1,720,994 research outputs found
1- D Relaxation from hyperbolic to parabolic systems with variable coefficients
No full textIn this paper we study the relaxation of semilinear hyperbolic systems to parabolic sys- tem. The singular limits are studied using G ́erard’s generalized compensated compactness
Dispersive shocks in quantum hydrodynamics with viscosity
No full textIn this paper we study existence and stability of shock profiles for a 1-D compressible Euler system in the context of quantum hydrodynamic models. The dispersive term is originated by the quantum effects described through the Bohm potential; moreover we introduce a (linear) viscosity to analyze its interplay with the former while proving existence, monotonicity and stability of traveling waves connecting a Lax shock for the underlying Euler system. The existence of monotone profiles is proved for sufficiently small shocks; while the case of large shocks leads to the (global) existence for an oscillatory profile, where dispersion plays a significant role. The spectral analysis of the linearized problem about a profile is also provided. In particular, we derive a sufficient condition for the stability of the essential spectrum and we estimate the maximum modulus of the eigenvalues in the unstable plane, using a careful analysis of the Evans function
Dissipative martingale solutions of the stochastically forced Navier–Stokes–Poisson system on domains without boundary
No full textWe construct solutions to the randomly-forced Navier–Stokes–Poisson system in periodic three-dimensional domains or in the whole three-dimensional Euclidean space. These solutions are weak in the sense of PDEs and also weak in the sense of probability. As such, they satisfy the system in the sense of distributions and the underlying probability space and the stochastic driving force are also unknowns of the problem. Additionally, these solutions dissipate energy, satisfies a relative energy inequality in the sense of Dafermos (1979) and satisfy a renormalized form of the continuity equation in the sense of DiPerna and Lions (1989)
A comparison of two mathematical models of the cerebrospinal fluid dynamics
No full textIn this paper we provide the numerical simulations of two cerebrospinal fluid dynamics models by comparing our results with the real data available in literature (see Section 4). The models describe different processes in the cerebrospinal fluid dynamics: the cerebrospinal flow in the ventricles of the brain and the reabsorption of the fluid. In the appendix we show in detail the mathematical analysis of both models and we identify the set of initial conditions for which the solutions of the systems of equations do not exhibit blow up. We investigate step by step the accuracy of these theoretical outcomes with respect to the real cerebrospinal physiology and dynamics. The plan of the paper is provided in Section 1.5
A model of synchronization over quantum networks
No full textWe investigate a non-Abelian generalization of the Kuramoto model proposed by Lohe and given by N quantum oscillators ('nodes') connected by a quantum network where the wavefunction at each node is distributed over quantum channels to all other connected nodes. It leads to a system of Schrö dinger equations coupled by nonlinear self-interacting potentials given by their correlations. We give a complete picture of synchronization results, given on the relative size of the natural frequency and the coupling constant, for two non-identical oscillators and show complete phase synchronization for arbitrary N > 2 identical oscillators. Our results are mainly based on the analysis of the ODE system satisfied by the correlations and on the introduction of a quantum order parameter, which is analogous to the one defined by Kuramoto in the classical model. As a consequence of the previous results, we obtain the synchronization of the probability and the current densities defined via the Madelung transformations
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
Numerical investigations of dispersive shocks and spectral analysis for linearized quantum hydrodynamics
No full textThe aim of this paper is to study solutions of one dimensional compressible Euler system with dissipation–dispersion terms, where the dispersive term is originated by the quantum effects described through the Bohm potential, as customary in quantum hydrodynamic models. We shall investigate numerically the sensitivity of the profiles with respect to the viscosity parameter, in particular in terms of their monotonicity properties. In addition, we shall also pinpoint numerically how the profile becomes more oscillatory as the end states approach the vacuum. The analysis of spectral properties of the linearized system around constant states is also provided, as well as the (numerical) localization of the point spectrum of the linearization along a profile. The latter investigation is carried out through the Evans function method
The cauchy problem for the maxwell–Schrödinger system with a power-type nonlinearity
No full textWe discuss some results on the Maxwell–Schrödinger system with a nonlinear power-like potential. We prove the local well-posedness in H2(R3)× H3/2R3) and the global existence of finite energy weak solutions. Then we apply these results to the analysis of finite energy weak solutions for Quantum Magnetohydrodynamic systems. Our interest in this problem is motivated by some models arising in quantum plasma dynamics
Splash singularity for a free-boundary incompressible viscoelastic fluid model
No full textNumerical computations in viscoelasticity show the failure of many numerical schemes when the Weissenberg number is beyond a critical value Keunings (J Non-Newtonian Fluid Mech 20:209–226, 1986, [6]). The existence of singularities in the continuum model could be the way to explain instability appearing in numerical simulations. We consider here a 2D Oldroyd-B type model at high Weissenberg number, and we show the existence of the so-called splash singularities (namely, points where the free boundary remains smooth but self-intersects). In our case, we assume physically realistic boundary conditions given by the static equilibrium of all the force fields acting at the interface. Our strategy is based on local existence and stability results applied to a family of smooth suitable initial configurations, we show they will evolve into a self-intersecting configuration, and then necessarily there exists a positive time t=t*, where the configuration has a splash singularity. To prove local existence and stability, we first apply a conformal transformation to the 2D domain, in order to separate the contact point with splash, and then we pass into Lagrangian coordinates to fix our domain, inspired by a Thomas Beale’s paper on the initial value problem for the Navier–Stokes equations with a free surface
Nonlinear Maxwell–Schrödinger system and quantum magneto-hydrodynamics in
No full textMotivated by some models arising in quantum plasma dynamics, in this paper we study the Maxwell-Schrodinger system with a power-type nonlinearity. We show the local well-posedness in H-2(R-3) x H-3/2(R-3) and the global existence of finite energy weak solutions, these results are then applied to the analysis of finite energy weak solutions for Quantum Magnetohydrodynamic systems
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