We are concerned with the global bifurcation analysis of positive solutions to free boundary problems arising in plasma physics. We show that in general, in the sense of domain variations, the following alternative holds: either the shape of the branch of solutions resembles the monotone one of the model case of the two-dimensional disk, or it is a continuous simple curve without bifurcation points which ends up at a point where the boundary density vanishes. On the other hand, we deduce a general criterion ensuring the existence of a free boundary in the interior of the domain. Application to a classic nonlinear eigenvalue problem is also discussed.
The quasi-neutral limit of the Navier-Stokes-Poisson system modeling a viscous plasma with vanishing viscosity coefficients in the half-space $\mathbb{R}^{3}_{+}$ is rigorously proved under a Navier-slip boundary condition for velocity and the Dirichlet boundary condition for electric potential. This is achieved by establishing the nonlinear stability of the approximation solutions involving the strong boundary layer in density and electric potential, which comes from the break-down of the quasi-neutrality near the boundary, and dealing with the difficulty of the interaction of this strong bou…
For $Ω\subset \mathbb{R}^2$ a smooth and bounded domain, we derive a sharp universal energy estimate for non-negative solutions of free boundary problems on $Ω$ arising in plasma physics. As a consequence, we are able to deduce new universal estimates for this class of problems. We first come up with a sharp positivity threshold which guarantees that there is no free boundary inside $Ω$ or either, equivalently, with a sharp necessary condition for the existence of a free boundary in the interior of $Ω$. Then we derive an explicit bound for the $L^{\infty}$-norm of non-negative solutions and al…
We construct a unique global-in-time solution to the two species Vlasov-Poisson-Boltzmann system in convex domains with the diffuse boundary condition, which can be viewed as one of the ideal scattering boundary model. The construction follows a new $L^{2}$-$L^{\infty}$ framework in [4]. In our knowledge this result is the first construction of strong solutions for $\textit{two species}$ plasma models with $\textit{self-consistent field}$ in general bounded domains.
We investigate the formation of a plasma boundary layer (sheath) by considering the Vlasov--Poisson system on a half-line with the completely absorbing boundary condition. In an earlier paper by the first two authors, the solvability of the stationary problem is studied. In this paper, we study the nonlinear stability and instability of these stationary solutions of the Vlasov--Poisson system.
This contribution, built on the companion paper [1], is focused on the different mathematical approaches available for the analysis of the quasilinear approximation in plasma physics.
We consider the free boundary problem for a plasma--vacuum interface in ideal incompressible magnetohydrodynamics. Unlike the classical statement, where the vacuum magnetic field obeys the div-curl system of pre-Maxwell dynamics, we do not neglect the displacement current in the vacuum region and consider the Maxwell equations for electric and magnetic fields. Our aim is to construct highly oscillating surface wave solutions in weakly nonlinear regime to this plasma--vacuum interface problem. Under a necessary and sufficient stability condition for a piecewise constant background state, we con…
In the present paper, we aim to mathematically analyse the role of the displacement current and the Cattaneo's law on the boundary-layer theory of plasma, when the corresponding characteristic speed is relativistic. We restrict our analysis to two-dimensional flows and we study the asymptotic limit of the Navier-Stokes-Maxwell equations with Cattaneo's law near a bounding flat line, when the Hartmann, Reynolds and magnetic Reynolds numbers proportionally diverge to infinity. The goal of this paper is twofold. We first show that the extended version of the Navier-Stokes-Maxwell equations leads …
Minimization of an energy error functional to solve a Cauchy problem arising in plasma physics: the reconstruction of the magnetic flux in the vacuum surrounding the plasma in a Tokamak
