We consider a singular semilinear hyperbolic approximation to hydrodynamic equations in 2D, inspi... more We consider a singular semilinear hyperbolic approximation to hydrodynamic equations in 2D, inspired by the kinetic theory. We show the convergence of the vector-BGK model to the incompressible Navier-Stokes equations under the diffusive scaling. Our strategy is based on the use of local in time Sobolev estimates, combined with the relative entropy and the interpolation properties of the Sobolev spaces.
We consider a simple example of a partially dissipative hyperbolic system violating the Shizuta-K... more We consider a simple example of a partially dissipative hyperbolic system violating the Shizuta-Kawashima condition, i.e. such that some eigendirections do not exhibit dissipation at all. In the space-time resonances framework introduced by Germain, Masmoudi and Shatah, we prove that, when the source term has a Nonresonant Bilinear Form, as proposed by Pusateri and Shatah CPAM 2013, the formation of singular-ities is prevented, despite the lack of dissipation. This allows us to show that smooth solutions to this preliminary case-study model exist globally in time
Journal of Hyperbolic Differential Equations, 2017
We propose a model of a density-dependent compressible–incompressible fluid, which is intended as... more We propose a model of a density-dependent compressible–incompressible fluid, which is intended as a simplified version of models based on mixture theory as, for instance, those arising in the study of biofilms, tumor growth and vasculogenesis. Though our model is, in some sense, close to the density-dependent incompressible Euler equations, it presents some differences that require a different approach from an analytical point of view. In this paper, we establish a result of local existence and uniqueness of solutions in Sobolev spaces to our model, using the Leray projector. Besides, we show the convergence of both a continuous version of the Chorin–Temam projection method, viewed as a singular perturbation approximation, and the artificial compressibility method.
We consider a singular semilinear hyperbolic approximation to hydrodynamic equations in 2D, inspi... more We consider a singular semilinear hyperbolic approximation to hydrodynamic equations in 2D, inspired by the kinetic theory. We show the convergence of the vector-BGK model to the incompressible Navier-Stokes equations under the diffusive scaling. Our strategy is based on the use of local in time Sobolev estimates, combined with the relative entropy and the interpolation properties of the Sobolev spaces.
We consider a simple example of a partially dissipative hyperbolic system violating the Shizuta-K... more We consider a simple example of a partially dissipative hyperbolic system violating the Shizuta-Kawashima condition, i.e. such that some eigendirections do not exhibit dissipation at all. In the space-time resonances framework introduced by Germain, Masmoudi and Shatah, we prove that, when the source term has a Nonresonant Bilinear Form, as proposed by Pusateri and Shatah CPAM 2013, the formation of singular-ities is prevented, despite the lack of dissipation. This allows us to show that smooth solutions to this preliminary case-study model exist globally in time
Journal of Hyperbolic Differential Equations, 2017
We propose a model of a density-dependent compressible–incompressible fluid, which is intended as... more We propose a model of a density-dependent compressible–incompressible fluid, which is intended as a simplified version of models based on mixture theory as, for instance, those arising in the study of biofilms, tumor growth and vasculogenesis. Though our model is, in some sense, close to the density-dependent incompressible Euler equations, it presents some differences that require a different approach from an analytical point of view. In this paper, we establish a result of local existence and uniqueness of solutions in Sobolev spaces to our model, using the Leray projector. Besides, we show the convergence of both a continuous version of the Chorin–Temam projection method, viewed as a singular perturbation approximation, and the artificial compressibility method.
Gathering together some existing results, we show that the solutions to the one-dimensional Burge... more Gathering together some existing results, we show that the solutions to the one-dimensional Burgers equation converge for long times towards the stationary solutions to the steady Burgers equation, whose Fourier spectrum is not integrable. This is one of the main features of wave turbulence.
To cite this version: Roberta Bianchini, Gigliola Staffilani. Revisitation of a Tartar's result o... more To cite this version: Roberta Bianchini, Gigliola Staffilani. Revisitation of a Tartar's result on a semilinear hyperbolic system with null condition. Abstract We revisit a method introduced by Tartar for proving global well-posedness of a semilinear hyperbolic system with null quadratic source in one space dimension. A remarkable point is that, since no dispersion effect is available for 1D hyperbolic systems, Tartar's approach is entirely based on spatial localization and finite speed of propagation.
We provide a general framework to extend the relative entropy method to a class of diffusive rela... more We provide a general framework to extend the relative entropy method to a class of diffusive relaxation systems with discrete velocities. The methodology is detailed in the toy case of the 1D Jin-Xin model under the diffusive scaling, and provides a direct proof of convergence to the limit parabolic equation in any interval of time, in the regime where the solutions are smooth. Recently, the same approach has been successfully used to show the strong convergence of a vector-BGK model to the 2D incompressible Navier-Stokes equations.
We consider a simple example of a partially dissipative hyperbolic system violating the Shizuta-K... more We consider a simple example of a partially dissipative hyperbolic system violating the Shizuta-Kawashima condition, i.e. such that some eigendirections do not exhibit dissipation at all. In the space-time resonances framework introduced by Germain, Masmoudi and Shatah, we prove that, when the source term has a Nonresonant Bilinear Form, as proposed by Pusateri and Shatah CPAM 2013, the formation of singular-ities is prevented, despite the lack of dissipation. This allows us to show that smooth solutions to this preliminary case-study model exist globally in time.
Internal waves describe the (linear) response of an incompressible stably stratified luid to sma... more Internal waves describe the (linear) response of an incompressible stably stratified luid to small perturbations. The inclination of their group velocity with respect to the vertical is completely determined by their frequency. Therefore the reflection on a sloping boundary cannot follow Descartes' laws, and it is expected to be singular if the slope has the same inclination as the group velocity. In this paper, we prove that in this critical geometry the weakly viscous and weakly nonlinear wave equations have actually a solution which is well approximated by the sum of the in- cident wave packet, a reflected second harmonic and some boundary layer terms. This result confirms the prediction by Dauxois and Young, and provides precise estimates on the time of validity of this approximation.
Journal de Mathématiques Pures et Appliquées, 2019
The aim of this paper is to prove the strong convergence of the solutions to a vector-BGK model u... more The aim of this paper is to prove the strong convergence of the solutions to a vector-BGK model under the diffusive scaling to the incompressible Navier-Stokes equations on the two-dimensional torus. This result holds in any interval of time [0, T ], with T > 0. We also provide the global in time uniform boundedness of the solutions to the approximating system. Our argument is based on the use of local in time H s-estimates for the model, established in a previous work, combined with the L 2-relative entropy estimate and the interpolation properties of the Sobolev spaces.
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Papers by Roberta Bianchini
velocity with respect to the vertical is completely determined by their
frequency. Therefore the reflection on a sloping boundary cannot follow
Descartes' laws, and it is expected to be singular if the slope has the same
inclination as the group velocity. In this paper, we prove that in this
critical geometry the weakly viscous and weakly nonlinear wave equations
have actually a solution which is well approximated by the sum of the in-
cident wave packet, a reflected second harmonic and some boundary layer
terms. This result confirms the prediction by Dauxois and Young, and
provides precise estimates on the time of validity of this approximation.