We consider a class of one dimensional compressible systems with degenerate diffusion coefficients. We establish the fact that the solutions remain smooth as long as the diffusion coefficients do not vanish, and give local and global existence results. The models include the barotropic compressible Navier-Stokes equations, shallow water systems and the lubrication approximation of slender jets. In all these models the momentum equation is forced by the gradient of a solution-dependent potential: the active potential. The method of proof uses the Bresch-Desjardins entropy and the analysis of the evolution of the active potential.
We study the lagrangian structure for weak solutions of two dimensional Navier-Stokes equations for a non-barotropic compressible fluid, i.e. we show the uniqueness of particle trajectories for two dimensional compressible fluids including the energy equation. Our result extends partially the result obtained for barotropic fluids by Hoff and Santos (2008).
We study the system of equations which describes barotropic (isentropic) flows of viscous compressible multi-fluids (mixtures of fluids). We study the relations between pressure, densities, concentrations, viscosities and other parameters of the flow and of the medium.
We extend the weak-strong uniqueness principle to general models of compressible viscous fluids near/on the vacuum. In particular, the physically relevant case of positive density with polynomial decay at infinity is considered.
We study a possibility of existence of localized two-dimensional structures, both smooth and non-smooth, that can move without significant change of their shape in a leading stream of compressible barotropic fluid on a rotating plane.
In this paper, we rigorously derive the compressible one-fluid Navier-Stokes equation from the scaled compressible two-fluid Navier-Stokes-Maxwell equations locally in time under the assumption that the initial data are well prepared. We justify the singular limit by proving the uniform decay of the error system, which is obtained by elaborate energy estimates.
The aim of this paper is to show the existence of $\mathcal{R}$-bounded solution operator families for a generalized resolvent problem on the half-space arising from a compressible fluid model of Korteweg type. Such a compressible fluid model was derived by Dunn and Serrin (1985) and studied by Kotschote (2008) as an initial-boundary value problem.
The aim of this paper is to prove the existence of ${\mathcal R}$-bounded solution operator families for a resolvent problem on the upper half-space arising from a compressible fluid model of Korteweg type with free boundary condition. Such a compressible fluid model was derived by Dunn and Serrin (1985) and studied by Kotschote (2008) as a boundary value problem with non-slip boundary condition.
We identify a class maximal dissipative solutions to models of compressible viscous fluids that maximize the energy dissipation rate. Then we show that any maximal dissipative solution approaches an equilibrium state for large times.
In this paper we study the incompressible inviscid limit for a compressible micro-polar model. We prove that the weak solution of the compressible micro-polar system converges to the solution of the Navier-Stokes equations (Euler equations) in the limit of small Mach number (and vanishing viscosity).
In this paper we study the interaction of a small rigid body in a viscous compressible fluid. The system occupies a bounded three dimensional domain. The object it allowed to freely move and its dynamics follows the Newton's laws. We show that as the size of the object converges to zero the system fluid plus rigid body converges to the compressible Navier-Stokes system under some mild lower bound on the mass and the inertia momentum. It is a first result of homogenization in the case of fluid-structure interaction in the compressible situation. As a corollary we slightly improved the result on…
We consider the model of viscous compressible homogeneous multi-fluids with multiple velocities. We review different formulations of the model and the existence results for boundary value problems. We analyze crucial mathematical difficulties which arise during the proof of the existence theorems.
We consider the model of viscous compressible multi-fluids with multiple velocities. We review different formulations of the model and the existence results for boundary value problems. We analyze crucial mathematical difficulties which arise during the proof of the global existence theorems in 1D case.
In this paper we first employ the energetic variational method to derive a micro-macro model for compressible polymeric fluids. This model is a coupling of isentropic compressible Navier-Stokes equations with a nonlinear Fokker-Planck equation. We then prove the global in time existence of the smooth solution near the global equilibrium.
In this paper, we study the global existence of classical solutions to the three dimensional ideal invicid compressible and heat conductive fluids with radial symmetrical data in $H^s(\mathbb{R}^3)$. Our proof is based on the symmetric hyperbolic structure of the system.
In this work, we study the motion of a rigid body in a bounded domain which is filled with a compressible isentropic fluid. We consider the Navier-slip boundary condition at the interface as well as at the boundary of the domain. This is the first mathematical analysis of a compressible fluid-rigid body system where Navier-slip boundary conditions are considered. We prove existence of a weak solution of the fluid-structure system up to collision.
We consider the Navier-Stokes-Fourier system describing the motion of a compressible viscous fluid in a container with impermeable boundary subject to time periodic heating and under the action of a time periodic potential force. We show the existence of a time periodic weak solution for arbitrarily large physically admissible data.
In this paper, we study the limiting behavior of Riemann solutions to the Euler equations of one-dimensional compressible fluid flow as $γ$ tends to one. We show that the limit solution forms the delta wave to the pressureless Euler system of one-dimensional compressible fluid flow in the distribution sense. Some numerical results exhibiting the phenomena of concentration are also presented.
We use Feireisl-Lions theory to deduce the existence of weak solutions to a system describing the dynamics of a linear oscillator containing a Newtonian compressible fluid. The appropriate Navier-Stokes equation is considered on a domain whose movement has one degree of freedom. The equation is paired with the Newton law and we assume a no-slip boundary condition.
In this paper, we study a nonlinear interaction problem between compressible viscous fluids and plates. For this problem, we introduce relative entropy and relative energy inequality for the finite energy weak solutions (FEWS). First, we prove that for all FEWS, the structure displacement enjoys improved regularity by utilizing the dissipation effects of the fluid onto the structure and that all FEWS satisfy the relative energy inequality. Then, we show that all FEWS enjoy the weak-strong uniqueness property, thus extending the classical result for compressible Navier-Stokes system to this flu…
We study singular limit for scaled barotropic Euler system modelling a rotating, compressible and inviscid fluid, where Mach and Rossby numbers are proportional to a small parameter $ε$. If the fluid is confined to an infinite slab, the limit behaviour is identified as a horizontal motion of an incompressible inviscid system that is analogous to the Euler system. We consider a very general class of solutions, named dissipative solution for the scaled compressible Euler systems and will show that it converges to a strong solution of that incompressible inviscid system.
We consider the barotropic Navier--Stokes system describing the motion of a compressible Newtonian fluid in a bounded domain with in and out flux boundary conditions. We show that if the boundary velocity coincides with that of a rigid motion, all solutions converge to an equilibrium state for large times.
We introduce a new concept of dissipative varifold solution to models of two phase compressible viscous fluids. In contrast with the existing approach based on the Young measure description, the new formulation is variational combining the energy and momentum balance in a single inequality. We show the existence of dissipative varifold solutions for a large class of general viscous fluids with non--linear dependence of the viscous stress on the symmetric velocity gradient.
We develop a mathematical theory for a class of compressible viscoelastic rate-type fluids with stress diffusion. Our approach is based on the concepts used in the nowadays standard theory of compressible Newtonian fluids as renormalization, effective viscous flux identity, compensated compactness. The presence of the extra stress, however, requires substantial modification of these techniques, in particular, a new version of the effective viscous flux identity is derived. With help of these tools, we show the existence of global--in--time weak solutions for any finite energy initial data.
Consider two compressible immiscible fluids in 1D in the isentropic approximation. The first fluid is surrounded and in contact with the second one. As the Mach number of the first fluid vanishes, the coupled dynamics of the two fluids results as the compressible to incompressible limit and is known to satisfy an ODE--PDE system. Below, a characterization of this limit is provided, ensuring its uniqueness.
We propose a new approach to models of general compressible viscous fluids based on the concept of dissipative solutions. These are weak solutions satisfying the underlying equations modulo a defect measure. A dissipative solution coincides with the strong solution as long as the latter exists (weak-strong uniqueness) and they solve the problem in the classical sense as soon as they are smooth (compatibility). We consider general models of compressible viscous fluids with non-linear viscosity tensor and non-homogeneous boundary conditions, for which the problem of existence of global-in-time w…
Following Abbatiello et al. [ DCCDS-A (41), 2020], we introduce dissipative turbulent solutions to a simple model of a mixture of two non interacting compressible fluids {\tc filling a bounded domain with general non zero inflow/outflow boundary conditions.} We prove existence of such solutions for all adiabatic coefficients $γ>1$, their compatibility with classical solutions, the relative energy inequality, and the weak strong uniqueness principle in this class. The class of dissipative turbulent solutions is so far the largest class of generalized solutions which still enjoys the weak strong…
We consider the initial boundary value problem for a model system of one-dimensional equations which describe unsteady polytropic motions of a mixture of viscous compressible fluids. We prove the global existence and uniqueness theorem for the strong solution without restrictions on the structure of the viscosity matrix except standard properties of symmetry and positiveness.
In this paper, we consider the compressible fluid model of Korteweg type which can be used as a phase transition model. It is shown that the system admits a unique, global strong solution for small initial data in $\mathbb R^N$, $N \geq 3$. In this study, the main tools are the maximal $L_p$-$L_q$ regularity and $L_p$-$L_q$ decay properties of solutions to the linearized equations.
This paper considers the Cauchy problem of equations for the viscous compressible and heat-conductive fluids in the two-dimensional(2D) space. We establish the local existence theory of unique strong solution under some initial layer compatibility conditions. The initial data can be arbitrarily large, the initial density is allowed to vanish in any set and the far field state is assumed to be vacuum.
We consider the motion of an inviscid compressible fluid under the mutual interactions with magnetic field. We show that the initial value problem is ill--posed in the class of weak solutions for a large class of physically admissible data. We also consider the same problem for inviscid heat--conductive fluid and show the same result under certain restrictions imposed on the magnetic field. The main tool is the method of convex integration adapted to the Euler system with `variable coefficients'.
In this paper, we consider the three dimensional Cauchy problem of the compressible micropolar viscous flows, we prove the existence of unique global classical solution for smooth initial data with small initial energy but possibly large oscillations, the initial density may allowed to contain vacuum states. Furthermore, the large-time behavior of the solution is obtained.
We prove the existence of a semiflow selection with range the space of càglàd, i.e. left--continuous and having right--hand limits functions defined on $[0,\infty)$ and taking values in a Hilbert space. Afterwards, we apply this abstract result to the system arising from a compressible viscous fluid with a barotropic pressure of the type $a\varrho^γ$, $γ\geq 1$, with a viscous stress tensor being a nonlinear function of the symmetric velocity gradient.
We consider a model describing the steady flow of compressible heat-conducting chemically-reacting multi-component mixture. We show the existence of strong solutions under the additional assumption that the mixture is sufficiently dense. We work in the $L^p$-setting combining the methods for the weak solutions with the method of decomposition. The result is a generalization of our previous papers, where the case of single-constituted fluid was studied.
We show the relative energy inequality for the compressible Navier-Stokes system driven by a stochastic forcing. As a corollary, we prove the weak-strong uniqueness property (pathwise and in law) and convergence of weak solutions in the inviscid-incompressible limit. In particular, we establish a Yamada-Watanabe type result in the context of the compressible Navier-Stokes system, that is, pathwise weak--strong uniqueness implies weak--strong uniqueness in law.
We prove existence of global in time weak solutions to a compressible two-fluid Stokes system with a single velocity field and algebraic closure for the pressure law. The constitutive relation involves densities of both fluids through an implicit function. The system appears to be outside the class of problems that can be treated using the classical Lions-Feireisl approach. Adapting the novel compactness tool developed by the first author and P.-E. Jabin in the mono-fluid compressible Navier-Stokes setting, we first prove the weak sequential stability of solutions. Next, we construct weak solu…
We consider a singular limit problem for the complete compressible Euler system in the low Mach and strong stratification regime. We identify the limit problem - the anelastic Euler system - in the case of well prepared initial data. The result holds in the large class of the dissipative measure-valued solutions of the primitive system. Applications are discussed to the driven shallow water equations.
We investigate a compressible two-fluid Navier-Stokes type system with a single velocity field and algebraic closure for the pressure law. The constitutive relation involves densities of both fluids through an implicit function. We are interested in regular solutions in a $L_p-L_q$ maximal regularity setting. We show that such solutions exists locally in time and, under additional smallness assumptions on the initial data, also globally. Our proof rely on appropriate transformation of the original problem, application of Lagrangian coordinates and maximal regularity estimates for associated li…
In this work we will focus on the existence of weak solutions for a system describing a general compressible viscous fluid in the case of the pressure being a linear function of the density and the viscous stress tensor being a non-linear function of the symmetric velocity gradient. More precisely, we will first prove the existence of dissipative solutions and study under which conditions it is possible to guarantee the existence of weak solutions.
We are concerned with the Cahn-Hilliard/Navier-Stokes equations for the stationary compressible flows in a three-dimensional bounded domain. The governing equations consist of the stationary Navier-Stokes equations describing the compressible fluid flows and the stationary Cahn-Hilliard type diffuse equation for the mass concentration difference. We prove the existence of weak solutions when the adiabatic exponent $γ$ satisfies $γ>\frac{4}{3}$. The proof is based on the weighted total energy estimates and the new techniques developed to overcome the difficulties from the capillary stress.
We introduce a simple model of the time evolution of a binary mixture of compressible fluids including the thermal effects. Despite its apparent simplicity, the model is thermodynamically consistent admitting an entropy balance equation. We introduce a suitable weak formulation of the problem based on a combination of the entropy inequality with the total energy conservation principle. Finally, we show compactness of any bounded family of weak solutions and establish a global existence result.
We consider the free boundary problem of compressible isentropic neo-Hookean viscoelastic fluid equations with surface tension. Under the physical kinetic and dynamic conditions proposed on the free boundary, we investigate regularities of classical solutions to viscoelastic fluid equations in Sobolev spaces which are uniform in viscosity and justify the corresponding vanishing viscosity limits. The key ingredient of our proof is that the deformation gradient tensor in Lagrangian coordinates can be represented as a parameter in terms of flow map so that the inherent structure of the elastic te…
