Well-posedness and exponential decay of the energy in the nonlinear jordanmooregibsonthompson equation arising in high intensity ultrasound

We consider a third order in time equation which arises, e.g. as a model for wave propagation in viscous thermally relaxing fluids. This equation displays, even in the linear version, a variety of dynamical behaviors for its solution that depend on the physical parameters in the equation. These range from non-existence and instability to exponential stability (in time) as was shown for the constant coefficient case in Ref. 23. In case of vanishing diffusivity of the sound, there is a lack of generation of a semigroup associated with the linear dynamics. If diffusivity of the sound is positive, the linear dynamics is described by a strongly continuous hyperbolic-like evolution. This evolution is exponentially stable provided sufficiently large viscous damping is accounted for in the model. In this paper, we consider the full nonlinear model referred to as JordanMooreGibsonThompson equation. This model can be seen as a hyperbolic version of Kuznetsov\u27s equation, where the linearization of the latter corresponds to an analytic semigroup. This is no longer valid for the presently considered third-order model whose linearization is associated with a group structure. In order to carry out the analysis of the nonlinear model, we first consider time and space-dependent viscosity which then leads to evolution rather than semigroup generators. Decay rates for both natural and higher level energies are derived. Relevant physical parameters that are responsible for spectral behavior (continuous and point spectrum) are identified. The theoretical estimates proved in the paper are confirmed by numerical simulations. The derived energy estimates are then used in order to establish global well-posedness and exponential decay for the solutions to the nonlinear equation. © 2012 World Scientific Publishing Company

Cell Protrusions and Tethers: A Unified Approach

Low pulling forces applied locally to cell surface membranes produce viscoelastic cell surface protrusions. As the force increases, the membrane can locally separate from the cytoskeleton and a tether forms. Tethers can grow to great lengths exceeding the cell diameter. The protrusion-to-tether transition is known as the crossover. Here we propose a unified approach to protrusions and tethers providing, to our knowledge, new insights into their biomechanics. We derive a necessary and sufficient condition for a crossover to occur, a formula for predicting the crossover time, conditions for a tether to establish a dynamic equilibrium (characterized by constant nonzero pulling force and tether extension rate), a general formula for the tether material after crossover, and a general modeling method for tether pulling experiments. We introduce two general protrusion parameters, the spring constant and effective viscosity, valid before and after crossover. Their first estimates for neutrophils are 50 pN μm−1 and 9 pN s μm−1, respectively. The tether elongation after crossover is described as elongation of a viscoelastic-like material with a nonlinearly decaying spring (NLDs-viscoelastic material). Our model correctly describes the results of the published protrusion and tether pulling experiments, suggesting that it is universally applicable to such experiments