Thermistor systems of p(x)-Laplace-type with discontinuous exponents via entropy solutions

We show the existence of solutions to a system of elliptic PDEs, that was recently introduced to describe the electrothermal behavior of organic semiconductor devices. Here, two difficulties appear: (i) the elliptic term in the current-flow equation is of p(x)-Laplacian-type with discontinuous exponent p, which limits the use of standard methods, and (ii) in the heat equation, we have to deal with an a priori L1 term on the right hand side describing the Joule heating in the device. We prove the existence of a weak solution under very weak assumptions on the data. Our existence proof is based on Schauder's fixed point theorem and the concept of entropy solutions for the heat equation. Here, the crucial point is the continuous dependence of the entropy solutions on the data of the problem

Transport and Mobilization of Pathogenic Microbes and Microspheres in Unsaturated Fractured Media: Effect of Microbe Size, Soil Physical Heterogeneity, and Intermittent Flow and Effect of Redox Condit

The potential of intermittent rainfall to mobilize microbes of diverse size, shape, and taxa in heterogeneous soil systems is important in assessing groundwater contamination risks. We investigated the transport, retention, and mobilization of microbes through an intact soil core. Microbes (MS-2 bacteriophage, Pseudomonas stutzeri bacteria, and Cryptosporidium parvum oocysts), microspheres, and a bromide tracer were applied to the core and breakthrough was measured to resolve the effect of soil physical heterogeneity. After breakthrough, the core was subjected to intermittent rainfalls to mobilize the attached microbes and microspheres. This study demonstrated that intermittent flows mobilize attached microbes and microspheres in a physically heterogeneous soil regardless of size or taxa; however, the degree of mobilization was dependent on microbe size and shape. Mobilization of larger, spherical C. parvum oocysts was greater than that of smaller, spherical MS-2 bacteriophage and rod-shaped P. stutzeri bacteria. Cumulatively, the order of recovery was C. parvum oocysts &gt; microspheres &gt; MS-2 bacteriophage &gt; P. stutzeri cells. The release of coal ash because of a storage pond dike failure at the Tennessee Valley Authority's Kingston Fossil Plant in December 2008 is a concern because of the potential ecological and human health risks posed by the release of toxic elements from the coal ash into surrounding waters. The effects of redox conditions were examined in this study, which focused on how the differences between elements affect their release as a function of varied redox conditions. Using Emory River sediments and water, and Kingston coal ash, a batch reactor system was created. This experiment was conducted in three stages - oxidizing, transition, and reducing - in order to simulate redox changes that might occur during weathering of the released coal ash on riverbed sediments. The results suggest that the release of several trace elements, from the coal ash may present water quality issues based on their increasing concentrations during the reducing stage. The differences in release can be attributed to changes in chemical properties (e.g., pH, redox potential), presence and aromaticity of DOC, dissolution of mineral phases, competition or interaction with other species, precipitation of immobile phases, or release of mobile phases.</p

Existence and uniqueness of global weak solutions to strain-limiting viscoelasticity with Dirichlet boundary data

We consider a system of evolutionary equations that is capable of describing certain viscoelastic effects in linearized yet nonlinear models of solid mechanics. The constitutive relation, involving the Cauchy stress, the small strain tensor and the symmetric velocity gradient, is given in an implicit form. For a large class of these implicit constitutive relations, we establish the existence and uniqueness of a global-in-time large-data weak solution. Then we focus on the class of so-called limiting strain models, i.e., models for which the magnitude of the strain tensor is known to remain small a priori, regardless of the magnitude of the Cauchy stress tensor. For this class of models, a new technical difficulty arises. The Cauchy stress is only an integrable function over its domain of definition, resulting in the underlying function spaces being nonreflexive and thus the weak compactness of bounded sequences of elements of these spaces is lost. Nevertheless, even for problems of this type we are able to provide a satisfactory existence theory, as long as the initial data have finite elastic energy and the boundary data fulfil natural compatibility conditions

Large data analysis for Kolmogorovâ s two-equation model of turbulence

We establish long-time and large-data existence of a suitable weak solution to three-dimensional internal unsteady flows described by Kolmogorovâ s two-equation model of turbulence. The governing system of equations is completed by initial and boundary conditions; concerning the velocity we consider generalized stickâ slip boundary conditions. The fact that the admissible class of boundary conditions includes various types of slipping mechanisms on the boundary makes the result robust from the point of view of possible applications.Non UBCUnreviewedAuthor affiliation: Charles UniversityResearche

On evolutionary Navier-Stokes-Fourier type systems in three spatial dimensions

International audienc

A boundary regularity result for minimizers of variational integrals with nonstandard growth

We prove global Lipschitz regularity for a wide class of convex variational integrals among all functions in $W^{1,1}$ with prescribed (sufficiently regular) boundary values, which are not assumed to satisfy any geometrical constraint (as for example bounded slope condition). Furthermore, we do not assume any restrictive assumption on the geometry of the domain and the result is valid for all sufficiently smooth domains. The result is achieved with a suitable approximation of the functional together with a new construction of appropriate barrier functions

Existence and uniqueness of global weak solutions to strain-limiting viscoelasticity with Dirichlet boundary data

We consider a system of evolutionary equations that is capable of describing certain viscoelastic effects in linearized yet nonlinear models of solid mechanics. The constitutive relation, involving the Cauchy stress, the small strain tensor and the symmetric velocity gradient, is given in an implicit form. For a large class of these implicit constitutive relations, we establish the existence and uniqueness of a global-in-time large-data weak solution. Then we focus on the class of so-called limiting strain models, i.e., models for which the magnitude of the strain tensor is known to remain small a priori, regardless of the magnitude of the Cauchy stress tensor. For this class of models, a new technical difficulty arises. The Cauchy stress is only an integrable function over its domain of definition, resulting in the underlying function spaces being nonreflexive and thus the weak compactness of bounded sequences of elements of these spaces is lost. Nevertheless, even for problems of this type we are able to provide a satisfactory existence theory, as long as the initial data have finite elastic energy and the boundary data fulfil natural compatibility conditions