329 research outputs found
BV solutions to a hyperbolic system of balance laws with logistic growth
We study BV solutions for a system of hyperbolic balance laws. We
show that when initial data have small total variation on
and small amplitude, and decay sufficiently fast to a constant equilibrium
state as , a Cauchy problem (with generic data) has a
unique admissible BV solution defined globally in time. Here the solution is
admissible in the sense that its shock waves satisfy the Lax entropy condition.
We also study asymptotic behavior of solutions. In particular, we obtain a time
decay rate for the total variation of the solution, and a convergence rate of
the solution to its time asymptotic solution. Our system is a modification of a
Keller-Segel type chemotaxis model. Its flux function possesses new features
when comparing to the well-known model of Euler equations with damping. This
may help to shed light on how to extend the study to a general system of
hyperbolic balance laws in the future
Hyperbolic Balance Laws: modeling, analysis, and numerics (hybrid meeting)
This workshop brought together
leading experts, as well as the most
promising young researchers, working on nonlinear
hyperbolic balance laws. The meeting focused on addressing new cutting-edge research in
modeling, analysis, and numerics. Particular topics included ill-/well-posedness,
randomness and multiscale modeling, flows in a moving domain, free boundary problems,
games and control
Continuum Mechanics and Thermodynamics in the Hamilton and the Godunov-type Formulations
Continuum mechanics with dislocations, with the Cattaneo type heat
conduction, with mass transfer, and with electromagnetic fields is put into the
Hamiltonian form and into the form of the Godunov type system of the first
order, symmetric hyperbolic partial differential equations (SHTC equations).
The compatibility with thermodynamics of the time reversible part of the
governing equations is mathematically expressed in the former formulation as
degeneracy of the Hamiltonian structure and in the latter formulation as the
existence of a companion conservation law. In both formulations the time
irreversible part represents gradient dynamics. The Godunov type formulation
brings the mathematical rigor (the well-posedness of the Cauchy initial value
problem) and the possibility to discretize while keeping the physical content
of the governing equations (the Godunov finite volume discretization)
Modeling Shallow Water Flows on General Terrains
A formulation of the shallow water equations adapted to general complex
terrains is proposed. Its derivation starts from the observation that the
typical approach of depth integrating the Navier-Stokes equations along the
direction of gravity forces is not exact in the general case of a tilted curved
bottom. We claim that an integration path that better adapts to the shallow
water hypotheses follows the "cross-flow" surface, i.e., a surface that is
normal to the velocity field at any point of the domain. Because of the
implicitness of this definition, we approximate this "cross-flow" path by
performing depth integration along a local direction normal to the bottom
surface, and propose a rigorous derivation of this approximation and its
numerical solution as an essential step for the future development of the full
"cross-flow" integration procedure. We start by defining a local coordinate
system, anchored on the bottom surface to derive a covariant form of the
Navier-Stokes equations. Depth integration along the local normals yields a
covariant version of the shallow water equations, which is characterized by
flux functions and source terms that vary in space because of the surface
metric coefficients and related derivatives. The proposed model is discretized
with a first order FORCE-type Godunov Finite Volume scheme that allows
implementation of spatially variable fluxes. We investigate the validity of our
SW model and the effects of the bottom geometry by means of three synthetic
test cases that exhibit non negligible slopes and surface curvatures. The
results show the importance of taking into consideration bottom geometry even
for relatively mild and slowly varying curvatures
Pointwise Green's function bounds and stability of relaxation shocks
We establish sharp pointwise Green's function bounds and consequent
linearized and nonlinear stability for smooth traveling front solutions, or
relaxation shocks, of general hyperbolic relaxation systems of dissipative
type, under the necessary assumptions ([G,Z.1,Z.4]) of spectral stability,
i.e., stable point spectrum of the linearized operator about the wave, and
hyperbolic stability of the corresponding ideal shock of the associated
equilibrium system. This yields, in particular, nonlinear stability of weak
relaxation shocks of the discrete kinetic Jin--Xin and Broadwell models. The
techniques of this paper should have further application in the closely related
case of traveling waves of systems with partial viscosity, for example in
compressible gas dynamics or MHD.Comment: 120 pages. Changes since original submission. Corrected typos, esp.
energy estimates of Section 7, corrected bad forward references, expanded
Remark 1.17, end of introductio
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