Generalized uniformly continuous semigroups and semilinear hyperbolic systems with regularized derivatives

We adopt the theory of uniformly continuous operator semigroups for use in Colombeau generalized function spaces. The main objective is to find a unique solution to a class of semilinear hyperbolic systems with singularities. The idea of regularized derivatives is to transform unbounded differential operators into bounded, integral ones. This idea is used here to permit working with uniformly continuous operators.     

Generalized solutions to semilinear hyperbolic systems

In this article semilinear hyperbolic first order systems in two variables are considered, whose nonlinearity satisfies a global Lipschitz condition. It is shown that these systems admit unique global solutions in the Colombeau algebraG(²). In particular, this provides unique generalized solutions for arbitrary distributions as initial data. The solution inG(²) is shown to be consistent with the locally integrable or the distributional solutions, when they exist.     

Domination of uniformly continuous semigroups

We prove that a bounded operator on a Banach lattice, satisfying a growth condition, is regular. Also, we prove that the generator of a C ₀-semigroup on such a lattice for which such an operator exists is bounded.     

Homogenization for semilinear hyperbolic systems with oscillatory data

The behavior of multi-dimensional discrete Boltzmann systems with highly oscillatory data is studied. Homogenized equations for the mean solutions are obtained. Uniform convergence of the oscillatory solutions of the discrete Boltzmann equations to the solutions of the corresponding homogenized equations is established. Moreover, we find that the weak limits of the oscillatory solutions for a model of Broadwell type are not continuous functions of the discrete velocities. Generalization of the above results to problems with multiple-scale initial data is also established.     

Stochastic dilations of uniformly continuous completely positive semigroups

For an arbitrary uniformly continuous completely positive semigroup ( _t :t0) on the space of bounded operators on a Hilbert space, we construct a family (U(t)t0) of unitary operators on a Hilbert space and a conditional expectation from to, such that, for arbitraryt0,. The unitary operatorsU(t) satisfy a stochastic differential equation involving a noncommutative generalisation of infinite dimensional Brownian motion. They do not form a semigroup.Part of this work was completed when the first author was visiting research associate at the Center for Relativity, Physics Department, The University of Texas at Austin, Austin, TX 78712, U.S.A., supported in part by NSF PHY 81-01381.     

Generalized solution to a semilinear hyperbolic system with a non-Lipshitz nonlinearity

Let

Global wellposedness for quasi-linear symmetric hyperbolic systems with regularized coefficients

Following the abstract setting of [8] and using the global results of [2], global wellposedness and regularity results are proved for the solutions of quasi-linear symmetric hyperbolic systems with bounded coefficients which are regularized by a convolution in the space variables with a regularizing function. In the case of unbounded regularized coefficients, local existence of classical solutions is proved, as well as uniqueness and regularity of (not necessarily existing) global weak solutions with initial value in a Sobolev space. As the regularizing function tends to Dirac's δ, local-in-time convergence to the classical solution of the non-regularized problem is proved.     

Estimates on the weak solution of semilinear hyperbolic systems

The Cauchy problem for a semilinear hyperbolic system of the type

Characteristic problems with normal derivatives for hyperbolic systems

We consider characteristic problems with normal derivatives for a hyperbolic systemwith two independent variables. Using the Riemann method, we obtain solvability conditions for these problems accurate to several arbitrary constants.     

Convergence of singular limits for multi-D semilinear hyperbolic systems to parabolic systems

In this paper we investigate the diffusive zero-relaxation limit of the following multi-D semilinear hyperbolic system in pseudodifferential form:

We analyze the singular convergence, as , in the case which leads to a limit system of parabolic type. The analysis is carried out by using the following steps:

(i)

We single out algebraic ``structure conditions' on the full system, motivated by formal asymptotics, by some examples of discrete velocity models in kinetic theories.

(ii)

We deduce ``energy estimates ', uniformly in , by assuming the existence of a symmetrizer having the so-called block structure and by assuming ``dissipativity conditions' on .

(iii)

We assume a Kawashima type condition and perform the convergence analysis by using generalizations of compensated compactness due to Tartar and Gérard.

Finally, we include examples that show how to use our theory to approximate any quasilinear parabolic systems, satisfying the Petrowski parabolicity condition, or general reaction diffusion systems, including Chemotaxis and Brusselator type systems.

     

Reflection of transversal progressing waves for semilinear strictly hyperbolic systems

We discuss the reflection of weak singularities of solutions to semilinear hyperbolic systems by using the method of energy estimates in the space of conormal distributions, and consequently obtain the general results on the reflection of weak singularities of the solution to a high order semilinear strictly hyperbolic equation. Research partially supported by the National Natural Science Foundation of China     

Weakly hyperbolic systems with Hölder continuous coefficients

We study the Cauchy Problem for a hyperbolic system with multiple characteristics and non-smooth coefficients depending on time. We prove in particular that, if the leading coefficients are α-Hölder continuous, and the system has size m?3, then the Problem is well posed in each Gevrey class of exponent s<1+α/m.     

Automatic extensions of local regularized semigroups and local regularized cosine functions

This paper establishes automatic extensions for local regularized semigroups and local regularized cosine functions in a certain sense and applies the results to abstract Cauchy problems.

     

Robust controller for systems with exponentially stable strongly continuous semigroups

The theory of robust controllers is extended to the case where we have boundary and/or distributed control and the system operator is an infinitesimal generator of a strongly continuous exponentially stable semigroup. An example on hyperbolic systems is presented.     

On the radius of spatial analyticity for semilinear symmetric hyperbolic systems

We study the problem of propagation of analytic regularity for semi-linear symmetric hyperbolic systems. We adopt a global perspective and we prove that if the initial datum extends to a holomorphic function in a strip of radius (= width) _ε0${\epsilon }_0$, the same happens for the solution u(t,⋅)$u(t,\cdot )$ for a certain radius ε(t)$\epsilon (t)$, as long as the solution exists. Our focus is on precise lower bounds on the spatial radius of analyticity ε(t)$\epsilon (t)$ as t grows.     

Global existence and nonexistence results for a class of semilinear hyperbolic systems

In this paper, we investigate the Cauchy problem for a class of the system of semilinear hyperbolic equations with damping. With the use of the L_p → L_q type estimation for the corresponding linear problem and the method of comparison of functional, the existence and nonexistence criteria of global solutions are found. Copyright © 2012 John Wiley & Sons, Ltd.