Unexpected solutions of first and second order partial differential equations.
Unicité pour certains problèmes de Cauchy non-linéaires, complexes, du premier ordre
Unicité pour les équations non linéaires du premier ordre
Uniqueness and approximation of Solutions of first order non linear equations.
Uniqueness and Nonuniqueness for Nonsmooth Divergence Free Transport
Uniqueness and weak stability for multi-dimensional transport equations with one-sided Lipschitz coefficient
The Cauchy problem for a multidimensional linear transport equation with discontinuous coefficient is investigated. Provided the coefficient satisfies a one-sided Lipschitz condition, existence, uniqueness and weak stability of solutions are obtained for either the conservative backward problem or the advective forward problem by duality. Specific uniqueness criteria are introduced for the backward conservation equation since weak solutions are not unique. A main point is the introduction of a generalized...
Viability and invariance for differential games with applications to Hamilton-Jacobi-Isaacs equations
Viscous Shock Capturing in a Time-Explicit Discontinuous Galerkin Method
We present a novel, cell-local shock detector for use with discontinuous Galerkin (DG) methods. The output of this detector is a reliably scaled, element-wise smoothness estimate which is suited as a control input to a shock capture mechanism. Using an artificial viscosity in the latter role, we obtain a DG scheme for the numerical solution of nonlinear systems of conservation laws. Building on work by Persson and Peraire, we thoroughly justify the...
Wave fronts of solutions of some classes of non-linear partial differential equations
1. This paper is devoted to the study of wave fronts of solutions of first order symmetric systems of non-linear partial differential equations. A short communication was published in [4]. The microlocal point of view enables us to obtain more precise information concerning the smoothness of solutions of symmetric hyperbolic systems. Our main result is a generalization to the non-linear case of Theorem 1.1 of Ivriĭ [3]. The machinery of paradifferential operators introduced by Bony [1] together...
Weakly continuous operators. Applications to differential equations
The paper is a supplement to a survey by J. Franců: Monotone operators, A survey directed to differential equations, Aplikace Matematiky, 35(1990), 257–301. An abstract existence theorem for the equation with a coercive weakly continuous operator is proved. The application to boundary value problems for differential equations is illustrated on two examples. Although this generalization of monotone operator theory is not as general as the M-condition, it is sufficient for many technical applications....
Well posed Cauchy problems for complex nonlinear equations must be semilinear.
Well-posedness and regularity of hyperbolic boundary control systems on a one-dimensional spatial domain
We study a class of hyperbolic partial differential equations on a one dimensional spatial domain with control and observation at the boundary. Using the idea of feedback we show these systems are well-posed in the sense of Weiss and Salamon if and only if the state operator generates a C0-semigroup. Furthermore, we show that the corresponding transfer function is regular, i.e., has a limit for s going to infinity.
Zentrale Distributionen auf Exponentialgruppen.
Zur Integration der partiellen Differentialgleichungen erster Ordnung
Zur Theorie der vollständigen Lösungen und der Transformation der partiellen Differentialgleichungen erster Ordnung
Zur Unlösbarkeit linearer partieller Differentialgleichungen.
-устойчивость классов отображений и системы линейных уравнений с частными производными.
Глобальные решения нестационарных кинетических уравнений
Дифференциальные уравнения на графах
Дифференцируемые решения симметричных систем в плоских областях с нерегулярной границей