Analysis and Applications

In this paper, we treat the general strongly elliptic systems with a class of singular potentials on a bounded Lipschitz domain Ω ⊂ ℝ^d, d ≥ 3. We establish the L^p resolvent estimates on Ω for the above systems with vanishing Dirichlet type or Neumann type boundary value condition, where 2d/(d + 2) - ϵ < p < 2d/(d - 2) + ϵ with some positive constant ϵ = ϵ(Ω).

In this paper, we study a quantum Boltzmann equation with a harmonic oscillator for isotropic gases of bosons and fermions, respectively. This model comes from physics literatures (see, e.g., [M. Holland, J. Williams and J. Cooper, Bose–Einstein condensation: Kinetic evolution obtained from simulated trajectories, Phys. Rev. A 55 (1997) 3670–3677]). The distribution function, i.e. the solution, is discrete in the energy variable. We give the classification of equilibria of the equation for bosons and fermions, respectively, and prove the global existence, uniqueness and the strong convergence to equilibrium for solutions of the equation.

We are concerned with a class of nonlinear Schrödinger-type equations with a reaction term and a differential operator that involves a variable exponent. By using related variational methods, we establish several existence results.

The existence of classical solutions to the Cauchy problem for the Boltzmann equation without angular cutoff has been extensively studied in the framework when the solution has Maxwellian decay in the velocity variable, cf. [6, 8] and the references therein. In this paper, we prove local existence of solutions with polynomial decay in the velocity variable for the Boltzmann equation with soft potential. In the proof, the singular change of variables between post- and pre-collision velocities plays an important role, as well as the regular one introduced in the celebrated cancelation lemma by Alexandre–Desvillettes–Villani–Wennberg [Entropy dissipation and long-range interactions, Arch. Ration. Mech. Anal.  152 (2000) 327–355].

Donati compatibility conditions on a surface allow to reformulate the minimization problem for a linearly elastic shell through the intrinsic approach, i.e. as a quadratic minimization problem with the linearized change of metric and change of curvature tensors of the middle surface of the shell as the new unknowns. Such compatibility conditions typically take the form of variational equations with divergence-free tensor fields as test-functions. In a previous work, the first author and Oana Iosifescu have identified and justified Donati compatibility conditions for shells modeled by Koiter's equations. In this paper, Donati compatibility conditions are identified and justified for two specific classes of linearly elastic shells, the so-called elliptic membrane shells and flexural shells.