part decreases and the stability approaches a Lipschitz stability. The increasing stability estimate reveals that ill-posedness of the inverse problems can be overcome by the use of multiwavenumber data. With multi-wavenumber data, the analysis does not resort to the construction of CGO solutions and thus can be used to deal with the two-dimensional case. A key ingredient in the analysis is employing scattering theory to obtain an analytic region and estimates in this region for the resolvent of the elliptic operator. This method is further utilized to investigate the inverse scattering problem of determining both the magnetic and the electric potentials for the magnetic Schro"\dinger equation. This problem is more challenging due to the nonlinearity and first-order perturbation.
This paper is concerned with the study of elliptic differential problems involving fractional variable exponent double phase operators with logarithmic perturbation (-\Delta)s \scrH generated by \scrH(x, y, t) = [tp(x,y) p(x,y) +\mu(x, y) tq(x,y) q(x,y) ] log(e+\alphat). In the first part, we study fractional double phase elliptic inclusions with a generalized multivalued mapping and a maximal monotone operator which is formulated by the convex subdifferential of the indicator function to a convex set. Based on the subsupersolution method along with truncation techniques and nonsmooth analysis we show an existence result and give an application construction such a pair of sub-supersolution. Additionally, under lattice conditions, we establish the compactness and the directedness of the solution set within a pair of suband supersolutions. In the second part, we consider a type of fractional Kirchhoff double phase problems governed by the operator (-\Delta)s\scrH. Applying variational methods, the Poincare'\--Miranda existence theorem together with the quantitative deformation lemma, we prove a multiplicity result which says that the problem has at least a positive solution, a negative solution, and a sign-changing solution.
Abstract. There are several errors in [SIAM J. Math. Anal., 9, 1, (1978), pp. 76–86]. In particular, the kernel form for the integral representation of the product for two Jacobi functions of the second kind in (2.10) has an incorrect lower-bound of integration and an incorrect multiplicative factor of [Formula: see text]. There are also several typographical errors in the Nicholson-type formula for Jacobi functions and Laguerre functions in the trigonometric context. In this erratum we explain what these errors are and how they should be fixed.
Abstract. This paper presents a proof of generic structural stability for Riemann solutions to [Formula: see text] system of hyperbolic conservation laws in one spatial variable, without diffusive terms. This means that for almost every left and right state, shocks and rarefaction solutions of the same type are preserved via perturbations of the flux functions, the left state, and the right state. The main assumptions for this proof involve standard assumptions on strict hyperbolicity and genuine nonlinearity, a technical assumption on directionality of rarefaction curves, and the regular manifold (submersion) assumption motivated by concepts in differential topology. We show that the structural stability of the Riemann solutions is related to the transversality of the Hugoniot loci and rarefaction curves in the state space. The regular manifold assumption is required to invoke a variant of a theorem from differential topology, Thom’s parametric transversality theorem, to show the genericity of transversality of these curves. This in turn implies the genericity of structural stability. We then apply this theorem to two examples: the p-system and a [Formula: see text] system governing the evolution of gravity-driven monodisperse particle-laden thin films. In particular, we illustrate how one can verify all the above assumptions for the former, and apply the theorem to different numerical and physical aspects of the system governing the latter.
Abstract. We consider the global-in-time well-posedness of Cauchy problem for the inhomogeneous Navier–Stokes system in critical spaces [Formula: see text] with small data in the sense of Kato’s solution [ 14 ]. When the initial velocity [Formula: see text] is small in the critical space [Formula: see text], and when the initial density [Formula: see text] with small fluctuation is away from vacuum, the global-in-time solutions in [Formula: see text] to the inhomogeneous Navier–Stokes system have been constructed with the help of the [Formula: see text]regularity argument, a refined estimate of pressure and a covering argument.
This work concerns both the classical and the ionic size-modified Poisson--Boltzmann (PB) models of the continuum electrostatics for an ionic solution. A unified approach is developed to analyze the minimizers of the PB electrostatic free-energy functionals of ionic concentrations and the solutions to the corresponding PB and the generalized PB equations. Key results of the analysis are the uniform positive bounds for the equilibrium concentrations and the uniform bounds for the solutions of the PB equations. Penalized and constraint-free PB energy functionals are constructed that can be used for solving the underlying variational problems and partial differential equations by machine learning with application to complex charged molecular systems. In addition to the existence and uniqueness of minimizers of such new functionals, uniform bounds with respect to the penalization parameters are obtained for such minimizers. The convergence of the penalized models is finally established.
This note is the natural continuation of what was started in [I. Birindelli, A. Briani, and H. Ishii, Test Function Approach to Fully Nonlinear Equations in Thin Domains, arxiv:2404.19577, 2024], i.e., the extension to fully nonlinear operators of the well-known result on thin domains of Hale and Raugel [J. Math. Pures Appl. (9), 71 (1992), pp. 33--95]. Here we consider oblique boundary conditions and find some new phenomena, in particular the limit equations contain ``new terms"" in the second and first order terms which don't have an equivalent in the Neumann case.
