We propose the new notion of Visco-Energetic solutions to rate-independent systems $(X,\mathcal E,\mathsf d)$ driven by a time dependent energy $\mathcal E$ and a dissipation quasi-distance $\mathsf d$ in a general metric-topological space $X$. As for the classic Energetic approach, solutions can be obtained by solving a modified time Incremental Minimization Scheme, where at each step the dissipation (quasi-)distance $\mathsf d$ is incremented by a viscous correction $δ$ (e.g.~proportional to the square of the distance $\mathsf d$), which penalizes far distance jumps by inducing a localized version of the stability condition.
We prove a general convergence result and a typical characterization by Stability and Energy Balance in a setting comparable to the standard energetic one, thus capable to cover a wide range of applications. The new refined Energy Balance condition compensates the localized stability and provides a careful description of the jump behavior: at every jump the solution follows an optimal transition, which resembles in a suitable variational sense the discrete scheme that has been implemented for the whole construction.
A boundary control problem for the viscous Cahn-Hilliard equations with possibly singular potentials and dynamic boundary conditions is studied and first order necessary conditions for optimality are proved. Key words: Cahn-Hilliard equation, dynamic boundary conditions, phase separation, singular potentials, optimal control, optimality conditions, adjoint state system
This paper is concerned with the asymptotic stability of a composite wave consisting of two viscous shock waves to the Cauchy problem for a one-dimensional system of heat-conductive ideal gas without viscosity. We extend the results by Huang-Matsumura \cite{Huang-Matsumura} where they treated the equation of viscous and heat-conductive ideal gas. That is, even forthe non-viscous and heat-conductive case, we show that if the strengths of the viscous shock waves and the initial perturbation are suitably small, the unique global solution in time exists and asymptotically tends toward the correspo…
In the present paper, it is shown that the large amplitude viscous shock wave is nonlinearly stable for isentropic Navier-Stokes equations, in which the pressure could be general and includes $γ$-law, and the viscosity coefficient is a smooth function of density. The strength of shock wave could be arbitrarily large. The proof is given by introducing a new variable, which can formulate the original system into a new one, and the elementary energy method introduced in Matsumura-Nishihara [On the stability of travelling wave solutions of a one-dimensional model system for compressible viscous ga…
This paper is concerned with the Cauchy problem of heat-conductive ideal gas without viscosity. We show that, for the non-viscous case, if the strengths of the wave patterns and the initial perturbation are suitably small, the unique global-in-time solution exists and asymptotically tends toward the corresponding the viscous contact wave or the composition of a viscous contact wave with rarefaction waves determined by the initial condition, which extended the results by Huang-Li-Matsumura[13], where they treated the viscous and heat-conductive ideal gas.
This paper is concerned with the inflow problem for the one-dimensional compressible Navier-Stokes equations. For such a problem, F. M. Huang, A. Matsumura and X. D. Shi showed that there exists viscous shock wave solution to the inflow problem and both the boundary layer solution, the viscous shock wave, and their superposition are time-asymptotically nonlinear stable under small initial perturbation. The main purpose of this paper is to show that similar stability results still hold for a class of large initial perturbation which can allow the initial density to have large oscillations. The …
A classical model of fluid dynamics is considered which describes the shape evolution of a viscous liquid droplet on a homogeneous substrate. All equilibria are characterized and their stability is analyzed by a geometric reduction argument.
In this work we will study the dynamics of a thin layer of a viscous fluid which is embedded in the interior of another viscous fluid. The resulting flow can be approximated by means of the solutions of a free boundary problem for the Stokes equation in which one of the unknowns is the shape of a curve which approximates the geometry of the thin layer of fluid. We also derive the equation yielding the thickness of this fluid. This model, that will be termed as the Geometric Free Boundary Problem, will be derived using matched asymptotic expansions. We will prove that the Geometric Free Boundar…
Cagnetti, Gomes, Mitake and Tran (2013) introduced a new idea to study the large time behavior for degenerate viscous Hamilton--Jacobi equations. In this paper, we apply the method to study the large-time behavior of the solution to the obstacle problem for degenerate viscous Hamilton--Jacobi equations. We establish the convergence result under rather general assumptions.
In this paper we consider the motion of a rigid body in a viscous incompressible fluid when some Navier slip conditions are prescribed on the body's boundary. The whole `viscous incompressible fluid + rigid body' system is assumed to occupy the full plane $\R^{2}$. We prove the existence of global-in-time weak solutions with constant non-zero circulation at infinity.
The large time behavior of solutions to Cauchy problem for viscous Hamilton-Jacobi equation is classified. The large time asymptotics are given by very singular self-similar solutions on one hand and by self-similar viscosity solutions on the other hand
A free boundary problem for the one-dimensional compressible Navier-Stokes equations is investigated. The asymptotic stability of the viscous shock wave is established under some smallness conditions. The proof is given by an elementary energy estimate.
Extending investigations of Métivier&Zumbrun in the hyperbolic case, we treat stability of viscous shock and boundary layers for viscous perturbations of multidimensional hyperbolic systems with characteristics of variable multiplicity, specifically the construction of symmetrizers in the low-frequency regime where variable multiplicity plays a role. At the same time, we extend the boundary-layer theory to ``real'' or partially parabolic viscosities, Neumann or mixed-type parabolic boundary conditions, and systems with nonconservative form, in addition proving a more fundamental version of the…
We prove the existence and the uniqueness of strong solutions for the viscous Hamilton-Jacobi Equation with Neumann boundary condition and initial data a continious function. Then, we study the large time behavior of the solutions.
We study by a combination of analytical and numerical Evans function techniques multi-D viscous and inviscid stability and associated transverse bifurcation of planar slow Lax MHD shocks in a channel with periodic boundary conditions. Notably, this includes the first multi-D numerical Evans function study for viscous MHD. Our results suggest that, rather than a planar shock, a nonplanar traveling wave with the same normal velocity is the typical mode of propagation in the slow Lax mode. Moreover, viscous and inviscid stability transitions appear to agree, answering (for this particular model a…
We consider the equations which describe polytropic one-dimensional flows of viscous compressible multifluids. We prove global existence and uniqueness of a solution to the initial-boundary value problem which corresponds to the flow in a bounded space domain.
This paper is concerned with the time-asymptotic behavior of strong solutions to an initial-boundary value problem of the compressible Navier-Stokes-Korteweg system on the half line $\mathbb{R}^+$. The asymptotic profile of the problem is shown to be a shifted viscous shock profile, which is suitably away from the boundary. Moreover, we prove that if the initial data around the shifted viscous shock profile and the strength of the shifted viscous shock profile are sufficiently small, then the problem has a unique global strong solution, which tends to the shifted viscous shock profile as time …
We establish global-posedness in time for the viscous Boussinesq equations in two dimensions of space with temperature-dependent diffusivity in the framework of a smooth vortex patch. We also provide the inviscid limit for velocity, temperature, and associated flow toward the system studied very recently in \cite{Paicu-Zhu} as soon as the viscosity goes to zero, and quantify the rate of convergence.
In this paper, we investigate the asymptotic behavior of solutions to the Cauchy problem for the scalar non-viscous diffusive dispersive conservation laws where the far field states are prescribed. We proved that the solution of the Cauchy problem tends toward the constant state as time goes to infinity.
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.
We consider the Cauchy problem for a class of nonlinear degenerate parabolic equa- tion with forcing. By using the vanishing viscosity method we obtain generalized solutions. We prove some regularity results about this generalized solutions.
We consider the barotropic Navier-Stokes system describing the motion of a compressible viscous fluid confined to a bounded domain driven by time periodic inflow/outflow boundary conditions. We show that the problem admits a time periodic solution in the class of weak solutions satisfying the energy inequality.
We consider a class of viscous fluids with a general monotone dependence of the viscous stress on the symmetric velocity gradient. We introduce the concept of dissipative solution to the associated initial boundary value problem inspired by the measure-valued solutions for the inviscid (Euler) system. We show the existence as well as the weak-strong uniqueness property in the class of dissipative solutions. Finally, the dissipative solution enjoying certain extra regularity coincides with a strong solution of the same problem.
