A numerical experiment on a second‐order partial differential equation of mixed type

Mathematical Aspects of the Cold Plasma Model

“…This polynomial has two real solutions; considering that the characteristic equation (2.75) has two roots, one concludes [26] that four characteristic lines must pass through the origin -two more than pass through any other hyperbolic point. This motivates the suspicion that solutions of at least the equatioñ Lu = 0 will tend to be singular at the origin.…”

Unique Solutions to Boundary Value Problems in the Cold Plasma Model

“…In the former case, data are prescribed on a proper subset of the boundary, whereas in the latter case, data are prescribed on the entire boundary. It is shown in section 3 of [10] that if κ = 1/2, the closed Dirichlet problem is overdetermined for the equation x − y 2 u xx + u yy + κu x = 0 (1.1) on a typical domain, where u (x, y) is required to be twice-continuously differentiable on the domain. However, this equation arises in a qualitative model for electromagnetic wave propagation in zero-temperature plasma [18, equation (81)]; see also [16, equation (9)].…”

“…Physical reasoning suggests that the closed Dirichlet problem for (1.1) should be well posed in a suitable function space, at least for some choice of lower-order terms; so the result reported in [10] would appear to represent a serious deficiency in the physical model. See section 1 of [10] for a discussion, in which the problem of formulating a closed Dirichlet-like problem that is well posed in an appropriate sense is characterized as an "outstanding and significant problem for the cold plasma model. "…”

A boundary value problem for cold plasma dynamics

“…In all these cases σ(y) is proportional to y 2 , but this specific restriction is not imposed by the physics; concerning the physical model, see [11,12]. If u 1 = ψ x , u 2 = ψ y , σ(y) = y 2 , and f = (0, 0), the system reduces to a scalar equation introduced in [7,Section 3]. In the context of this equation, condition (1.5) corresponds to imposing constant boundary conditions on the scalar solution ψ(x, y).…”

“…The system is elliptic for x > σ(y) and hyperbolic for x < σ(y). Following [7], we emphasize the analogy to fluid dynamics by calling the curve x = σ(y) the sonic curve.…”