Well/Ill Posedness for the Euler-Korteweg-Poisson System and Related Problems

Donatelli, Donatella; Feireisl, Eduard; Marcati, Pierangelo

doi:10.1080/03605302.2014.972517

Cited by 66 publications

(67 citation statements)

References 20 publications

(32 reference statements)

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“…The arguments are adaptations of the method developed by De Lellis and Székelyhidi [12,13] to compressible setting in the spirit of [9]. We refer to [7,14,19,21] for similar results on other physical models of related to compressible fluids. The MHD case, however, is more delicate than the situation studied in [9] as there is no obvious way how to control the L ∞ -norm of the magnetic field B.…”

Section: Preliminaries the Main Results In The General Casementioning

confidence: 99%

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On global-in-time weak solutions to the magnetohydrodynamic system of compressible inviscid fluids

Feireisl

2019

Nonlinearity

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We consider the motion of an inviscid compressible fluid under the mutual interactions with magnetic field. We show that the initial value problem is ill-posed in the class of weak solutions for a large class of physically admissible data. We also consider the same problem for inviscid heat-conductive fluid and show the same result under certain restrictions imposed on the magnetic field. The main tool is the method of convex integration adapted to the Euler system with "variable coefficients".

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Section: Preliminaries the Main Results In The General Casementioning

confidence: 99%

“…The oscillatory lemma was later adapted to the case of compressible flows by Chiodaroli [7]. Here, we report a generalized version from [14].…”

Section: Oscillatory Lemmamentioning

confidence: 95%

On global-in-time weak solutions to the magnetohydrodynamic system of compressible inviscid fluids

Feireisl

2019

Nonlinearity

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“…is an open set, it is possible, similarly to [13,Section 4.3] to replace F m by F as long as the approximation is uniform. Specifically, for any δ > 0, there is m = m(δ) such that for some deterministic constant δ > 0.…”

Section: Now Since S[e]mentioning

confidence: 99%

On solvability and ill-posedness of the compressible Euler system subject to stochastic forces

2020

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We consider the (barotropic) Euler system describing the motion of a compressible inviscid fluid driven by a stochastic forcing. Adapting the method of convex integration we show that the initial value problem is ill-posed in the class of weak (distributional) solutions. Specifically, we find a sequence τ M → ∞ of positive stopping times for which the Euler system admits infinitely many solutions originating from the same initial data. The solutions are weak in the PDE sense but strong in the probabilistic sense, meaning, they are defined on an a priori given stochastic basis and adapted to the driving stochastic process.

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“…Lemma 1. For any P − > 0 there is an ǫ > 0, such that if s > 0 (s < 0), |P + − P − | < ǫ and P + < P − (P − < P + ), there exists a heteroclinic for the system (8)- (9), connecting [P − , 0] to [P + , 0].…”

Section: Existence Of Shock Profilesmentioning

confidence: 99%

Dispersive shocks in quantum hydrodynamics with viscosity

Lattanzio

Marcati

Zhelyazov

2020

Physica D: Nonlinear Phenomena

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In 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 travelling 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.

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Well/Ill Posedness for the Euler-Korteweg-Poisson System and Related Problems

Cited by 66 publications

References 20 publications

On global-in-time weak solutions to the magnetohydrodynamic system of compressible inviscid fluids

On global-in-time weak solutions to the magnetohydrodynamic system of compressible inviscid fluids

On solvability and ill-posedness of the compressible Euler system subject to stochastic forces

Dispersive shocks in quantum hydrodynamics with viscosity

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