Quantum Navier Stokes Equations Research Articles

Global weak solutions to a quantum kinetic-fluid model with large initial data

In this paper, we consider a quantum kinetic-fluid model in one-dimensional torus. This model consists of the Vlasov–Fokker–Planck equation coupled with the compressible quantum Navier–Stokes equations via a nonlinear friction force depends on the density of the fluid. The quantum fluid with degenerate viscosity is assumed to be isentropic (adiabatic coefficient γ>1). We establish the global weak solutions to this model with arbitrary large initial data. This is the first result on existence of solutions to kinetic-fluid models with density-dependent friction force and degenerate viscosity.

Global weak solutions for quantum isothermal fluids

We construct global weak solutions to isothermal quantum Navier–Stokes equations, with or without Korteweg term, in the whole space of dimension at most three. Instead of working on the initial set of unknown functions, we consider an equivalent reformulation, based on a time-dependent rescaling, that we introduced in a previous paper to study the large time behavior, and which provides suitable a priori estimates, as opposed to the initial formulation where the potential energy is not signed. We proceed by working on tori whose size eventually becomes infinite. On each fixed torus, we consider the equations in the presence of drag force terms. Such equations are solved by regularization, and the limit where the drag force terms vanish is treated by resuming the notion of renormalized solution developed by I. Lacroix-Violet and A. Vasseur. We also establish global existence of weak solutions for the isothermal Korteweg equation (no viscosity), when initial data are well-prepared, in the sense that they stem from a Madelung transform.

Logarithmic Schrödinger equation and isothermal fluids

We consider the large time behavior in two types of equations, posed on the whole space \mathbb{R}^d : the Schrödinger equation with a logarithmic nonlinearity on the one hand; compressible, isothermal, Euler, Korteweg and quantum Navier–Stokes equations on the other hand. We explain some connections between the two families of equations, and show how these connections may help having an insight in all cases. We insist on some specific aspects only, and refer to the cited articles for more details, and more complete statements. We try to give a general picture of the results, and present some heuristical arguments that can help the intuition, which are not necessarily found in the mentioned articles.

Global existence of finite energy weak solutions to the Quantum Navier-Stokes equations with non-trivial far-field behavior

We prove global existence of finite energy weak solutions to the quantum Navier-Stokes equations in the whole space with non trivial far-field condition in dimensions d=2,3. The vacuum regions are included in the weak formulation of the equations. Our method consists in an invading domains approach. More precisely, by using a suitable truncation argument we construct a sequence of approximate solutions. The energy and the BD entropy bounds allow for the passage to the limit in the truncated formulation leading to a finite energy weak solution. Moreover, the result is also valid in the case of compressible Navier-Stokes equations with degenerate viscosity.

Existence of global weak solution for quantum Navier–Stokes system

In this paper, the barotropic compressible quantum Navier–Stokes equations with a density-dependent viscosity in a three-dimensional torus is studied. By introducing a cold pressure to handle the convection term, we prove the global-in-time existence of weak solutions to quantum Navier–Stokes equations for large data in the sense of standard definition.

Global existence of weak solutions to the compressible quantum Navier-Stokes equations with degenerate viscosity

We study the compressible quantum Navier-Stokes (QNS) equations with degenerate viscosity in the three dimensional periodic domains. On the one hand, we consider QNS with additional damping terms. Motivated by the recent works of Li and Xin [e-print arXiv:1504.06826] and P. Antonelli and S. Spirito [Arch. Ration. Mech. Anal. 225, 1161–1199 (2017)], we construct a suitable approximate system which has smooth solutions satisfying the energy inequality and the BD entropy estimate. Using this system, we obtain the global existence of weak solutions to the compressible QNS equations with damping terms for large initial data. Moreover, we obtain some new a priori estimates, which can avoid using the assumption that the gradient of the velocity is a well-defined function, which was indeed used directly in the work of Vasseur and Yu [SIAM J. Math. Anal. 48, 1489–1511 (2016); Invent. Math. 206, 935–974 (2016)]. On the other hand, in the absence of damping terms, we also prove the global existence of weak solutions to the compressible QNS equations without the lower bound assumption on the dispersive coefficient, which improves the previous result of P. Antonelli and S. Spirito [Arch. Ration. Mech. Anal. 225, 1161–1199 (2017)].

Blow-up of smooth solutions to compressible quantum Navier-Stokes equations

This paper investigates the blow-up phenomena of smooth solutions to the Cauchy problem of multi-dimensional compressible quantum Navier-Stokes equations. Different from the blow-up results of the smooth solutions to the classical Navier-Stokes equations, for any adiabatic exponent bigger than one, we prove the smooth solutions to the compressible quantum Navier-Stokes equations will blow up in finite time under some reasonable restrictions of initial data, when the viscous coefficient is small relative to the Planck constant.

Rigidity results in generalized isothermal fluids

We investigate the long-time behavior of solutions to the isothermal Euler, Korteweg or quantum Navier Stokes equations, as well as generalizations of these equations where the convex pressure law is asymptotically linear near vacuum. By writing the system with a suitable time-dependent scaling we prove that the densities of global solutions display universal dispersion rate and asymptotic profile. This result applies to weak solutions defined in an appropriate way. In the exactly isothermal case, we establish the compactness of bounded sets of such weak solutions, by introducing modified entropies adapted to the new unknown functions.

Exponential Decay in Time of Density of One-dimensional Quantum Navier-Stokes Equations

The long-time behavior of the particle density of the compressible quantum Navier-Stokes equations in one space dimension is studied. It is shown that the particle density converges exponentially fast to the constant thermal equilibrium state as the time tends to infinity, the decay rate is also obtained. The results hold regardless of either the bigger of the scaled Planck constant or the viscosity constant. This improves the decay results of [5] by removing the crucial assumption that the scaled Planck constant is bigger than the viscosity constant. The proof is based on the entropy dissipation method and the Bresch-Desjardins type of entropy.

A remark on the global existence of weak solutions to the compressible quantum Navier–Stokes equations

In this paper, we obtain the global existence of weak solutions to the compressible quantum Navier–Stokes equations. By virtue of a useful identity and an interesting estimate, we solve the critical case that the viscosity equals the dispersive coefficient. This result removes the restrictions on the coefficients and improves the recent work of Antonell and Spirito (2017) in some senses.

On the compactness of finite energy weak solutions to the quantum Navier–Stokes equations

We consider the quantum Navier–Stokes (QNS) system in three space dimensions. We prove compactness of finite energy weak solutions for large initial data. The main novelties are that vacuum regions are included in the weak formulation and no extra terms, like damping or cold pressure, are considered in the equations in order to define the velocity field. Our argument uses an equivalent formulation of the system in terms of an effective velocity, in order to eliminate the third-order terms in the new system. This will allow to obtain the same compactness properties as for the Navier–Stokes equations with degenerate viscosity.

Global Existence of Finite Energy Weak Solutions of Quantum Navier–Stokes Equations

In this paper we consider the Quantum Navier-Stokes system both in two and in three space dimensions and prove global existence of finite energy weak solutions for large initial data. In particular, the notion of weak solutions is the standard one. This means that the vacuum region are included in the weak formulations. In particular, no extra term like damping or cold pressure are added to the system in order to define the velocity field in the vacuum region. The main contribution of this paper is the construction of a regular approximating system consistent with the effective velocity transformation needed to get necessary a priori estimates.

Remarks on classical solutions to steady quantum Navier-Stokes equations

The paper of Dong [Dong, J. Classical solutions to one-dimensional stationary quantum Navier–Stokes equations, J. Math Pure Appl. 2011] which proved the existence of classical solutions to one-dimensional steady quantum Navier–Stokes equations, when the nonzero boundary value u0 satisfies some conditions. In this paper, we obtain a different version of existence theorem without restriction to u0. As a byproduct, we get the existence result of classical solutions to the stationary quantum Navier–Stokes equations.

About the barotropic compressible quantum Navier–Stokes equations

In this paper we consider the barotropic compressible quantum Navier–Stokes equations with a linear density dependent viscosity and its limit when the scaled Planck constant vanishes. Following recent works on degenerate compressible Navier–Stokes equations, we prove the global existence of weak solutions by the use of a singular pressure close to vacuum. With such singular pressure, we can use the standard definition of global weak solutions which also allows to justify the limit when the scaled Planck constant denoted by ε tends to 0.

Asymptotic limits of Navier–Stokes equations with quantum effects

This paper is concerned with the combined incompressible limit and semiclassical limit of the weak solutions to the barotropic quantum Navier–Stokes equations of compressible flows. By using the relative entropy method, we show that for well-prepared initial data, the weak solutions of the compressible quantum Navier–Stokes model converge to the strong solution of the incompressible Navier–Stokes equations as long as the latter exists. Furthermore, the convergence rates are also obtained.

On the blowing up of solutions to one-dimensional quantum Navier-Stokes equations

The blow-up in finite time for the solutions to the initial-boundary value problem associated to the one-dimensional quantum Navier-Stokes equations in a bounded domain is proved. The model consists of the mass conservation equation and a momentum balance equation, including a nonlinear third-order differential operator, with the quantum Bohm potential, and a density-dependent viscosity. It is shown that, under suitable boundary conditions and assumptions on the initial data, the solution blows up after a finite time, if the viscosity constant is not bigger than the scaled Planck constant. The proof is inspired by an observable constructed by Gamba, Gualdani and Zhang, which has been used to study the blowing up of solutions to quantum hydrodynamic models.

A New Derivation of the Quantum Navier–Stokes Equations in the Wigner–Fokker–Planck Approach

A quantum Navier–Stokes system for the particle, momentum, and energy densities is formally derived from the Wigner–Fokker–Planck equation using a moment method. The viscosity term depends on the particle density with a shear viscosity coefficient which equals the quantum diffusion coefficient of the Fokker–Planck collision operator. The main idea of the derivation is the use of a so-called osmotic momentum operator, which is the sum of the phase-space momentum and the gradient operator. In this way, a Chapman–Enskog expansion of the Wigner function, which typically leads to viscous approximations, is avoided. Moreover, we show that the osmotic momentum emerges from local gauge theory.

Classical solutions to one-dimensional stationary quantum Navier–Stokes equations

The existence of classical solutions to the stationary quantum Navier–Stokes equations in one space dimension is studied. The main idea of the proof is to reformulate the quantum Navier–Stokes equations as the viscous quantum Euler system. The existence of classical solutions to the stationary viscous quantum Euler system is shown by using an exponential variable transformation and the Leray–Schauder fixed-point theorem. As a consequence, the existence of classical solutions to the stationary quantum Navier–Stokes equations can be deduced.

Effective velocity in compressible Navier–Stokes equations with third-order derivatives

A formulation of certain barotropic compressible Navier–Stokes equations with third-order derivatives as a viscous Euler system is proposed by using an effective velocity variable. The equations model, for instance, viscous Korteweg or quantum Navier–Stokes flows. The formulation in the new variable allows for the derivation of an entropy identity, which is known as the BD (Bresch–Desjardins) entropy equation. As a consequence of this estimate, a new global-in-time existence result for the one-dimensional quantum Navier–Stokes equations with strictly positive particle densities is proved.

A remark on weak solutions to the barotropic compressible quantum Navier–Stokes equations

In [A. Jüngel, Global weak solutions to compressible Navier–Stokes equations for quantum fluids, SIAM J. Math. Anal. 42 (2010) 1025–1045], Jüngel proved the global existence of the barotropic compressible quantum Navier–Stokes equations for when the viscosity constant is bigger than the scaled Planck constant. Recently, Dong [J. Dong, A note on barotropic compressible quantum Navier–Stokes equations, Nonlinear Anal. TMA 73 (2010) 854–856] extended Jüngel’s result to the case where the viscosity constant is equal to the scaled Planck constant by using a new estimate of the square root of the solutions. In this paper we show that Jüngel’s existence result still holds when the viscosity constant is bigger than the scaled Planck constant. Consequently, with our result, the existence for all physically interesting cases of the scaled Planck and viscosity constants is obtained.