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We consider a second order linear evolution equation with a dissipative term multiplied by a time-dependent coefficient. Our aim is to design the coefficient in such a way that all solutions decay in time as fast as possible. We discover…
In a first order theory of dissipative hydrodynamics, we have simulated hydrodynamic evolution of QGP fluid with dissipation due to shear viscosity only. Simulation confirms that compared to an ideal fluid, energy density or temperature of…
In this paper we study smooth solutions to a fractional mean curvature flow equation. We establish a comparison principle and consequences such as uniqueness and finite extinction time for compact solutions. We also establish evolutions…
In this note we study the singular vanishing-viscosity limit of a gradient flow set in a finite-dimensional Hilbert space and driven by a smooth, but possibly non convex, time-dependent energy functional. We resort to ideas and techniques…
Recently, fractional derivatives have been employed to analyze various systems in engineering, physics, finance and hidrology. For instance, they have been used to investigate anomalous diffusion processes which are present in different…
As a counterpoint to recent numerical methods for crystal surface evolution, which agree well with microscopic dynamics but suffer from significant stiffness that prevents simulation on fine spatial grids, we develop a new numerical method…
We present a new temporal discretization paradigm for developing energy-production-rate preserving numerical approximations to thermodynamically consistent partial differential equation systems, called the supplementary variable method. The…
Scaling and structural evolutions are contemplated in a new perspective for turbulent channel flows. The total integrated turbulence kinetic energy remains constant when normalized by the friction velocity squared, while the total…
Shape optimization based on surface gradients and the Hadarmard-form is considered for a compressible viscous fluid. Special attention is given to the difference between the 'function composition' approach involving local shape derivatives…
An experiment was performed using SPIV in the LMFL boundary layer facility to determine all the derivative moments needed to estimate the average dissipation rate of the turbulence kinetic energy, $\varepsilon = 2 \nu \langle s_{ij}s_{ij}…
We revisit the two body problem, where one body can be deformed under the action of tides raised by the companion. Tidal deformation and consequent dissipation result in spin and orbital evolution of the system. In general, the equations of…
Phase field models for two-phase flow with a surfactant soluble in possibly both fluids are derived from balance equations and an energy inequality so that thermodynamic consistency is guaranteed. Via a formal asymptotic analysis, they are…
The emergence of a classical spacetime from any quantum gravity model is still a subtle and only partially understood issue. If indeed spacetime is arising as some sort of large scale condensate of more fundamental objects then it is…
In this paper we consider two-dimensional, stratified, steady water waves propagating over an impermeable flat bed and with a free surface. The motion is assumed to be driven by capillarity (that is, surface tension) on the surface and a…
This paper is devoted to the global (in time) regularity problem for a family of active scalar equations with fractional dissipation. Each component of the velocity field $u$ is determined by the active scalar $\theta$ through $\mathcal{R}…
Macroscopic equations arising out of stochastic particle systems in detailed balance (called dissipative systems or gradient flows) have a natural variational structure, which can be derived from the large-deviation rate functional for the…
We consider the governing equations for the motion of the inviscid fluids in two moving domains and an evolving surface from an energetic point of view. We employ our energetic variational approaches to derive inviscid multiphase flow…
This paper studies a class of linear unconditionally energy stable schemes for the gradient flows. Such schemes are built on the SAV technique and the general linear time discretization (GLTD) as well as the linearization based on the…
In this chapter we provide an introduction to fractional dissipative partial differential equations (PDEs) with a focus on trying to understand their dynamics. The class of PDEs we focus on are reaction-diffusion equations but we also…
The mechanisms promoting scalar dissipation through scalar gradient production are scrutinized in terms of vorticity alignment with respect to strain principal axes. For that purpose, a stochastic Lagrangian model for the velocity gradient…