We prove the existence of solution in a class H^2(Ω) x H^1(Ω) to steady compressible Oseen system with slip boundary conditions in a two dimensional, convex domain with the boundary of class H^{5/2}. The method is to regularize a weak solution obtained via the Galerkin method. The problem of regularization is reduced to a problem of solvability of a certain transport equation by application of the Helmholtz decomposition. The method works under additional assumption on the geometry of the boundary.
In this study, we consider a linearized compressible flow structure interaction PDE model for which the interaction interface is under the effect of material derivative term. While the linearization takes place around a constant pressure and density components in structure equation, the flow linearization is taken with respect to a non-zero, fixed, variable ambient vector field. This process produces extra "convective derivative" and "material derivative" terms which causes the coupled system to be nondissipative. We analyze the long time dynamics in the sense of asymptotic (strong) stability …
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 compressible Navier-Stokes system on time-dependent domains with prescribed motion of the boundary, supplemented with slip boundary conditions for the velocity. We derive the relative entropy inequality in the spirit of Feireisl et al. for the system on moving domain and use it to prove the weak-strong uniqueness property.
An initial boundary value problem for compressible Magnetohydrodynamics (MHD) is considered on an exterior domain (with the first Betti number vanishes) in $R^3$ in this paper. The global existence of smooth solutions near a given constant state for compressible MHD with the boundary conditions of Navier-slip for the velocity filed and perfect conduction for the magnetic field is established. Moreover the explicit decay rate is given. In particular, the results obtained in this paper also imply the global existence of classical solutions for the full compressible Navier-Stokes equations with N…
We consider a model of chemically reacting heat conducting compressible mixture. We investigate the corresponding system of partial differential equations in the steady regime with slip boundary conditions for the velocity and, in dependence on the model parameters, we establish existence of either weak or variational entropy solutions. The results extend the range of parameters for which the existence of weak solutions is known in the case of homogeneous Dirichlet boundary conditions for the velocity.
The inviscid limit for the two-dimensional compressible viscoelastic equations on the half plane is considered under the no-slip boundary condition. When the initial deformation tensor is a perturbation of the identity matrix and the initial density is near a positive constant, we establish the uniform estimates of solutions to the compressible viscoelastic flows in the conormal Sobolev spaces. It is well-known that for the corresponding inviscid limit of the compressible Navier-Stokes equations with the no-slip boundary condition, one does not expect the uniform energy estimates of solutions …
We consider the incompressible Navier-Stokes equations in a two-dimensional exterior domain, with no-slip boundary conditions. We assume that the initial velocity is a finite-energy and L^q-summable perturbation of the Oseen vortex with circulation α, where 1 < q < 2. If α is sufficiently small, we show that the solution behaves asymptotically in time like the self-similar Oseen vortex with circulation α. This is a global stability result, which holds for arbitrarily large perturbations of the Oseen vortex, and our smallness assumption on the circulation is independent of the domain under cons…
The author treats the system of motion for an incompressible non-Newtonian fluids of the stress tensor described by $p-$potential function subject to slip boundary conditions in $\mathbb{R}^3_+$. Making use of the Oseen-type approximation to this model and the $L^\infty$-truncation method, one can establish the existence theorem of weak solutions for $p-$potential flow with $p\in(\frac{8}{5},2]$ provided that large initial data.
We consider the initial-boundary-value problem of the isentropic compressible Navier-Stokes-Poisson equations subject to large and non-flat doping profile in 3D bounded domain with slip boundary condition and vacuum. The global well-posedness of classical solution is established with small initial energy but possibly large oscillations and vacuum. The steady state (except velocity) and the doping profile are allowed to be of large variation.
We study the barotropic compressible Navier-Stokes equations with Navier-type boundary condition in a two-dimensional simply connected bounded domain with $C^{\infty}$ boundary $\partialΩ.$ By some new estimates on the boundary related to the Navier-type slip boundary condition, the classical solution to the initial-boundary-value problem of this system exists globally in time provided the initial energy is suitably small even if the density has large oscillations and contains vacuum states. Futhermore, we also prove that the oscillation of the density will grow unboundedly in the long run wit…
In this paper we study the initial-boundary-value problem for the barotropic compressible magnetohydrodynamic system with slip boundary conditions in three-dimensional exterior domain. We establish the global existence and uniqueness of classical solutions to the exterior domain problem with the regular initial data that are of small energy but possibly large oscillations with constant state as far field which could be either vacuum or nonvacuum. In particular, the initial density of such a classical solution is allowed to have large oscillations contain vacuum states. Moreover, the large-time…
In this paper, we investigate the uniform regularity for the isentropic compressible Navier-Stokes system with general Navier-slip boundary conditions (1.6) and the inviscid limit to the compressible Euler system. It is shown that there exists a unique strong solution of the compressible Navier-Stokes equations with general Navier-slip boundary conditions in an interval of time which is uniform in the vanishing viscosity limit. The solution is uniformly bounded in a conormal Sobolev space and is uniform bounded in $W^{1,\infty}$. It is also shown that the boundary layer for the density is weak…
Maximal regularity for a compressible fluid-structure interaction system with Navier-slip boundary conditions
Institute of Mathematics, Czech Academy of Sciences（捷克科学院数学研究所）；Department of Mathematics, CY Cergy Paris University, CNRS UMR 8088（CY塞吉-巴黎大学数学系，法国国家科学研究中心联合研究单位8088）
We are concerned with the global existence of classical solutions to the barotropic compressible Navier-Stokes equations with slip boundary condition in a three-dimensional (3D) exterior domain. We demonstrate that the classical solutions exists globally in time provided that the initial total energy is suitably small. It is worth noting that the initial density is allowed to have large oscillations and contain vacuum states. For our purpose, some new techniques and methods are adopted to obtain necessary a priori estimates, especially the estimates on the boundary. Moveover, we also give the …
We deal with the barotropic compressible magnetohydrodynamic equations in three-dimensional (3D) bounded domain with slip boundary condition and vacuum. By a series of a priori estimates, especially the boundary estimates, we prove the global well-posedness of classical solution and the exponential decay rate to the initial-boundary-value problem of this system for the regular initial data with small energy but possibly large oscillations. The initial density of such a classical solution is allowed to contain vacuum states. Moreover, it is also shown that the oscillation of the density will gr…
Global classical solution for three-dimensional compressible isentropic magneto-micropolar fluid equations with Coulomb force and slip boundary condition in bounded domains
We deal with the barotropic compressible Navier-Stokes equations subject to large external potential forces with slip boundary condition in a 3D simply connected bounded domain, whose smooth boundary has a finite number of 2D connected components. The global existence of strong or classical solutions to the initial boundary value problem of this system is established provided the initial energy is suitably small. Moreover, the density has large oscillations and contains vacuum states. Finally, we show that the global strong or classical solutions decay exponentially in time to the equilibrium …
We consider the compressible Navier-Stokes system on time-dependent domains with prescribed motion of the boundary. For both the no-slip boundary conditions as well as slip boundary conditions we prove local-in-time existence of strong solutions. These results are obtained using a transformation of the problem to a fixed domain and an existence theorem for Navier-Stokes like systems with lower order terms and perturbed boundary conditions. We also show the weak-strong uniqueness principle for slip boundary conditions which remained so far open question.
This paper concerns the global existence for classical solutions problem to the 3D Density-Dependent Viscosity barotropic compressible Navier-Stokes in $Ω$ with slip boundary condition, where $Ω$ is a simply connected bounded $C^{\infty}$ domain in $\mathbb{R}^3$ and its boundary only has a finite number of 2-dimensional connected components. By a series of a priori estimates, we show that the classical solution to the system exists globally in time under the assumption that the initial energy is suitably small. The initial density of such a classical solution is allowed to have large oscillat…
Existence and Exponential Growth of Global Classical Solutions to the Compressible Navier-Stokes Equations with Slip Boundary Conditions in 3D Bounded Domains
Global stability for compressible isentropic Navier-Stokes equations in 3D bounded domains with Navier-slip boundary conditions
School of Mathematics and Statistics, Xiamen University of Technology（厦门理工学院数学与统计学院）；School of Mathematics and Statistics, Southwest University（西南大学数学与统计学院）
We prove the existence and uniqueness of strong solutions to the steady isentropic compressible Navier-Stokes equations with inflow boundary conditions for density and mixed boundary conditions for the velocity around a shear flow. In particular, the Dirichlet boundary condition on inflow and outflow part of the boundary while the full Navier boundary conditions on the wall $Γ_{0}$ for the velocity filed are considered. For our result, there are no restrictions on the amplitude of friction coefficients $α$, and the appropriately large hypothesis for the viscous coefficients $μ$ is enough. One …
Nishida-Smoller type large solutions and exponential growth for the compressible Navier-Stokes equations with slip boundary conditions in 3D bounded domain
This paper concerns the low Mach number limit of weak solutions to the compressible Navier-Stokes equations for isentropic fluids in a bounded domain with a Navier-slip boundary condition. In \cite{DGLM99}, it has been proved that if the velocity is imposed the homogeneous Dirichlet boundary condition, as the Mach number goes to 0, the velocity of the compressible flow converges strongly in $L^2$ under the geometrical assumption (H) on the domain. We justify the same strong convergence when the slip length in the Navier condition is the reciprocal of the square root of the Mach number.
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.
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 consider the compressible (barotropic) Navier-Stokes system on time-dependent domains, supplemented with slip boundary conditions. Our approach is based on penalization of the boundary behaviour, viscosity, and the pressure in the weak formulation. Global-in-time weak solutions are obtained.
This work is devoted to study the global behavior of viscous flows contained in a symmetric domain with complete slip boundary. In such scenario the boundary no longer provides friction and therefore the perturbation of angular velocity lacks decaying structure. In fact, we show the existence of uniformly rotating solutions as steady states for the compressible Navier-Stokes equations. By manipulating the conservation law of angular momentum, we establish a suitable Korn's type inequality to control the perturbation and show the asymptotic stability of the uniformly rotating solutions with sma…
The Navier-Stokes equations for compressible barotropic flow in the stationary three dimensional case are considered. It is assumed that a fluid occupies a bounded domain and satisfies the no-slip boundary condition. The existence of a weak solution under the assumption that the adiabatic exponent satisfies $γ>1$ is proved. These results cover the cases of monoatomic, diatomic, and polyatomic gases.
We consider a system of partial differential equations describing the steady flow of a compressible heat conducting Newtonian fluid in a three-dimensional channel with inflow and outflow part. We show the existence of a strong solution provided the data are close to a constant, but nontrivial flow with sufficiently large dissipation in the energy equation.
We consider inviscid flow with isentropic coefficient greater than one. For flow along smooth infinite protruding corners we attempt to impose a nonzero limit for velocity at infinity at the upstream wall. We prove that the problem does not have any irrotational uniformly subsonic solutions, whereas rotational flows do exist. This can be considered a case of a slip-condition solid "generating" vorticity in inviscid flow.
The three-dimensional (3D) full compressible magnetohydrodynamic system is studied in a general bounded domain with slip boundary condition for the velocity filed, adiabatic condition for the temperature and perfect conduction for the magnetic field. For the regular initial data with small energy but possibly large oscillations, the global existence of classical and weak solution as well as the exponential decay rate to the initial-boundary-value problem of this system is obtained. In particular, the density and temperature of such a classical solution are both allowed to vanish initially. Mor…
This paper studies the incompressible limit of global strong solutions to the three-dimensional compressible Navier-Stokes equations associated with Navier's slip boundary condition, provided that the time derivatives, up to first order, of solutions are bounded initially. The main idea is to derive a differential inequality with decay, so that the estimates are bounded uniformly both in the Mach number 0<ε<=ε0 for some ε0>0 and the time t>=0.
In this paper, we consider the initial-boundary value problem of three-dimensional isentropic compressible Navier-Stokes equations with rotating effect terms in an exterior domain with Navier-slip boundary condition and with far-field vacuum. This problem is related to the motion of the compressible viscous flow past a rotating obstacle. We establish the global existence and uniqueness of classical solutions, provided that the initial mass is small. The initial data and the angular velocity of the obstacle are allowed to have large oscillations.
We study the barotropic compressible Navier-Stokes system where the shear viscosity is a positive constant and the bulk one proportional to a power of the density with the power bigger than one and a third. The system is subject to the Navier-slip boundary conditions in a general two-dimensional bounded simply connected domain. For initial density allowed to vanish, we establish the global existence of strong and weak solutions without any restrictions on the size of initial value. To get over the difficulties brought by boundary, on the one hand, we apply Riemann mapping theorem and the pull …
In this paper, we consider the local well-posedness of the Prandtl boundary layer equations that describe the behavior of boundary layer in the small viscosity limit of the compressible isentropic Navier-Stokes equations with non-slip boundary condition. Under the strictly monotonic assumption on the tangential velocity in the normal variable, we apply the Nash-Moser-Hörmander iteration scheme and further develop the energy method introduced in [1] to obtain the well-posedness of the equations locally in time.
We aim at proving existence of weak solutions to the stationary compressible Navier-Stokes system coupled with the Allen-Cahn equation. The model is studied in a bounded three dimensional domain with slip boundary conditions for the momentum equations and the Neumann condition for the Allen-Cahn model. The main result establishes existence of weak solutions with bounded densities. The construction is possible assuming sufficiently large value of the heat capacity ratio $γ$ ($p \sim ρ^γ$). As a corollary we obtain weak solutions for a less restrictive case losing point-wise boundedness of the d…
The paper examines the issue of stability of Poiseuille type flows in regime of compressible Navier-Stokes equations in a three dimensional finite pipe-like domain. We prove the existence of stationary solutions with inhomogeneous Navier slip boundary conditions admitting nontrivial inflow condition in the vicinity of constructed generic flows. Our techniques are based on an application of a modification of the Lagrangian coordinates. Thanks to such approach we are able to overcome difficulties coming from hyperbolicity of the continuity equation, constructing a maximal regularity estimate for…
We are concerned with the uniform regularity estimates of solutions to the two dimensional compressible non-resistive magnetohydrodynamics (MHD) equations with the no-slip boundary condition on velocity in the half plane. Under the assumption that the initial magnetic field is transverse to the boundary, the uniform conormal energy estimates are established for the solutions to compressible MHD equations with respect to small viscosity coefficients. As a direct consequence, we proved the inviscid limit of solutions from viscous MHD systems to the ideal MHD systems in $L^\infty$ sense. It shows…
Validity of Prandtl Expansion for Steady Compressible Navier-Stokes-Fourier Flows
The Division of Applied Mathematics, Brown University（布朗大学应用数学系）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
