Dr. Irina Kmit
Profil
Forschungsthemen1
Modelling, Analysis, and Approximationstheorie mit Anwendungen in Inversen Problemen und Tomographie
Quelle ↗Förderer: Volkswagen Stiftung Zeitraum: 06/2016 - 12/2019 Projektleitung: Dr. Irina Kmit
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Publikationen25
Top 25 nach Zitationen — Quelle: OpenAlex (BAAI/bge-m3 embedded für Matching).
International Journal of Dynamical Systems and Differential Equations · 39 Zitationen · DOI
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).
Journal of Mathematical Analysis and Applications · 19 Zitationen · DOI
Journal of Differential Equations · 18 Zitationen · DOI
Journal of Mathematical Analysis and Applications · 14 Zitationen · DOI
Differential and Integral Equations · 12 Zitationen · DOI
This paper concerns linear first-order hyperbolic systems in one space dimension of the type $$ \partial_tu_j + a_j(x,t)\partial_xu_j + \sum\limits_{k=1}^nb_{jk}(x,t)u_k = f_j(x,t),\; x \in (0,1),\; j=1,\ldots,n, $$ with periodicity conditions in time and reflection boundary conditions in space. We state a non-resonance condition (depending on the coefficients $a_j$ and $b_{jj}$ and the boundary reflection coefficients), which implies Fredholm solvability of the problem in the space of continuous functions. Further, we state one more non-resonance condition (depending also on $\partial_ta_j$), which implies $C^1$-solution regularity. Moreover, we give examples showing that both non-resonance conditions cannot be dropped, in general. Those conditions are robust under small perturbations of the problem data. Our results work for many non-strictly hyperbolic systems, but they are new even in the case of strict hyperbolicity.
arXiv (Cornell University) · 12 Zitationen
Zeitschrift für Analysis und ihre Anwendungen · 12 Zitationen · DOI
In this paper we study initial-boundary value problems for first-order semilinear hyperbolic systems where the boundary conditions are non-local. We focus on situations involving strong singularities, of the Dirac delta type, in the initial data as well as in the boundary conditions. In such cases we prove an existence and uniqueness result in an algebra of generalized functions. Furthermore, we investigate the existence and structure of delta waves, i.e., distributional limits of solutions to the regularized systems. Due to the additional singularities in the boundary data the search for delta waves requires a delicate splitting of the solution into a linearly evolving singular part and a regular part satisfying a nonlinear equation. A new feature in the splitting procedure used here, compared to delta waves in pure initial value problems, is the dependence of the singular part also on part of the regular part due to singularities enetering from the boundary. Finally, we include simple examples where the existence of delta waves breaks down.
Applicable Analysis · 11 Zitationen · DOI
We consider initial-boundary problems for general linear first-order strictly hyperbolic systems with local or nonlocal nonlinear boundary conditions. While boundary data are supposed to be smooth, initial conditions can contain distributions of any order of singularity. It is known that such problems have a unique continuous solution if the initial data are continuous. In the case of strongly singular initial data we prove the existence of a (unique) delta wave solution. In both cases, we say that a solution is smoothing if it eventually becomes k-times continuously differentiable for each k. Our main result is a criterion allowing us to determine whether or not the solution is smoothing. In particular, we prove a rather general smoothingness result in the case of classical boundary conditions.
10 Zitationen · DOI
Journal of Differential Equations · 10 Zitationen · DOI
Solution regularity and smooth dependence for abstract equations and applications to hyperbolic PDEs
2015Journal of Differential Equations · 9 Zitationen · DOI
Journal of Differential Equations · 8 Zitationen · DOI
Journal of Differential Equations · 7 Zitationen · DOI
Journal of Mathematical Analysis and Applications · 7 Zitationen · DOI
Ukrainian Mathematical Journal · 7 Zitationen · DOI
Periodic Solutions to Dissipative Hyperbolic Systems. I: Fredholm Solvability of Linear Problems
2011arXiv (Cornell University) · 6 Zitationen · DOI
This paper concerns linear first-order hyperbolic systems in one space dimension of the type $$ \partial_tu_j + a_j(x,t)\partial_xu_j + \sum\limits_{k=1}^nb_{jk}(x,t)u_k = f_j(x,t),\; x \in (0,1),\; j=1,\ldots,n, $$ with periodicity conditions in time and reflection boundary conditions in space. We state a non-resonance condition (depending on the coefficients $a_j$ and $b_{jj}$ and the boundary reflection coefficients), which implies Fredholm solvability of the problem in the space of continuous functions. Further, we state one more non-resonance condition (depending also on $\partial_ta_j$), which implies $C^1$-solution regularity. Moreover, we give examples showing that both non-resonance conditions cannot be dropped, in general. Those conditions are robust under small perturbations of the problem data. Our results work for many non-strictly hyperbolic systems, but they are new even in the case of strict hyperbolicity.
Ukrainian Mathematical Journal · 5 Zitationen · DOI
arXiv (Cornell University) · 5 Zitationen · DOI
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).
Bulletin Classe des sciences mathematiques et natturalles · 4 Zitationen · DOI
We prove the existence and uniqueness of global generalized solutions in a Colombeau algebra of generalized functions to semilinear hyperbolic systems with nonlinear boundary conditions. Our analysis covers the case of non-Lipschitz nonlinearities both in the differential equations and in the boundary conditions. We admit strong singularities in the differential equations as well as in the initial and boundary conditions. AMS Mathematics Subject Classification (2000): 35L50, 35L67, 35D05.
Zeitschrift für Analysis und ihre Anwendungen · 4 Zitationen · DOI
We investigate the existence and the singular structure of delta wave solutions to a semilinear 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.
On a nonlocal problem for a quasilinear first-order hyperbolic system with two independent variables
1993Ukrainian Mathematical Journal · 4 Zitationen · DOI
arXiv (Cornell University) · 3 Zitationen · DOI
We address nonautonomous initial boundary value problems for decoupled linear\nfirst-order one-dimensional hyperbolic systems, investigating the phenomenon of\nfinite time stabilization. We establish sufficient and necessary conditions\nensuring that solutions stabilize to zero in a finite time for any initial\n$L^2$-data. In the nonautonomous case we give a combinatorial criterion stating\nthat the robust stabilization occurs if and only if the matrix of reflection\nboundary coefficients corresponds to a directed acyclic graph. An equivalent\nrobust algebraic criterion is that the adjacency matrix of this graph is\nnilpotent. In the autonomous case we also provide a spectral stabilization\ncriterion, which is nonrobust with respect to perturbations of the coefficients\nof the hyperbolic system.\n
Journal of Mathematical Analysis and Applications · 3 Zitationen · DOI
Czech digital mathematics library · 3 Zitationen
summary:We study one-dimensional linear hyperbolic systems with $L^{\infty}$-coeffici\-ents 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.
Time-periodic Second-order Hyperbolic Equations: Fredholmness, Regularity, and Smooth Dependence
2015Operator theory · 2 Zitationen · DOI
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Stammdaten
Identität, Organisation und Kontakt aus HU-FIS.
- Name
- Dr. Irina Kmit
- Titel
- Dr.
- Fakultät
- Mathematisch-Naturwissenschaftliche Fakultät
- Institut
- Institut für Mathematik
- Telefon
- +49 30 2093-45380
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- 26.4.2026, 01:07:38