On the piecewise smoothness of entropy solutions to scalar conservation laws

The behavior and structure of entropy solutions of scalar convex conservation laws are studied. It is well known that such entropy solutions consist of at most countable number of C¹-smooth regions. We obtain new upper. bounds on the higher order derivatives of the entropy solution in any one of its C¹-smoothness regions. These bounds enable us to measure the high order piecewise smoothness of the entropy solution. To this end we introduce an appropriate new Cⁿ-semi norm - localized to the smooth part of the entropy solution, and we show that the entropy solution is stable with respect to this norm. We also address the question regarding the number of C¹-smoothness pieces; we show that if the initial speed has a finite number of decreasing inflection points then it bounds the number of future shock discontinuities. Loosely speaking this says that in the case of such generic initial data the entropy solution consists of a finite number of smooth pieces, each of which is as smooth as the data permits. It is this type of piecewise smoothness which is assumed - sometime implicitly - in many finite-dimensional computations for such discontinuous problems.     

Physical picture behind the oscillating sign of drag in high Landau levels

We consider the oscillating sign of the drag resistivity and its anomalous temperature dependence discovered experimentally in a bilayer system in the regime of the integer quantum Hall effect. We attribute the oscillating sign to the effect of disorder on the relation between an adiabatic momentum transfer to an electron and the displacement of its position. While in the absence of any Landau level mixing a momentum transfer implies a displacement of (with being the magnetic length), Landau level mixing induced by short range disorder adds a potentially large displacement that depends on the electron's energy, with the sign being odd with respect to the distance of that energy from the center of the Landau level. We show how the oscillating sign of drag disappears when the disorder is smooth and when the electronic states are localized.     

Line energy,line tension and drop size

The relation between drop radius, r, the force to move the three phase contact line and the advancing and receding contact angles θ_A and θ_R is studied. To keep the line energy (energy per 2πr, also named line tension) independent of r, the modified Young equation predicts that the advancing and receding contact angles, θ_A and θ_R, change considerably with r. As shown by many investigators, θ_A and θ_R change negligibly, if at all, with r. We quantify recent evidences showing that the line energy is a function of the Laplace pressure and show that this way the modified Young equation is correct and still θ_A and θ_R should hardly change with r. According to our model, the small surface deformation associated with the unsatisfied normal component of the Young equation results in higher intermolecular interactions at the three phase contact line which corresponds to a higher retention force. This time increasing effect is supported by recent experiments.     

Hierarchical Construction of Bounded Solutions in Critical Regularity Spaces

We construct uniformly bounded solutions for the equations div U = f and U = f in the critical cases and , respectively. Criticality in this context manifests itself by the lack of a linear solution operator mapping . Thus, the intriguing aspect here is that although the problems are linear, construction of their solutions is not. Our constructions are special cases of a general framework for solving linear equations of the form , where is a linear operator densely defined in Banach space with a closed range in a (proper subspace) of Lebesgue space , and with an injective dual . The solutions are realized in terms of a multiscale hierarchical representation, , interesting for its own sake. Here, u _j's are constructed recursively as minimizers of where the residuals are resolved in terms of a dyadic sequence of scales with large enough . The nonlinear aspect of this construction is a counterpart of the fact that one cannot linearly solve in critical regularity spaces.© 2016 Wiley Periodicals, Inc.     

A Unified Interpretation of Stress in Molecular Systems

The microscopic definition for the Cauchy stress tensor has been examined in the past from many different perspectives. This has led to different expressions for the stress tensor and consequently the “correct” definition has been a subject of debate and controversy. In this work, a unified framework is set up in which all existing definitions can be derived, thus establishing the connections between them. The framework is based on the non-equilibrium statistical mechanics procedure introduced by Irving, Kirkwood and Noll, followed by spatial averaging. The Irving–Kirkwood–Noll procedure is extended to multi-body potentials with continuously differentiable extensions and generalized to non-straight bonds, which may be important for particles with internal structure. Connections between this approach and the direct spatial averaging approach of Murdoch and Hardy are discussed and the Murdoch–Hardy procedure is systematized. Possible sources of non-uniqueness of the stress tensor, resulting separately from both procedures, are identified and addressed. Numerical experiments using molecular dynamics and lattice statics are conducted to examine the behavior of the resulting stress definitions including their convergence with the spatial averaging domain size and their symmetry properties.     

Wirelessly powered dielectrophoresis of metal oxide particles using spark-gap Tesla coil

Wirelessly powered dielectrophoresis (DEP) of metal oxide particles was performed using a spark-gap Tesla coil (TC). The main contribution of this work is the simplification of the conventional DEP setup that requires attaching wires directly to the electrodes. Wireless power from the TC generates a high output frequency and voltage, which corresponds to that used for the DEP. Therefore, a spark-gap TC was built and utilized to conduct the DEP process. Metal oxides (ZnO and Fe₂O₃) were used as targets for the assembly. The results showed that the wirelessly powered DEP technique via a TC was successful in assembling the metal oxide particles. Positive and negative DEP phenomena were observed. Positive DEP occurred during ZnO assembly, making particles chain grow 0.92 mm toward the sparks within 60 s. Negative DEP was observed during Fe₂O₃ assembly, where the repulsion of particles formed a void around the sparks with a 1.45 mm radius. The mechanism of this wireless DEP system is discussed.