scholarly journals Perturbative Cauchy theory for a flux-incompressible Maxwell-Stefan system

2022 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Andrea Bondesan ◽  
Marc Briant
Keyword(s):  
Equilibrium State ◽  
Existence And Uniqueness ◽  
Quasilinear Parabolic Equation ◽  
Strong Solutions ◽  
Cross Diffusion ◽  
Anisotropic Norm ◽  
Exponential Trend ◽  
Perturbative Regime ◽  
Global Existence And Uniqueness ◽  
Quasilinear Parabolic

<p style='text-indent:20px;'>Recently, the authors proved [<xref ref-type="bibr" rid="b2">2</xref>] that the Maxwell-Stefan system with an incompressibility-like condition on the total flux can be rigorously derived from the multi-species Boltzmann equation. Similar cross-diffusion models have been widely investigated, but the particular case of a perturbative incompressible setting around a non constant equilibrium state of the mixture (needed in [<xref ref-type="bibr" rid="b2">2</xref>]) seems absent of the literature. We thus establish a quantitative perturbative Cauchy theory in Sobolev spaces for it. More precisely, by reducing the analysis of the Maxwell-Stefan system to the study of a quasilinear parabolic equation on the sole concentrations and with the use of a suitable anisotropic norm, we prove global existence and uniqueness of strong solutions and their exponential trend to equilibrium in a perturbative regime around any macroscopic equilibrium state of the mixture. As a by-product, we show that the equimolar diffusion condition naturally appears from this perturbative incompressible setting.</p>

scholarly journals Global existence of strong solutions for incompressible hydrodynamic flow of liquid crystals with vacuum

Filomat ◽  
2013 ◽  
Vol 27 (7) ◽  
pp. 1247-1257 ◽  
Author(s):  
Shijin Ding ◽  
Jinrui Huang ◽  
Fengguang Xia
Keyword(s):  
Cauchy Problem ◽  
Liquid Crystals ◽  
Global Existence ◽  
Existence And Uniqueness ◽  
Strong Solutions ◽  
Hydrodynamic Flow ◽  
Three Dimensions ◽  
Small Initial Data ◽  
The Cauchy Problem ◽  
Global Existence And Uniqueness

We consider the Cauchy problem for incompressible hydrodynamic flow of nematic liquid crystals in three dimensions. We prove the global existence and uniqueness of the strong solutions with nonnegative p0 and small initial data.

scholarly journals EXISTENCE AND UNIQUENESS OF STRONG SOLUTIONS FOR A COMPRESSIBLE MULTIPHASE NAVIER–STOKES MISCIBLE FLUID-FLOW PROBLEM IN DIMENSION n = 1

2009 ◽  
Vol 19 (03) ◽  
pp. 443-476 ◽  
Author(s):  
C. MICHOSKI ◽  
A. VASSEUR
Keyword(s):  
Diffusion Coefficient ◽  
Fluid Flow ◽  
Existence And Uniqueness ◽  
Flow Problem ◽  
Stokes Equations ◽  
Strong Solutions ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Global Existence And Uniqueness ◽  
Miscible Fluid

We prove the global existence and uniqueness of strong solutions for a compressible multifluid described by the barotropic Navier–Stokes equations in dim = 1. The result holds when the diffusion coefficient depends on the pressure. It relies on a global control in time of the L2 norm of the space derivative of the density, via a new kind of entropy.

scholarly journals Decay of the 3D Quasilinear Hyperbolic Equations with Nonlinear Damping

2017 ◽  
Vol 2017 ◽  
pp. 1-13 ◽  
Author(s):  
Hongjun Qiu ◽  
Yinghui Zhang
Keyword(s):  
Low Temperature ◽  
Existence And Uniqueness ◽  
Hyperbolic Equations ◽  
Initial Perturbation ◽  
Strong Solutions ◽  
Nonlinear Damping ◽  
Decay Rates ◽  
Quasilinear Hyperbolic Equations ◽  
Global Existence And Uniqueness ◽  
Very Low Temperature

We investigate the 3D quasilinear hyperbolic equations with nonlinear damping which describes the propagation of heat wave for rigid solids at very low temperature, below about 20 K. The global existence and uniqueness of strong solutions are obtained when the initial data is near its equilibrium in the sense of H3-norm. Furthermore, if, additionally, Lp-norm (1≤p<6/5) of the initial perturbation is finite, we also prove the optimal Lp-L2 decay rates for such a solution without the additional technical assumptions for the nonlinear damping f(v) given by Li and Saxton.

Pseudo-parabolic regularization of forward-backward parabolic equations: Power-type nonlinearities

2016 ◽  
Vol 2016 (712) ◽  
Author(s):  
Michiel Bertsch ◽  
Flavia Smarrazzo ◽  
Alberto Tesei
Keyword(s):  
Parabolic Equations ◽  
Existence And Uniqueness ◽  
Quasilinear Parabolic Equation ◽  
Singular Part ◽  
Qualitative Properties ◽  
Power Type ◽  
Radon Measures ◽  
Entropy Inequalities ◽  
Backward Parabolic Equations ◽  
Quasilinear Parabolic

AbstractWe study a quasilinear parabolic equation of forward-backward type, under assumptions on the nonlinearity which hold for a wide class of mathematical models, using a pseudo-parabolic regularization of power type. We prove existence and uniqueness of positive solutions of the regularized problem in a space of Radon measures. It is shown that these solutions satisfy suitable entropy inequalities. We also study their qualitative properties, in particular proving that the singular part of the solution with respect to the Lebesgue measure is constant in time.

REGULARIZATION OF QUASILINEAR HEAT EQUATIONS BY A FRACTIONAL NOISE

2004 ◽  
Vol 04 (02) ◽  
pp. 201-221 ◽  
Author(s):  
DAVID NUALART ◽  
YOUSSEF OUKNINE
Keyword(s):  
Parabolic Equation ◽  
White Noise ◽  
Existence And Uniqueness ◽  
Integrability Condition ◽  
Quasilinear Parabolic Equation ◽  
One Dimension ◽  
Heat Equations ◽  
Fractional Noise ◽  
Uniqueness Of A Solution ◽  
Quasilinear Parabolic

We show the existence and uniqueness of a solution for a quasilinear parabolic equation in one dimension driven by an additive fractional white noise, assuming that the drift is measurable and satisfies a suitable integrability condition. The proof is based on Girsanov theorem and lower estimates of the density of the solution of the equation without drift.

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