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We investigate the dynamics of a nonlinear model for tumor growth within a cellular medium. In this setting the 'tumor' is viewed as a multiphase flow consisting of cancerous cells in either proliferating phase or quiescent phase and a collection of cells accounting for the 'waste' and/or dead cells in the presence of a nutrient. Here, the tumor is thought of as a growing continuum $$\Omega $$ with boundary $$\partial \Omega $$ both of which evolve in time. In particular, the evolution of the boundary $$\partial \Omega $$ is prescibed by a given velocity $${{{\varvec{V}}}.}$$ The key characteristic of the present model is that the total density of cancerous cells is allowed to vary, which is often the case within cellular media. We refer the reader to the articles (Enault in Mathematical study of models of tumor growth, 2010; Li and Lowengrub in J Theor Biol, 343:79-91, 2014) where compressible type tumor growth models are investigated. Global-in-time weak solutions are obtained using an approach based on penalization of the boundary behavior, diffusion, viscosity and pressure in the weak formulation, as well as convergence and compactness arguments in the spirit of Lions (Mathematical topics in fluid dynamics. Compressible models, 1998) [see also Donatelli and Trivisa (J Math Fluid Mech 16: 787-803, 2004), Feireisl (Dynamics of viscous compressible fluids, 2014)]. [ABSTRACT FROM AUTHOR]

Polymeric fluids arise in many practical applications in biotechnology, medicine, chemistry, industrial processes, and atmospheric sciences. In this paper, the Doi model for the suspensions of rod-like molecules in a compressible fluid is investigated. The model under consideration describes the interaction between the orientation of rod-like polymer molecules on the microscopic scale and the macroscopic properties of the fluid in which these molecules are contained. Prescribing arbitrarily the initial density of the fluid, the initial velocity, and the initial orientation distribution in suitable spaces, we establish the global-in-time existence of a weak solution to our model defined on a bounded domain in the three-dimensional space. The proof relies on the construction of an approximate sequence of solutions by introducing appropriate regularization and the establishment of compactness. [ABSTRACT FROM AUTHOR]

This article deals with a fluid-particle interaction model for the evolution of particles dispersed in a viscous compressible fluid within the physical space Ω ⊂ R&sup3'. The system is expressed by the continuity equation, the balance of momentum and the so-called Smoluchowski equation for the evolution of particles. The coupling between the dispersed and dense phases is obtained through the drag forces that the fluid and the particles exert mutually by the action-reaction principle. Using the relative entropy method of Dafermos and DiPerna, the global-in-time existence of suitable weak solutions is presented under reasonable physical assumptions on the initial data, the physical domain, and the external potential. In addition, the low Mach number and low stratification limits of the system are established rigorously for both bounded and unbounded domains. [ABSTRACT FROM AUTHOR]

We establish a low Mach number limit for classical solutions over the whole space of a compressible fluid dynamic system that includes dispersive corrections to the Navier-Stokes equations. The limiting system is similar to a ghost effect system [Y. Sone, Kinetic Theory and Fluid Dynamics, Model. Simul. Sci. Eng. Technol., Birkhäuser, Boston, 2002]. Our analysis builds upon the framework developed by Métivier and Schochet [Arch. Ration. Mech. Anal., 158 (2001), pp. 61-90] and Alazard [Arch. Ration. Mech. Anal., 180 (2006), pp. 1-73] for nondispersive systems. The strategy involves establishing a priori estimates for the slow motion as well as a priori estimates for the fast motion. The desired convergence is obtained by establishing the local decay of the energy of the fast motion. [ABSTRACT FROM AUTHOR]

Abstract: A model for the dynamics of gaseous stars is introduced and formulated by the Navier–Stokes–Poisson system for compressible, reacting gases. The combined quasineutral and inviscid limit of the Navier–Stokes–Poisson system in the torus is investigated. The convergence of the Navier–Stokes–Poisson system to the incompressible Euler equations is proven for the global weak solution and for the case of general initial data. [Copyright &y& Elsevier]

We prove the global existence of solutions of the Navier-Stokes equations describing the dynamic combustion of a compressible, exothermically reacting fluid, and we study the large-time behavior of solutions, giving necessary and sufficient conditions for complete combustion in certain cases. The adiabatic constants and specific heats of the burned (product) and unburned (reactant) fluids may differ, and the initial data may be large and discontinuous. [ABSTRACT FROM AUTHOR]