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We consider the modified Zakharov-Kuznetsov (mZK) equation in two space dimensions in both focusing and defocusing cases. Using the I-method, we prove the global well-posedness of the H s solutions for s > 3 4 for any data in the defocusing case and under the assumption that the mass of the initial data is less than the mass of the ground state solution of ∆ϕ − ϕ + ϕ 3 = 0 in the focusing case. This improves the global well-posedness result of Linares and Pastor [20].2010 Mathematics Subject Classification. Primary: 35Q53, 35Q51, 37K40.

We consider load controlled quasistatic evolution. Well posedness results for the nonlocal continuum model related to peridynamics are established. We show local existence and uniqueness of quasistatic evolution for load paths originating at critical points associated with energy minima. These are local minima among the convex set of deformations belonging to the strength domain of the material. The evolution of the displacements however is not constrained to lie inside the strength domain of the material. The load-controlled evolution is shown to exhibit energy balance.

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