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Eduard Feireisl - One of the best experts on this subject based on the ideXlab platform.
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asymptotic preserving error estimates for numerical solutions of compressible navier stokes equations in the low mach number regime
Multiscale Modeling & Simulation, 2018Co-Authors: Eduard Feireisl, Antonin Novotný, Maria Lukacovamedviďova, Sarka Necasova, Bangwei SheAbstract:We study the convergence of numerical solutions of the compressible Navier--Stokes system to its Incompressible Limit. The numerical solution is obtained by a combined finite element--finite volume method based on the linear Crouzeix--Raviart finite element for the velocity and piecewise constant approximation for the density. The convective terms are approximated using upwinding. The distance between a numerical solution of the compressible problem and the strong solution of the Incompressible Navier--Stokes equations is measured by means of a relative energy functional. For barotropic pressure exponent $\gamma \geq 3/2$ and for well-prepared initial data we obtain uniform convergence of order ${\cal O}(\sqrt{\Delta t}, h^a, \varepsilon)$, $a = \min \{ \frac{2 \gamma - 3 }{ \gamma}, 1\}$. Extensive numerical simulations confirm that the numerical solution of the compressible problem converges to the solution of the Incompressible Navier--Stokes equations as the discretization parameters $\Delta t$, $h$ a...
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Incompressible Limit for Compressible Fluids with Stochastic Forcing
Archive for Rational Mechanics and Analysis, 2016Co-Authors: Dominic Breit, Eduard Feireisl, Martina HofmanováAbstract:We study the asymptotic behavior of the isentropic Navier–Stokes system driven by a multiplicative stochastic forcing in the compressible regime, where the Mach number approaches zero. Our approach is based on the recently developed concept of a weak martingale solution to the primitive system, uniform bounds derived from a stochastic analogue of the modulated energy inequality, and careful analysis of acoustic waves. A stochastic Incompressible Navier–Stokes system is identified as the Limit problem.
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the Incompressible Limit of the full navier stokes fourier system on domains with rough boundaries
Nonlinear Analysis-real World Applications, 2009Co-Authors: Eduard Feireisl, Dorin BucurAbstract:Abstract We study the Incompressible Limit of the full Navier–Stokes–Fourier system on condition that the boundary of the spatial domain oscillates with the amplitude and wave length proportional to the Mach number. Assuming the fluid satisfies the complete slip boundary conditions on the oscillating boundary, we identify the asymptotic Limit, and, in particular, establish strong (pointwise) convergence of the velocities towards a solenoidal vector field.
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on the Incompressible Limit for the navier stokes fourier system in domains with wavy bottoms
Mathematical Models and Methods in Applied Sciences, 2008Co-Authors: Eduard Feireisl, Antonin Novotný, Hana PetzeltovaAbstract:In this paper, the Oberbeck–Boussinesq approximation is identified as a singular Limit of the full Navier–Stokes–Fourier system provided the Mach and Froude numbers tend to zero. The result holds for any ill-prepared initial data and without any restrictions imposed on the length of the time interval. In particular, it is shown that the velocity converges almost everywhere, the oscillations of the sound waves being effectively damped by the presence of a "wavy bottom" of the physical domain.
Piotr B. Mucha - One of the best experts on this subject based on the ideXlab platform.
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compressible navier stokes system large solutions and Incompressible Limit
Advances in Mathematics, 2017Co-Authors: Raphaël Danchin, Piotr B. MuchaAbstract:Abstract Here we prove the existence of global in time regular solutions to the two-dimensional compressible Navier–Stokes equations supplemented with arbitrary large initial velocity v 0 and almost constant density ϱ 0 , for large volume (bulk) viscosity. The result is generalized to the higher dimensional case under the additional assumption that the strong solution of the classical Incompressible Navier–Stokes equations supplemented with the divergence-free projection of v 0 , is global. The systems are examined in R d with d ≥ 2 , in the critical B ˙ 2 , 1 s Besov spaces framework.
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Compressible Navier-Stokes system : large solutions and Incompressible Limit
Advances in Mathematics, 2017Co-Authors: Raphaël Danchin, Piotr B. MuchaAbstract:Here we prove the existence of global in time regular solutions to the two-dimensional compressible Navier-Stokes equations supplemented with arbitrary large initial velocity $v_0$ and almost constant density $\varrho_0$, for large volume (bulk) viscosity. The result is generalized to the higher dimensional case under the additional assumption that the strong solution of the classical Incompressible Navier-Stokes equations supplemented with the divergence-free projection of $v_0,$ is global. The systems are examined in $R^d$ with $d \geq 2$, in the critical $\dot B^s_{2,1}$ Besov spaces framework.
Raphaël Danchin - One of the best experts on this subject based on the ideXlab platform.
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compressible navier stokes system large solutions and Incompressible Limit
Advances in Mathematics, 2017Co-Authors: Raphaël Danchin, Piotr B. MuchaAbstract:Abstract Here we prove the existence of global in time regular solutions to the two-dimensional compressible Navier–Stokes equations supplemented with arbitrary large initial velocity v 0 and almost constant density ϱ 0 , for large volume (bulk) viscosity. The result is generalized to the higher dimensional case under the additional assumption that the strong solution of the classical Incompressible Navier–Stokes equations supplemented with the divergence-free projection of v 0 , is global. The systems are examined in R d with d ≥ 2 , in the critical B ˙ 2 , 1 s Besov spaces framework.
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Compressible Navier-Stokes system : large solutions and Incompressible Limit
Advances in Mathematics, 2017Co-Authors: Raphaël Danchin, Piotr B. MuchaAbstract:Here we prove the existence of global in time regular solutions to the two-dimensional compressible Navier-Stokes equations supplemented with arbitrary large initial velocity $v_0$ and almost constant density $\varrho_0$, for large volume (bulk) viscosity. The result is generalized to the higher dimensional case under the additional assumption that the strong solution of the classical Incompressible Navier-Stokes equations supplemented with the divergence-free projection of $v_0,$ is global. The systems are examined in $R^d$ with $d \geq 2$, in the critical $\dot B^s_{2,1}$ Besov spaces framework.
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The Incompressible Limit in $L^p$ type critical spaces
Mathematische Annalen, 2016Co-Authors: Raphaël DanchinAbstract:This paper aims at justifying the low Mach number convergence to the Incompressible Navier-Stokes equations for viscous compressible flows in the ill-prepared data case. The fluid domain is either the whole space, or the torus. A number of works have been dedicated to this classical issue, all of them being, to our knowledge, related to $L^2$ spaces and to energy type arguments. In the present paper, we investigate the low Mach number convergence in the $L^p$ type critical regularity framework. More precisely, in the barotropic case, the divergence-free part of the initial velocity field just has to be bounded in the critical Besov space $\dot B^{d/p-1}_{p,r}\cap\dot B^{-1}_{\infty,1}$ for some suitable $(p,r)\in[2,4]\times[1,+\infty].$ We still require $L^2$ type bounds on the low frequencies of the potential part of the velocity and on the density, though, an assumption which seems to be unavoidable in the ill-prepared data framework, because of acoustic waves. In the last part of the paper, our results are extended to the full Navier-Stokes system for heat conducting fluids.
Nicolas Vauchelet - One of the best experts on this subject based on the ideXlab platform.
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Incompressible Limit for a two species model with coupling through brinkman s law in any dimension
Journal de Mathématiques Pures et Appliquées, 2021Co-Authors: Tomasz Debiec, Benoît Perthame, Markus Schmidtchen, Nicolas VaucheletAbstract:Abstract We study the Incompressible Limit for a two-species model with applications to tissue growth in the case of coupling through the so-called Brinkman's law in any space dimensions. The coupling through this elliptic equation accounts for viscosity effects among the individual species. In a recent paper Debiec & Schmidtchen established said result in one spacial dimension, with their proof hinging on being able to establish uniform BV-bounds. This approach is fundamentally different from the one-species case in arbitrary dimension, established by Perthame & Vauchelet . Their result relies on a kinetic reformulation to obtain strong compactness of the pressure. In this paper we fill this gap in the literature and present the Incompressible Limit for the system in arbitrary space dimension. The difficulty stems from jump discontinuities in the pressure not only at the boundary of the support of the two species but also at internal layers giving rise to the question as to how compactness can be obtained. The answer is a combination of techniques consisting of the application of the compactness method of Bresch & Jabin , an adaptation of the aforementioned kinetic reformulation, and several parallels to the one dimensional strategy. The main result of this paper establishes a rigorous bridge between the population dynamics of growing tissue at a density level and a geometric model thereof.
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Incompressible Limit of a continuum model of tissue growth for two cell populations
Networks and Heterogeneous Media, 2020Co-Authors: Pierre Degond, Sophie Hecht, Nicolas VaucheletAbstract:This paper investigates the Incompressible Limit of a system modelling the growth of two cells population. The model describes the dynamics of cell densities, driven by pressure exclusion and cell proliferation. It has been shown that solutions to this system of partial differential equations have the segregation property, meaning that two population initially segregated remain segregated. This work is devoted to the Incompressible Limit of such system towards a free boundary Hele Shaw type model for two cell populations.
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Incompressible Limit of a mechanical model for tissue growth with non-overlapping constraint.
Communications in Mathematical Sciences, 2017Co-Authors: Sophie Hecht, Nicolas VaucheletAbstract:A mathematical model for tissue growth is considered. This model describes the dynamics of the density of cells due to pressure forces and proliferation. It is known that such cell population model converges at the Incompressible Limit towards a Hele-Shaw type free boundary problem. The novelty of this work is to impose a non-overlapping constraint. This constraint is important to be satisfied in many applications. One way to guarantee this non-overlapping constraint is to choose a singular pressure law. The aim of this paper is to prove that, although the pressure law has a singularity, the Incompressible Limit leads to the same Hele-Shaw free boundary problem.
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Incompressible Limit of the Navier-Stokes model with a growth term
Nonlinear Analysis, 2017Co-Authors: Nicolas Vauchelet, Ewelina ZatorskaAbstract:Starting from isentropic compressible Navier-Stokes equations with growth term in the continuity equation, we rigorously justify that performing an Incompressible Limit one arrives to the two-phase free boundary fluid system.
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Incompressible Limit of a mechanical model of tumour growth with viscosity
Philosophical transactions. Series A Mathematical physical and engineering sciences, 2015Co-Authors: Benoît Perthame, Nicolas VaucheletAbstract:Various models of tumor growth are available in the litterature. A first class describes the evolution of the cell number density when considered as a continuous visco-elastic material with growth. A second class, describes the tumor as a set and rules for the free boundary are given related to the classical Hele-Shaw model of fluid dynamics. Following the lines of previous papers where the material is described by a purely elastic material, or when active cell motion is included, we make the link between the two levels of description considering the 'stiff pressure law' Limit. Even though viscosity is a regularizing effect, new mathematical difficulties arise in the visco-elastic case because estimates on the pressure field are weaker and do not imply immediately compactness. For instance, traveling wave solutions and numerical simulations show that the pressure may be discontinous in space which is not the case for the elastic case.
Ramon Codina - One of the best experts on this subject based on the ideXlab platform.
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a mixed three field fe formulation for stress accurate analysis including the Incompressible Limit
Computer Methods in Applied Mechanics and Engineering, 2015Co-Authors: Michele Chiumenti, Miguel Cervera, Ramon CodinaAbstract:Abstract In previous works, the authors have presented the stabilized mixed displacement/pressure formulation to deal with the incompressibility constraint. More recently, the authors have derived stable mixed stress/displacement formulations using linear/linear interpolations to enhance stress accuracy in both linear and non-linear problems. In both cases, the Variational Multi Scale (VMS) stabilization technique and, in particular, the Orthogonal Subgrid Scale (OSS) method allows the use of linear/linear interpolations for triangular and tetrahedral elements bypassing the strictness of the inf–sup condition on the choice of the interpolation spaces. These stabilization procedures lead to discrete problems which are fully stable, free of volumetric locking or stress oscillations. This work exploits the concept of mixed finite element methods to formulate stable displacement/stress/pressure finite elements aimed for the solution of nonlinear problems for both solid and fluid finite element (FE) analyses. The final goal is to design a finite element technology able to tackle simultaneously problems which may involve isochoric behavior (preserve the original volume) of the strain field together with high degree of accuracy of the stress field. These two features are crucial in nonlinear solid and fluid mechanics, as used in most numerical simulations of industrial manufacturing processes. Numerical benchmarks show that the results obtained compare very favorably with those obtained with the corresponding mixed displacement/pressure formulation.
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a mixed three field fe formulation for stress accurate analysis including the Incompressible Limit
Computer Methods in Applied Mechanics and Engineering, 2015Co-Authors: Michele Chiumenti, Miguel Cervera, Ramon CodinaAbstract:In previous works, the authors have presented the stabilized mixed displacement/pressure formulation to deal with the incompressibility constraint. More recently, the authors have derived stable mixed stress/displacement formulations using linear/linear interpolations to enhance stress accuracy in both linear and non-linear problems. In both cases, the Variational Multi Scale (VMS) stabilization technique and, in particular, the Orthogonal Subgrid Scale (OSS) method allows the use of linear/linear interpolations for triangular and tetrahedral elements bypassing the strictness of the inf-sup condition on the choice of the interpolation spaces. These stabilization procedures lead to discrete problems which are fully stable, free of volumetric locking or stress oscillations.; This work exploits the concept of mixed finite element methods to formulate stable displacement/stress/pressure finite elements aimed for the solution of nonlinear problems for both solid and fluid finite element (FE) analyses. The final goal is to design a finite element technology able to tackle simultaneously problems which may involve isochoric behavior (preserve the original volume) of the strain field together with high degree of accuracy of the stress field. These two features are crucial in nonlinear solid and fluid mechanics, as used in most numerical simulations of industrial manufacturing processes.; Numerical benchmarks show that the results obtained compare very favorably with those obtained with the corresponding mixed displacement/pressure formulation. (C) 2014 Elsevier B.V. All rights reserved.
