skip to main content

Attention:

The NSF Public Access Repository (PAR) system and access will be unavailable from 11:00 PM ET on Thursday, January 16 until 2:00 AM ET on Friday, January 17 due to maintenance. We apologize for the inconvenience.

Explore Research Products in the PAR
It may take a few hours for recently added research products to appear in PAR search results.


Title: Lp time asymptotic decay for general hyperbolic–parabolic balance laws with applications
We study the time asymptotic decay of solutions for a general system of hyperbolic–parabolic balance laws in one space dimension. The system has a physical viscosity matrix and a lower-order term for relaxation, damping or chemical reaction. The viscosity matrix and the Jacobian matrix of the lower-order term are rank deficient. For Cauchy problem around a constant equilibrium state, existence of solution global in time has been established recently under a set of reasonable assumptions. In this paper, we obtain optimal [Formula: see text] decay rates for [Formula: see text]. Our result is general and applies to models such as Keller–Segel equations with logarithmic chemotactic sensitivity and logistic growth, and gas flows with translational and vibrational non-equilibrium. Our result also recovers or improves the existing results in literature on the special cases of hyperbolic–parabolic conservation laws and hyperbolic balance laws, respectively.  more » « less
Award ID(s):
1908195
PAR ID:
10155910
Author(s) / Creator(s):
Date Published:
Journal Name:
Journal of Hyperbolic Differential Equations
Volume:
16
Issue:
04
ISSN:
0219-8916
Page Range / eLocation ID:
663 to 700
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. ( , Journal of Hyperbolic Differential Equations)
    We consider a [Formula: see text] system of hyperbolic balance laws that is the converted form under inverse Hopf–Cole transformation of a Keller–Segel type chemotaxis model. We study Cauchy problem when Cauchy data connect two different end-states as [Formula: see text]. The background wave is a diffusive contact wave of the reduced system. We establish global existence of solution and study the time asymptotic behavior. In the special case where the cellular population initially approaches its stable equilibrium value as [Formula: see text], we obtain nonlinear stability of the diffusive contact wave under smallness assumption. In the general case where the population initially does not approach to its stable equilibrium value at least at one of the far fields, we use a correction function in the time asymptotic ansatz, and show that the population approaches logistically to its stable equilibrium value. Our result shows two significant differences when comparing to Euler equations with damping. The first one is the existence of a secondary wave in the time asymptotic ansatz. This implies that our solutions converge to the diffusive contact wave slower than those of Euler equations with damping. The second one is that the correction function logistically grows rather than exponentially decays. 
    more » « less
  2. ; ; ( , Journal of Hyperbolic Differential Equations)

    In this paper, we study the [Formula: see text]-contraction property of small extremal shocks for 1-d systems of hyperbolic conservation laws endowed with a single convex entropy, when subjected to large perturbations. We show that the weight coefficient [Formula: see text] can be chosen with amplitude proportional to the size of the shock. The main result of this paper is a key building block in the companion paper [G. Chen, S. G. Krupa and A. F. Vasseur, Uniqueness and weak-BV stability for [Formula: see text] conservation laws, Arch. Ration. Mech. Anal. 246(1) (2022) 299–332] in which uniqueness and BV-weak stability results for [Formula: see text] systems of hyperbolic conservation laws are proved.

     
    more » « less
  3. ; ; ; ; ; ( , Journal of Algebra and Its Applications)

    [Formula: see text]A hyperbolic code is an evaluation code that improves a Reed–Muller code because the dimension increases while the minimum distance is not penalized. We give necessary and sufficient conditions, based on the basic parameters of the Reed–Muller code, to determine whether a Reed–Muller code coincides with a hyperbolic code. Given a hyperbolic code [Formula: see text], we find the largest Reed–Muller code contained in [Formula: see text] and the smallest Reed–Muller code containing [Formula: see text]. We then prove that similar to Reed–Muller and affine Cartesian codes, the [Formula: see text]th generalized Hamming weight and the [Formula: see text]th footprint of the hyperbolic code coincide. Unlike for Reed–Muller and affine Cartesian codes, determining the [Formula: see text]th footprint of a hyperbolic code is still an open problem. We give upper and lower bounds for the [Formula: see text]th footprint of a hyperbolic code that, sometimes, are sharp.

     
    more » « less
  4. ; ( , International Journal of Algebra and Computation)
    Motivated by results about “untangling” closed curves on hyperbolic surfaces, Gupta and Kapovich introduced the primitivity and simplicity index functions for finitely generated free groups, [Formula: see text] and [Formula: see text], where [Formula: see text], and obtained some upper and lower bounds for these functions. In this paper, we study the behavior of the sequence [Formula: see text] as [Formula: see text]. Answering a question from [17], we prove that this sequence is unbounded and that for [Formula: see text], we have [Formula: see text]. By contrast, we show that for all [Formula: see text], one has [Formula: see text]. In addition to topological and group-theoretic arguments, number-theoretic considerations, particularly the use of asymptotic properties of the second Chebyshev function, turn out to play a key role in the proofs. 
    more » « less
  5. ; ; ; ( , Journal of Rheology)
    Monodisperse suspensions of Brownian colloidal spheres crystallize at high densities, and ordering under shear has been observed at densities below the crystallization threshold. We perform large-scale simulations of a model suspension containing over [Formula: see text] particles to quantitatively study the ordering under shear and to investigate its link to the rheological properties of the suspension. We find that at high rates, for [Formula: see text], the shear flow induces an ordering transition that significantly decreases the measured viscosity. This ordering is analyzed in terms of the development of layering and planar order, and we determine that particles are packed into hexagonal crystal layers (with numerous defects) that slide past each other. By computing local [Formula: see text] and [Formula: see text] order parameters, we determine that the defects correspond to chains of particles in a squarelike lattice. We compute the individual particle contributions to the stress tensor and discover that the largest contributors to the shear stress are primarily located in these lower density, defect regions. The defect structure enables the formation of compressed chains of particles to resist the shear, but these chains are transient and short-lived. The inclusion of a contact friction force allows the stress-bearing structures to grow into a system-spanning network, thereby disrupting the order and drastically increasing the suspension viscosity. 
    more » « less
  • Citation Formats
  • MLA
  • APA
  • Chicago
  • BibTeX
  • Save / Share this Record
  • Send to Email