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In [D. Feng and W. Huang, J. Math. Pures Appl. 106, 411–452 (2016)], the authors studied weighted topological pressure and established a variational principle for it. In this paper, we introduce the notion of local weighted topological pressure and generalize Feng and Huang's main results to localized version. [ABSTRACT FROM AUTHOR]

In this paper, we consider a random one-dimensional Schrödinger operator H ω on the half line which has, as its potential term, Gaussian white noise multiplied by a decaying factor. Although the potential term is not an ordinary function, but a distribution, it is possible to realize H ω as a symmetric operator in L2([0, ∞); dt) as was pointed out by the present author [Minami, Lect. Notes Math. 1299, 298 (1986)], and it will be shown that H ω is actually self-adjoint with probability one. When the white noise in H ω is replaced by random functions of a specific type, [Kotani and Ushiroya, Commun. Math. Phys. 115, 247 (1988)] made a precise analysis of the positive part of the spectrum. According to them, if the decaying factor is not square integrable, the positive part of the spectrum typically consists of singular continuous and dense pure-point parts, which are separated by a threshold number. On the other hand, the positive part of the spectrum is purely absolutely continuous when the decaying factor is square integrable. In this work, we shall focus on the case of square integrable decaying factor, and prove the absolute continuity of the positive part of the spectrum of H ω . We shall further prove that the negative part of the spectrum of H ω is discrete, with no accumulation points other than 0. [ABSTRACT FROM AUTHOR]

In this paper, we establish the existence of small-amplitude unique solutions near the Maxwellian for the Fokker–Planck–Boltzmann equation in a finite channel with specular reflection boundary conditions. The solution space we consider is denoted as L k ̄ 1 L T ∞ L x 1 , v 2 , introduced in Duan et al. [Commun. Pure Appl. Math. 74(5), 932–1020 (2021)]. In addition, we investigate the long-time behavior of solutions for both hard and soft potentials. [ABSTRACT FROM AUTHOR]

In this paper, we consider all possible variants of Choi matrices of linear maps, and show that they are determined by non-degenerate bilinear forms on the domain space. We will do this in the setting of finite dimensional vector spaces. In case of matrix algebras, we characterize all variants of Choi matrices which retain the usual correspondences between k-superpositivity and Schmidt number ≤ k as well as k-positivity and k-block-positivity. We also compare de Pillis' definition [Pac. J. Math. 23, 129–137 (1967)] and Choi's definition [Linear Algebra Appl. 10, 285–290 (1975)], which arise from different bilinear forms. [ABSTRACT FROM AUTHOR]

In this paper, we will show that the Dubrovin–Frobenius prepotentials on the orbit space of the Coxeter group B n constructed by Arsie et al. [Sel. Math. New Ser. 29, 1 (2023)] coincide with the solutions of Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations associated with the constrained KP hierarchy. [ABSTRACT FROM AUTHOR]

In this paper, we prove the uniqueness of an energy conservative Hölder continuous weak solution to a general quasilinear wave equation by the analysis of characteristics. This result has no restriction on the size of solutions, i.e., it is a large data result. This paper is a companion paper of Cai, Chen, and Shen [J. Math. Pures Appl. (submitted); arXiv:2007.15201] addressing the Lipschitz continuous dependence of solution. [ABSTRACT FROM AUTHOR]

In this paper, we consider the long-time behavior of the nonclassical diffusion equation with perturbed parameter and memory on a bounded domain Ω ⊂ R n (n ≥ 3). The main feature of this model is that the equation contains a dissipative term with perturbation parameters −νΔu and the nonlinearity f satisfies the polynomial growth of arbitrary order. By using the nonclassical operator method and a new analytical method (or technique) (Lemma 2.7), the existence and regularity of uniform attractors generated for this equation are proved. Furthermore, we also get the upper semicontinuity of the uniform attractors when the perturbed parameter ν → 0. It is remarkable that if ν = 0, we can get the same conclusion as in the works of Toan et al. [Acta Appl. Math. 170, 789–822 (2020)] and Conti et al. [Commun. Pure Appl. Anal. 19, 2035–2050 (2020)], but the nonlinearity is critical. [ABSTRACT FROM AUTHOR]

We consider the initial-boundary value problem of compressible Navier–Stokes–Vlasov equations under a local alignment regime in a one-dimensional bounded domain. Based on the relative entropy method and compactness argument, we prove that a weak solution of the initial-boundary value problem converges to a strong solution of the limiting two-phase fluid system. This work extends in some sense the previous work of Choi and Jung [Math. Models Methods Appl. Sci. 31(11), 2213–2295 (2021)], which considered the diffusive term ∂ξξfɛ in the kinetic equation. Note that the diffusion term was not considered in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Cauchy problem of tri-dimensional compressible Navier–Stokes equations and construct global smooth solutions by choosing a class of new special initial velocity and density whose B ̇ 2 , ∞ − s -norm can be arbitrarily large and improve the previous result in Li et al. [J. Math. Fluid Mech. 24, 22 (2022)]. Our main idea is splitting the linearized equations from the compressible Navier–Stokes equations and exploring the damping effect of the linearized system. [ABSTRACT FROM AUTHOR]

The aim of this paper is to get an estimation of decay rates to first-order and second-order derivatives of space for large-solutions to 3D compressible magnetohydrodynamic system. While the condition (σ0 − 1, u0, Q0) ∈ L1 ∩ H2 is satisfied via a classical energy method and Fourier splitting method, first-order and second-order derivatives of space for large-solutions tending to 0 by L2-rate (1 + t) − 5 4 are shown. It is a necessary supplement to the result of Gao, Wei, and Yao [Appl. Math. Lett. 102, 106100 (2020)] in which they only obtained an estimation of decay rates to magnetic fields. Meanwhile, compared with the work of Gao, Wei, and Yao [Physica D 406, 132506 (2020)], we find that the appearance of magnetic fields does not have any bad effect on the estimation of decay rates to both the velocity field and density. [ABSTRACT FROM AUTHOR]

Given a random subspace Hn chosen uniformly in a tensor product of Hilbert spaces Vn ⊗ W, we consider the collection Kn of all singular values of all norm one elements of Hn with respect to the tensor structure. A law of large numbers has been obtained for this random set in the context of W fixed and the dimension of Hn, Vn tending to infinity at the same speed by Belinschi, Collins, and Nechita [Commun. Math. Phys. 341(3), 885–909 (2016)]. In this paper, we provide measure concentration estimates in this context. The probabilistic study of Kn was motivated by important questions in quantum information theory and allowed us to provide the smallest known dimension for the dimension of an ancilla space, allowing for Minimum Output Entropy (MOE) violation. With our estimates, we are able, as an application, to provide actual bounds for the dimension of spaces where the violation of MOE occurs. [ABSTRACT FROM AUTHOR]

In this paper, we have illustrated the construction of a real structure on a fuzzy sphere S * 2 in its spin-1/2 representation. Considering the SU(2) covariant Dirac and chirality operator on S * 2 given by U. C. Watamura and Watamura [Commun. Math. Phys. 183, 365–382 (1997) and Commun. Math. Phys. 212, 395–413 (2000)], we have shown that the real structure is consistent with other spectral data for KO dimension-4 fulfilling the zero order condition, where we find it necessary to enlarge the symmetry group from SO(3) to the full orthogonal group O(3). However, the first order condition is violated, thus paving the way to construct a toy model for an SU(2) gauge theory to capture some features of physics beyond the standard model following Chamseddine et al. (J. High Energy Phys. 2013, 132). [ABSTRACT FROM AUTHOR]

This paper aims at studying the inelastic Boltzmann equation without Grad's angular cutoff assumption, where the well-posedness theory of the Cauchy problem is established for the Maxwellian molecules in a space of probability measure defined by Cannone and Karch [Commun. Pure Appl. Math. 63, 747–778 (2010)] via Fourier transform and the infinite energy solutions are not a priori excluded. The key strategy is to construct a brand new geometric relation of the inelastic collision mechanism to extend the result of Cannone and Karch from moderate singularity of the non-cutoff collision kernels to strong singularity and simultaneously handle more general restitution coefficients. Moreover, we extend the self-similar solution to the Boltzmann equation with infinite energy shown by Bobylev and Cercignani [J. Stat. Phys. 106, 1039–1071 (2002)] to the inelastic case by using a constructive approach, which is also proved to be the large-time asymptotic steady solution with the help of an asymptotic stability result in a certain sense. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of critical fractional Kirchhoff-type equations with singular potential. With a range of parameters, we propose several existence results. Our work extends the results of Li and Su [Z. Angew. Math. Phys. 66, 3147 (2015)]. [ABSTRACT FROM AUTHOR]

Whether the three dimensional isentropic compressible Navier–Stokes equations admit weak solutions for arbitrary initial data with adiabatic exponent γ > 1 remains a challenging problem. The only available results under γ > 1 were achieved by either assuming the initial data with small energy due to Hoff [J. Differ. Equations 120(1), 215–254 (1995)] or under the spherically symmetric condition by Jiang and Zhang [Commun. Math. Phys. 215, 559–581 (2001)] and Huang [J. Differ. Equations 262, 1341–1358 (2017)]. In this paper, we establish the existence of weak solutions with higher regularity of the three-dimensional periodic compressible isentropic Navier–Stokes equations in small time for the adiabatic exponent γ > 1 in the presence of vacuum. It can be viewed as a local version of Hoff's work and also extends the result of Desjardins [Commun. Partial Differ. Equations 22(5–6), 977–1008 (1997)] by removing the assumption of γ > 3. [ABSTRACT FROM AUTHOR]

We study emergent behaviors of Cucker–Smale (CS) flocks on the hyperboloid H d in any dimensions. In a recent work [Ha et al., J. Math. Phys. 61(4), 042701 (2020)], a first-order aggregation model on the hyperboloid was proposed and sufficient conditions for emergent dynamics were proposed in terms of initial configuration and system parameters. In this paper, we are interested in the second-order modeling of CS flocks on the hyperboloid. For this, we derive our second-order model from the abstract CS model on complete and smooth Riemannian manifolds via explicit identifications of geodesic and parallel transport. Velocity alignment has been shown by combining general velocity alignment estimates for the abstract CS model on manifolds and verifications of the a priori estimate of the second derivative of the energy functional. For the two-dimensional case H 2 , similar to the recent result by Ahn, Ha, and Shim [Kinet. Relat. Models 14(2), 323–351 (2021)], asymptotic flocking admits only two types of asymptotic scenarios, either convergence to a rest state or a state lying on the same plane (coplanar state). We also provide several numerical simulations to illustrate an aforementioned dichotomy on the asymptotic dynamics of the hyperboloid CS model on H 2 . [ABSTRACT FROM AUTHOR]

We thank Professor Jordan for his comments[1] on our paper.[2] Indeed, he is right in pointing that when I i tends to infinite and I i tends to infinite, but I i / I i remains finite, Eq. (3) of his comment[1] tends to a wave equation for I u i , if coefficient I i in the last term (non-linear term) is finite. The criticism by Jordan[1] related to the lack of the Joule effect has no relation with our work, because we did not consider any electric flux, but only heat propagation. However, his proposal [Eq. (4) in Ref. [1]] of considering that coefficient I i of the last term is proportional to I i or to I i is interesting and deserves analysis. [Extracted from the article]