arXiv subjects
Publications and source records attributed to Irina Kmit.
We prove the global classical solvability of initial-boundary problems for semilinear first-order hyperbolic systems subjected to local and nonlocal nonlinear boundary conditions. We also establish lower bounds for the order of nonlinearity demarkating a frontier between regular cases (classical solvability) and singular cases (blow-up of solutions).
We study spherically symmetric solutions of the Vlasov-Poisson system in the context of algebras of generalized functions. This allows to model highly concentrated initial configurations and provides a consistent setting for studying singular limits of the system. The proof of unique solvability in our approach depends on new stability properties of the system with respect to perturbations.
We study one-dimensional linear hyperbolic systems with $L^{\infty}$-coefficients subjected to periodic conditions in time and reflection boundary conditions in space. We derive a priori estimates and give an operator representation of solutions in the whole scale of Sobolev-type spaces of periodic functions. These spaces give an optimal regularity trade-off for our problem.
We consider a generalization of the Lotka-McKendrick problem describing the dynamics of an age-structured population with time-dependent vital rates. The generalization consists in allowing the initial and the boundary conditions to be derivatives of the Dirac measure. We construct a unique $\D'$-solution in the framework of intrinsic multiplication of distributions. We also investigate the regularity of this solution.
This paper concerns hyperbolic systems of two linear first-order PDEs in one space dimension with periodicity conditions in time and reflection boundary conditions in space. The coefficients of the PDEs are supposed to be time independent, but allowed to be discontinuous with respect to the space variable. We construct two scales of Banach spaces (for the solutions and for the right hand sides of the equations, respectively) such that the problem can be modeled by means of Fredholm operators of index zero between corresponding spaces of the two scales.
Using the framework of Colombeau algebras of generalized functions, we prove the existence and uniqueness results for global generalized solvability of semilinear hyperbolic systems with nonlinear nonlocal boundary conditions. We admit strong singularities in the differential equations as well as in the initial and boundary conditions. Our analysis covers the case of non-Lipshitz nonlinearities both in the differential equations and in the boundary conditions.
We use the framework of Colombeau algebras of generalized functions to study existence and uniqueness of global generalized solutions to mixed non-local problems for a semilinear hyperbolic system. Coefficients of the system as well as initial and boundary data are allowed to be strongly singular, as the Dirac delta function and derivatives thereof. To obtain the existence-uniqueness result we prove a criterion of invertibility in the full version of the Colombeau algebras.
This paper has been withdrawn by the author due to a gap in the proof of the main result.
We investigate the existence and the singular structure of delta wave solutions to a semilinear strictly hyperbolic equation with strongly singular initial and boundary conditions. The boundary conditions are given in nonlocal form with a linear integral operator involved. We construct a delta wave solution as a distributional limit of solutions to the regularized system. This determines the macroscopic behavior of the corresponding generalized solution in the Colombeau algebra $\G$ of generalized functions. We represent our delta wave as a sum of a purely singular part satisfying a linear system and a regular part satisfying a nonlinear system.