R. E. Showalter

Monotone operators in Banach space and nonlinear partial differential equations

PDE examples by type Linear problems...An introduction Nonlinear stationary problems Nonlinear evolution problems Accretive operators and nonlinear Cauchy problems Appendix Bibliography Index.

Pseudoparabolic Partial Differential Equations

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IMPLICIT DEGENERATE EVOLUTION EQUATIONS AND APPLICATIONS

The initial-value problem is studied for evolution equations in Hilbert space of the general form d se(u)+N(u) l:, dt where and are maximal monotone operators. Existence of a solution is proved when1 is a subgradient and either is strongly monotone or 9 is coercive; existence is established also in the case where 1 is strongly monotone and is subgradient. Uniqueness is provedwhen one of or is continuous self-adjoint and thesum is strictlymonotone; examples of nonuniqueness are given. Applications are indicated for various classes of degenerate nonlinear partial differential equations or systems of mixed elliptic-parabolic-pseudo- parabolic types and problems with nonlocal nonlinearity.

Diffusion models for fractured media

Two models for diffusion in fractured media are described; the compartment model as an example of a double-porosity system, and the micro-structure model as the limit by homogenization of local flux-coupled classical diffusion models which depend on the geometry. These two models are shown to be examples of a single evolution equation for which the appropriate initial-boundary-value problems are well-posed. This gives a unified theoretical basis for these two (as well as classical diffusion) models in which they can be compared and studied.

Diffusion of fluid in a fissured medium with microstructure

A system of quasilinear degenerate parabolic equations arising in the modeling of diffusion in a fissured medium is studied. There is one such equation in the local cell coordinates at each point of the medium, and these are coupled through a similar equation in the global coordinates. It is shown that the initial boundary value problems are well posed in the appropriate spaces.

Single Phase Flow in Partially Fissured Media

Totally fissured media in which the individual cells are isolated by the fissure system are effectively described by double porosity models with microstructure. Such models contain the geometry of the individual cells in the medium and the flux across their interface with the fissure system which surrounds them. We extend these results to a dual-permeability model which accounts for the secondary flux arising from direct cell-to-cell diffusion within the solid matrix. Homogenization techniques are used to construct a new macroscopic model for the flow of a single phase compressible fluid through a partially fissured medium from an exact but highly singular microscopic model, and it is shown that this macroscopic model is mathematically well posed. Preliminary numerical experiments illustrate differences in the behaviour of solutions to the partially fissured from that of the totally fissured case.

Micro-Structure Models of Diffusion in Fissured Media*

Abstract Two diffusion models are developed which recognize the local geometry of the individual cells or storage sites and the exchange of flux on the micro-scale of these cells. The Cauchy problems for these model systems are shown to be resolved by holomorphic semigroups, and various classical models are obtained as limits of these disributed micro-structure models.

A Nonlinear Parabolic-Sobolev Equation

Abstract Let M and L be (nonlinear) operators in a reflexive Banach space B for which Rg(M + L) = B and ¦(Mx − My) + α(Lx − Ly)¦ ⩾ | mx − My | for all α > 0 and pairs x, y in D(M) ∩ D(L). Then there is a unique solution of the Cauchy problem (Mu(t))′ + Lu(t) = 0, Mu(0) = v0. When M and L are realizations of elliptic partial differential operators in space variables, this gives existence and uniqueness of generalized solutions of boundary value problems for nonlinear partial differential equations of mixed parabolic-Sobolev type.

Sobolev equations for nonlinear dispersive systems

Global existence and stability results are obtained for a semifinear evolution equation of Sobolev type in a Banach space. The nonlinear term is assumed to be uniformly Lipschitz on each bounded set and to satisfy a dissipation-type inequality. Applications include various initial-boundary value problems for certain partial differential equations which have been used to model unidirectional long waves in nonlinear dispersive systems

Diffusion in Fissured Media

The nonlinear initial-boundary value problem \[\begin{gathered} \frac{{\partial u}}{{\partial t}} + \frac{1}{\varepsilon }(\alpha (u) - v) = f_1 ,\quad - {\operatorname{div}}\,(k{\operatorname{grad}}\,v) + \frac{1}{\varepsilon }(v - \alpha (u)) = f_2 \quad {\text{in }}G \times (0,T), \hfill \\ u(x,0) = u_0 (x)\quad {\text{in }}G,\quad v(s,t) = 0\quad {\text{on }}\partial G \times (0,T) \hfill \\ \end{gathered} \] is a well-posed model of diffusion in a fissured porous medium. Special features of the solution include the perseverance of local spatial continuity or singularities in the concentration u, the instantaneous propagation of the partially-saturated region throughout G, the delayed and limited advance of the fully-saturated region into G, and the concentration discontinuity on the boundary of the fully-saturated region. Weak maximum and order-comparison principles are obtained as