1,720,990 research outputs found
Global existence of weak solutions to the Navier-Stokes-Korteweg equations
Full text linkIn this paper we consider the Navier-Stokes-Korteweg equations for a viscous compressible fluid with capillarity effects in three space dimensions. We prove global existence of finite energy weak solutions for large initial data. Contrary to previous results regarding this system, vacuum regions are considered in the definition of weak solutions and no additional damping terms are considered. The convergence of the approximating solutions is obtained by introducing suitable truncations of the velocity field and the mass density at different scales in the momentum equations and use only the a priori bounds obtained by the energy and the Bresch-Desjardins entropy. Moreover, the approximating solutions enjoy only a limited amount of regularity, and the derivation of the truncations of the velocity and the density is performed by a suitable regularization procedure
On the compactness of weak solutions to the Navier–Stokes–Korteweg equations for capillary fluids
No full textIn this paper we consider the Navier–Stokes–Korteweg equations for a viscous compressible fluid with capillarity effects in three space dimensions. We prove compactness of finite energy weak solutions for large initial data. Incontrast with previous results regarding this system, vacuum regions are allowed in the definition of weak solutions and no additional damping terms are considered. The compactness is obtained by introducing suitable truncations of the velocity field and the mass density at different scales and use only the a priori bounds obtained by the energy and the BD entropy
On the Boussinesq system: regularity criteria and singular limits
No full textWe consider the 3D Boussinesq system and we prove several criteria, not involving
the density, for the continuation of smooth solutions. We give particular emphasis to the results
in bounded domains, under various boundary conditions. The results we prove are partially known
and we collect them in a unified framework, mainly with the perspective of understanding the stabi-
lization/smoothing required by numerical methods (especially by large scales methods). In the final
section, we also consider the vanishing viscosity/diffusivity limits, proving (locally-in-time) sharp
singular limits for smooth solutions of the Cauchy problem
A remark on the Euler equations in dimension two
No full textWe review some results concerning the global existence of weak solutions to the Euler equations in a two dimensional open bounded set. These results
are obtained by means of a suitable vanishing viscosity approximation, through the
Navier-Stokes equations equipped with Navier-type boundary conditions. Next, we
prove a theorem of existence for weak solutions with a given non-zero normal velocity, slightly relaxing with respect to the time variable the known assumptions on
the data of the problem
On the compactness of finite energy weak solutions to the quantum Navier–Stokes equations
No full text 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. </jats:p
Splash Singularities for a General Oldroyd Model with Finite Weissenberg Number
No full textIn this paper we study a 2D free-boundary Oldroyd-B model which describes the evolution of a viscoelastic fluid. We prove the existence of splash singularities, namely points where the free-boundary remains smooth but self-intersects. This paper extends the previous results obtained for the infinite Weissenberg number by the authors in Di Iorio et al. (Splash singularity for a free boundary incompressible viscoelastic fluid model, 2018. arXiv:1806.11089; Splash singularity for a 2D Oldroyd-B model with nonlinear Piola-Kirchhoff stress, Nonlinear Differ Equ Appl 24:60, 2017) to the more realistic physical case of any finite Weissenberg number. The main difficulty faced in this paper is due to the non-linear balance law of the elastic tensor, which cannot be reduced, as in the case of infinite Weissenberg, to a transport equation for the deformation gradient. Overcoming this difficulty requires a very accurate local existence theorem in terms of dependence on the Weissenberg number. The method in this case is based on the combined use of conformal transformations and lagrangian coordinates, whose formulation must however take into account the general balance law of the elastic tensor and its dependence on the Weissenberg number. The existence of the splash singularities is therefore guaranteed by an adequate choice of initial data, depending also on the elastic tensor, combined with stability estimates
On the Vanishing viscosity limit of 3D Navier-Stokes equations under slip boundary conditions in general domains
No full textWe consider the vanishing-viscosity limit for the Navier-Stokes equations with certain slip-without-friction boundary conditions in a bounded domain with non-flat boundary. In particular, we are able to show convergence in strong norms for a solution starting with initial data belonging to the special subclass of data with vanishing vorticity on the boundary. The proof is obtained by smoothing the initial data and by a perturbation argument with quite precise estimates for the equations of the vorticity and for that of the curl of the vorticity
Global existence of finite energy weak solutions to the Quantum Navier-Stokes equations with non-trivial far-field behavior
No full textWe 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
ON INVISCID LIMITS FOR THE NAVIER-STOKES EQUATIONS WITH SLIP BOUNDARY CONDITIONS INVOLVING THE VORTICITY
No full textIn this note we consider the inviscid limit for the Navier-Stokes equations under different slip boundary conditions of Navier's type and we show how this influences the convergence rate in the energy norm. The role of the initial data is also emphasized in connection with the vanishing viscosity limit
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