In order to prove our results, we develop a new characterization of the dissipation structure for the linearized Euler-Maxwell system with respect to the relaxation parameter $\varepsilon$. This is done by partitioning the frequency space into three distinct regimes: low, medium and high frequencies, each associated with a different behaviour of the solution. Then, in each regime, the use of efficient unknowns and Lyapunov functionals based on the hypocoercivity theory leads to uniform a priori estimates},
author = {Crin-Barat, Timothée and Shou, Ling-Yun and Xu, Jiang and Peng, Yue-Jun},
faupublication = {yes},
keywords = {Euler-Maxwell system; Drift-diffusion system; Diffusive relaxation limit; Non-symmetric relaxation; Partially dissipative systems; Critical regularity;},
note = {https://cris.fau.de/converis/publicweb/Publication/324982683},
peerreviewed = {automatic},
title = {{A} new characterization of the dissipation structure and the relaxation limit for the compressible {Euler}-{Maxwell} system},
url = {https://arxiv.org/abs/2407.00277},
year = {2024}
}
@unpublished{faucris.321081290,
author = {Crin-Barat, Timothée and Manea, Dragos},
faupublication = {yes},
note = {https://cris.fau.de/converis/publicweb/Publication/321081290},
peerreviewed = {automatic},
title = {{Asymptotic}-preserving finite difference method for partially dissipative hyperbolic systems},
year = {2024}
}
@article{faucris.289674511,
abstract = {We study the diffusive relaxation limit of the Jin-Xin system toward viscous conservation laws in the multi-dimensional setting. For initial data being small perturbations of a constant state in suitable homogeneous Besov norms, we prove the global well-posedness of strong solutions satisfying uniform estimates with respect to the relaxation parameter. Then, we justify the strong relaxation limit and exhibit an explicit convergence rate of the process. Our proof is based on an adaptation of the techniques developed in [12,13] to be able to deal with additional low-order nonlinear terms.},
author = {Crin-Barat, Timothée and Shou, Ling Yun},
doi = {10.1016/j.jde.2023.02.015},
faupublication = {yes},
journal = {Journal of Differential Equations},
keywords = {Besov spaces; Hyperbolic approximation; Jin-Xin approximation; Relaxation limit},
note = {CRIS-Team Scopus Importer:2023-02-24},
pages = {302-331},
peerreviewed = {Yes},
title = {{Diffusive} relaxation limit of the multi-dimensional {Jin}-{Xin} system},
volume = {357},
year = {2023}
}
@article{faucris.294849924,
abstract = {In this paper, we study a singular limit problem for a compressible one-velocity bifluid system. More precisely, we show that solutions of the Kapila system generated by initial data close to equilibrium are obtained in the pressure-relaxation limit from solutions of the Baer-Nunziato (BN) system. The convergence rate of this process is a consequence of our stability result. Besides the fact that the quasilinear part of the (BN) system cannot be written in conservative form, its natural associated entropy is only positive semi-definite such that it is not clear if the entropic variables can be used in the present case. Using an ad-hoc change of variables we obtain a reformulation of the (BN) system which couples, via low-order terms, an undamped mode and a non-symmetric partially dissipative hyperbolic system satisfying the Shizuta-Kawashima stability condition.},
author = {Burtea, Cosmin and Crin-Barat, Timothée and Tan, Jin},
doi = {10.1142/S0218202523500161},
faupublication = {yes},
journal = {Mathematical Models & Methods in Applied Sciences},
keywords = {Baer-Nunziato; Kapila system; non-conservative quasilinear systems; one-velocity bifluid system; pressure-relaxation limit},
note = {CRIS-Team Scopus Importer:2023-03-31},
peerreviewed = {Yes},
title = {{Pressure}-relaxation limit for a one-velocity {Baer}-{Nunziato} model to a {Kapila} model},
year = {2023}
}
@article{faucris.323020720,
abstract = {We derive a novel two-phase flow system in porous media as a relaxation limit of compressible multi-fluid systems. Considering a one-velocity Baer-Nunziato system with friction forces, we first justify its pressure-relaxation limit toward a Kapila model in a uniform manner with respect to the time-relaxation parameter associated with the friction forces. Then, we show that the diffusely rescaled solutions of the damped Kapila system converge to the solutions of the new two-phase porous media system as the time-relaxation parameter tends to zero. In addition, we also prove the convergence of the Baer-Nunziato system to the same two-phase porous media system as both relaxation parameters tend to zero. For each relaxation limit, we exhibit sharp rates of convergence in a critical regularity setting. Our proof is based on an elaborate low-frequency and high-frequency analysis via the Littlewood-Paley decomposition and includes three main ingredients: a refined spectral analysis of the linearized problem to determine the frequency threshold explicitly in terms of the time-relaxation parameter, the introduction of an effective flux in the low-frequency region to overcome the loss of parameters due to the overdamping phenomenon, and renormalized energy estimates in the high-frequency region to cancel higher-order nonlinear terms. To justify the convergence rates, we discover several auxiliary unknowns allowing us to recover crucial O ( ϵ ) bounds.},
author = {Crin-Barat, Timothée and Shou, Ling Yun and Tan, Jin},
doi = {10.1088/1361-6544/ad3f66},
faupublication = {yes},
journal = {Nonlinearity},
keywords = {35B40; 35Q35; 76N10; 76T17; Baer-Nunziato system; critical regularity; Kapila system; multi-fluid system; overdamping phenomenon; pressure-relaxation limit; two-phase flow in porous media},
note = {CRIS-Team Scopus Importer:2024-05-31},
peerreviewed = {Yes},
title = {{Quantitative} derivation of a two-phase porous media system from the one-velocity {Baer}-{Nunziato} and {Kapila} systems},
volume = {37},
year = {2024}
}
@unpublished{faucris.321209321,
author = {Crin-Barat, Timothée and Skondrić, Stefan and Violini, Alessandro},
faupublication = {yes},
note = {https://cris.fau.de/converis/publicweb/Publication/321209321},
peerreviewed = {automatic},
title = {{Relative} energy method for weak-strong uniqueness of the inhomogeneous {Navier}-{Stokes} equations},
year = {2024}
}
@article{faucris.315830818,
abstract = {We prove the nonlinear asymptotic stability of stably stratified solutions to the Incompressible Porous Media equation (IPM) for initial perturbations in H˙ ^{1}^{-}^{τ}(R^{2}) ∩ H˙ ^{s}(R^{2}) with s> 3 and for any 0 < τ< 1 . Such a result improves upon the existing literature, where the asymptotic stability is proved for initial perturbations belonging at least to H^{20}(R^{2}) . More precisely, the aim of the article is threefold. First, we provide a simplified and improved proof of global-in-time well-posedness of the Boussinesq equations with strongly damped vorticity in H^{1}^{-}^{τ}(R^{2}) ∩ H˙ ^{s}(R^{2}) with s> 3 and 0 < τ< 1 . Next, we prove the strong convergence of the Boussinesq system with damped vorticity towards (IPM) under a suitable scaling. Lastly, the asymptotic stability of stratified solutions to (IPM) follows as a byproduct. A symmetrization of the approximating system and a careful study of the anisotropic properties of the equations via anisotropic Littlewood-Paley decomposition play key roles to obtain uniform energy estimates. Finally, one of the main new and crucial points is the integrable time decay of the vertical velocity ‖u2(t)‖L∞(R2) for initial data only in H˙ ^{1}^{-}^{τ}(R^{2}) ∩ H˙ ^{s}(R^{2}) with s> 3 .},
author = {Bianchini, Roberta and Crin-Barat, Timothée and Paicu, Marius},
doi = {10.1007/s00205-023-01945-x},
faupublication = {yes},
journal = {Archive for Rational Mechanics and Analysis},
note = {CRIS-Team Scopus Importer:2023-12-22},
peerreviewed = {Yes},
title = {{Relaxation} {Approximation} and {Asymptotic} {Stability} of {Stratified} {Solutions} to the {IPM} {Equation}},
volume = {248},
year = {2024}
}
@unpublished{faucris.321081881,
author = {Crin-Barat, Timothée and Kawashima, Shuichi and Xu, Jiang},
faupublication = {yes},
note = {https://cris.fau.de/converis/publicweb/Publication/321081881},
peerreviewed = {automatic},
title = {{The} {Cattaneo}-{Christov} approximation of {Fourier} heat-conductive compressible fluids},
year = {2024}
}