Continuous Solutions of Distributional Cauchy Problems
نویسنده
چکیده
Existence of the smallest, greatest, minimal, maximal and unique continuous solutions to distributional Cauchy problems, as well as their dependence on the data, are studied. The main tools are a continuous primitive integral and fixed point results in function spaces.
منابع مشابه
Nvestigation of a Boundary Layer Problem for Perturbed Cauchy-Riemann Equation with Non-local Boundary Condition
Boundary layer problems (Singular perturbation problems) more have been applied for ordinary differential equations. While this theory for partial differential equations have many applications in several fields of physics and engineering. Because of complexity of limit and boundary behavior of the solutions of partial differential equations these problems considered less than ordinary case. In ...
متن کاملEulerian, Lagrangian and Broad continuous solutions to a balance law with non-convex flux. I
We discuss different notions of continuous solutions to the balance law ∂tu + ∂x(f(u)) = g with g bounded and f ∈ C, extending previous works relative to the flux f(u) = u. We establish the equivalence among distributional solutions and a suitable notion of Lagrangian solutions for general smooth fluxes. We eventually find that continuous solutions are Kružkov iso-entropy solutions, which yield...
متن کاملInstitute for Mathematical Physics Semilinear Geometric Optics for Generalized Solutions Semilinear Geometric Optics for Generalized Solutions
This paper is devoted to the study of nonlinear geometric optics in Colombeau algebras of generalized functions in the case of Cauchy problems for semilinear hyperbolic systems in one space variable. Extending classical results, we establish a generalized variant of nonlinear geometric optics. As an application, a nonlinear superposition principle is obtained when distributional initial data ar...
متن کاملA Linear First-order Hyperbolic Equation with a Discontinuous Coefficient: Distributional Shadows and Propagation of Singularities
It is well-known that distributional solutions to the Cauchy problem for ut +(b(t, x)u)x = 0 with b(t, x) = 2H(x− t), where H is the Heaviside function, are non-unique. However, it has a unique generalized solution in the sense of Colombeau. The relationship between its generalized solutions and distributional solutions is established. Moreover, the propagation of singularities is studied.
متن کاملSingular solutions to systems of conservation laws: shocks, δ- and δ′-shocks
Using the definitions of δand δ′-shocks for the systems of conservation laws [12], [13], [39], the Rankine–Hugoniot conditions for δand δ′-shocks are derived. We present a construction of solutions to the Cauchy problems admitting δand δ′-shocks. In particular, the Riemann problem admitting shocks, δ-shocks, δ′-shocks, and vacuum states is considered. The geometric aspects of δand δ′-shocks are...
متن کامل