We consider an energy-functional describing rotating superfluids at a rotating velocity ω, and prove similar results as for the Ginzburg-Landau functional of superconductivity: mainly the existence of branches of solutions with vortices, the existence of a critical ω above which energy-minimizers have vortices, evaluations of the minimal energy as a function of ω, and the derivation of a limiting free-boundary problem.
We consider the following singularly perturbed elliptic problemwhere satisfies some growth conditions, , and () is a smooth and bounded domain. The cases (Neumann problem) and (Dirichlet problem) have been studied by many authors in recent years. We show that, there exists a generic constant such that, as , the least energy solution has a spike near the boundary if , and has an interior spike near the innermost part of the domain if . Central to our study is the corresponding problem...
In this paper we study the null-controllability of an artificial advection-diffusion system in dimension n. Using a spectral method, we prove that the control cost goes to zero exponentially when the viscosity vanishes and the control time is large enough. On the other hand, we prove that the control cost tends to infinity exponentially when the viscosity vanishes and the control time is small enough.
For a nonlinear hyperbolic equation defined in a thin domain we prove the existence of a periodic solution with respect to time both in the non-autonomous and autonomous cases. The methods employed are a combination of those developed by J. K. Hale and G. Raugel and the theory of the topological degree.
Reaction-diffusion equations with degenerate nonlinear diffusion are in widespread use as models of biological phenomena. This paper begins with a survey of applications to ecology, cell biology and bacterial colony patterns. The author then reviews mathematical results on the existence of travelling wave front solutions of these equations, and their generation from given initial data. A detailed study is then presented of the form of smooth-front...
We investigate different asymptotic regimes for Vlasov equations modeling the evolution of a cloud of particles in a turbulent flow. In one case we obtain a convection or a convection-diffusion effective equation on the concentration of particles. In the second case, the effective model relies on a Vlasov-Fokker-Planck equation.
We investigate different asymptotic regimes for Vlasov equations modeling the evolution of a cloud of particles in a turbulent flow. In one case we obtain a convection or a convection-diffusion effective equation on the concentration of particles. In the second case, the effective model relies on a Vlasov-Fokker-Planck equation.
We study singularly perturbed 1D nonlinear Schrödinger equations (1.1). When has multiple critical points, (1.1) has a wide variety of positive solutions for small and the number of positive solutions increases to as . We give an estimate of the number of positive solutions whose growth order depends on the number of local maxima of . Envelope functions or equivalently adiabatic profiles of high frequency solutions play an important role in the proof.