WE established the existence of solutions to a class of initial value problems with severe tempor... more WE established the existence of solutions to a class of initial value problems with severe temporal irregularities (discussed below). The theory developed in that paper is applicable to nonlinear partial differential equations which preserve some regularity in the space variables. This includes parabolic equations; dispersive equations, such as the KdV equation; and hyperbolic equations, during the short time interval before shocks develop. In this paper, we shall briefly restate the main conclusions of , and then prove that the solutions depend continuously on a parameter. That is, we shall consider a sequence of evolution equations u;(t) =A,(& 4(t)); we shall show, roughly, that u,, --, U, if and only if JA,(s, .) ds+ JA,(s, *) ds in an appropriate weak sense. As corollaries we shall obtain a Trotter product formula and a result on compact perturbations.
By interpolating between Sobolev spaces we find that many partial differential operators become c... more By interpolating between Sobolev spaces we find that many partial differential operators become continuous when restricted to a sufficiently small domain. Hence some techniques from the theory of ordinary differential equations can be applied to some p.d.e.'s. Using these ideas, we study a class of nonlinear evolutions in a Banach space. We obtain some very simple existence and continuous dependence results. The theory is applicable to reactiondiffusion equations, dispersion equations, and hyperbolic equations before shocks develop.
We show that the Axiom of Choice is equivalent to each of the following statements: (i) A product... more We show that the Axiom of Choice is equivalent to each of the following statements: (i) A product of closures of subsets of topological spaces is equal to the closure of their product (in the product topology); (ii) A product of complete uniform spaces is complete.
WE established the existence of solutions to a class of initial value problems with severe tempor... more WE established the existence of solutions to a class of initial value problems with severe temporal irregularities (discussed below). The theory developed in that paper is applicable to nonlinear partial differential equations which preserve some regularity in the space variables. This includes parabolic equations; dispersive equations, such as the KdV equation; and hyperbolic equations, during the short time interval before shocks develop. In this paper, we shall briefly restate the main conclusions of , and then prove that the solutions depend continuously on a parameter. That is, we shall consider a sequence of evolution equations u;(t) =A,(& 4(t)); we shall show, roughly, that u,, --, U, if and only if JA,(s, .) ds+ JA,(s, *) ds in an appropriate weak sense. As corollaries we shall obtain a Trotter product formula and a result on compact perturbations.
By interpolating between Sobolev spaces we find that many partial differential operators become c... more By interpolating between Sobolev spaces we find that many partial differential operators become continuous when restricted to a sufficiently small domain. Hence some techniques from the theory of ordinary differential equations can be applied to some p.d.e.'s. Using these ideas, we study a class of nonlinear evolutions in a Banach space. We obtain some very simple existence and continuous dependence results. The theory is applicable to reactiondiffusion equations, dispersion equations, and hyperbolic equations before shocks develop.
We show that the Axiom of Choice is equivalent to each of the following statements: (i) A product... more We show that the Axiom of Choice is equivalent to each of the following statements: (i) A product of closures of subsets of topological spaces is equal to the closure of their product (in the product topology); (ii) A product of complete uniform spaces is complete.
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Papers by Eric Schechter