Global existence of weak solutions to unsaturated poroelasticity

We study unsaturated poroelasticity, i.e., coupled hydro-mechanical processes in variably saturated porous media, here modeled by a non-linear extension of Biot's well-known quasi-static consolidation model. The coupled elliptic-parabolic system of partial differential equations is a simplified version of the general model for multi-phase flow in deformable porous media, obtained under similar assumptions as usually considered for Richards' equation. In this work, existence of weak solutions is established in several steps involving a numerical approximation of the problem using a physically-motivated regularization and a finite element/finite volume discretization. Eventually, solvability of the original problem is proved by a combination of the Rothe and Galerkin methods, and further compactness arguments. This approach in particular provides the convergence of the numerical discretization to a regularized model for unsaturated poroelasticity. The final existence result holds under non-degeneracy conditions and natural continuity properties for the constitutive relations. The assumptions are demonstrated to be reasonable in view of geotechnical applications.

Finite Volume approximation of a two-phase two fluxes degenerate Cahn-Hilliard model

We study a time implicit Finite Volume scheme for degenerate Cahn-Hilliard model proposed in [W. E and P. Palffy-Muhoray. Phys. Rev. E , 55:R3844–R3846, 1997] and studied mathematically by the authors in [C. Canc\`es, D. Matthes, and F. Nabet. Arch. Ration. Mech. Anal. , 233(2):837-866, 2019]. The scheme is shown to preserve the key properties of the continuous model, namely mass conservation, positivity of the concentrations, the decay of the energy and the control of the entropy dissipation rate. This allows to establish the existence of a solution to the nonlinear algebraic system corresponding to the scheme. Further, we show thanks to compactness arguments that the approximate solution converges towards a weak solution of the continuous problems as the discretization parameters tend to 0. Numerical results illustrate the behavior of the numerical model.

Global weak solutions to the generalized mCH equation via characteristics

<p style='text-indent:20px;'>In this paper, we study the generalized modified Camassa-Holm (gmCH) equation via characteristics. We first change the gmCH equation for unknowns <inline-formula><tex-math id="M1">\begin{document}$ (u,m) $\end{document}</tex-math></inline-formula> into its Lagrangian dynamics for characteristics <inline-formula><tex-math id="M2">\begin{document}$ X(\xi,t) $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M3">\begin{document}$ \xi\in\mathbb{R} $\end{document}</tex-math></inline-formula> is the Lagrangian label. When <inline-formula><tex-math id="M4">\begin{document}$ X_\xi(\xi,t)&gt;0 $\end{document}</tex-math></inline-formula>, we use the solutions to the Lagrangian dynamics to recover the classical solutions with <inline-formula><tex-math id="M5">\begin{document}$ m(\cdot,t)\in C_0^k(\mathbb{R}) $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M6">\begin{document}$ k\in\mathbb{N},\; \; k\geq1 $\end{document}</tex-math></inline-formula>) to the gmCH equation. The classical solutions <inline-formula><tex-math id="M7">\begin{document}$ (u,m) $\end{document}</tex-math></inline-formula> to the gmCH equation will blow up if <inline-formula><tex-math id="M8">\begin{document}$ \inf_{\xi\in\mathbb{R}}X_\xi(\cdot,T_{\max}) = 0 $\end{document}</tex-math></inline-formula> for some <inline-formula><tex-math id="M9">\begin{document}$ T_{\max}&gt;0 $\end{document}</tex-math></inline-formula>. After the blow-up time <inline-formula><tex-math id="M10">\begin{document}$ T_{\max} $\end{document}</tex-math></inline-formula>, we use a double mollification method to mollify the Lagrangian dynamics and construct global weak solutions (with <inline-formula><tex-math id="M11">\begin{document}$ m $\end{document}</tex-math></inline-formula> in space-time Radon measure space) to the gmCH equation by some space-time BV compactness arguments.</p>

Singular weighted sharp Trudinger-Moser inequalities defined on $ \mathbb{R}^N $ and applications to elliptic nonlinear equations

<p style='text-indent:20px;'>This work comes to complete some previous ones of ours. Actually, in this paper, we establish some singular weighted inequalities of Trudinger-Moser type for radial functions defined on the whole euclidean space <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^N,\ N \geq 2. $\end{document}</tex-math></inline-formula> The weights considered are of logarithmic type. The singularity plays a capital role to prove the sharpness of the inequalities. These inequalities are later improved using some concentration-compactness arguments. The last part in this work is devoted to the application of the inequalities established to some singular elliptic nonlinear equations involving a new growth conditions at infinity of exponential type.</p>

Global zero-relaxation limit of the non-isentropic Euler–Poisson system for ion dynamics

This paper is concerned with smooth solutions of the non-isentropic Euler–Poisson system for ion dynamics. The system arises in the modeling of semi-conductor, in which appear one small parameter, the momentum relaxation time. When the initial data are near constant equilibrium states, with the help of uniform energy estimates and compactness arguments, we rigorously prove the convergence of the system for all time, as the relaxation time goes to zero. The limit system is the drift-diffusion system.

Borderline regularity for fully nonlinear equations in Dini domains

In this paper, we prove borderline gradient continuity of viscosity solutions to fully nonlinear elliptic equations at the boundary of a C^{1,\mathrm{Dini}}-domain. Our main result constitutes the boundary analogue of the borderline interior gradient regularity estimates established by P. Daskalopoulos, T. Kuusi and G. Mingione. We however mention that, differently from the approach used there which is based on W^{1,q} estimates, our proof is slightly more geometric and is based on compactness arguments inspired by the techniques in the fundamental works of Luis Caffarelli.

Linking extended and plateau emissions of short gamma-ray bursts

ABSTRACT Some short gamma-ray bursts (SGRBs) show a longer lasting emission phase, called extended emission (EE) lasting ${\sim}10^{2\!-\!3}\, \rm s$, as well as a plateau emission (PE) lasting ${\sim}10^{4\!-\!5}\, \rm s$. Although a long-lasting activity of the central engines is a promising explanation for powering both emissions, their physical origin and their emission mechanisms are still uncertain. In this work, we study the properties of the EEs and their connection with the PEs. First, we constrain the minimal Lorentz factor Γ of the outflows powering EEs, using compactness arguments and find that the outflows should be relativistic, Γ ≳ 10. We propose a consistent scenario for the PEs, where the outflow eventually catches up with the jet responsible for the prompt emission, injecting energy into the forward shock formed by the prior jet, which naturally results in a PE. We also derive the radiation efficiency of EEs and the Lorentz factor of the outflow within our scenario for 10 well-observed SGRBs accompanied by both EE and PE. The efficiency has an average value of ${\sim}3\, {{\ \rm per\ cent}}$ but shows a broad distribution ranging from ∼0.01 to ${\sim}100{{\ \rm per\ cent}}$. The Lorentz factor is ∼20–30, consistent with the compactness arguments. These results suggest that EEs are produced by a slower outflow via more inefficient emission than the faster outflow that causes the prompt emission with a high radiation efficiency.

On short GRBs similar to GRB 170817A detected by Fermi-GBM

ABSTRACT Von Kienlin et al. (2019) selected 11 short gamma-ray bursts (sGRBs) whose characteristics are similar to GRB 170817A. These bursts, like GRB 170817A, have a hard spike followed by a soft thermal tail. However, as their redshifts are unknown it is not clear if their luminosities are as low as that of GRB 170817A. Comparing the positions in the ϵp–Eγ,iso (spectral peak energy – isotropic-equivalent energy) plane and using compactness arguments to estimate the minimal Lorentz factor, Γ, we find that all the bursts in this sample are consistent with being regular sGRBs if they are located at $z$ ≃ 0.3–3. They are also consistent with being similar to GRB 170817A if they are located at $z$ ≲ 0.1. Even in the latter case, the events must involve at least mildly relativistic (Γ ≳ 2) motion within the sources. We, further, find that at most one or two bursts in the sample are consistent with the cocoon shock-breakout model. Finally, we calculate the event rate of off-axis emission either from a jet core or from a jet wing (surrounding the core). We find that the off-axis emission model as an origin of the sample is rejected as it predicts too small event rate. The wing model can be consistent with the observed rate but the model parameters cannot be constrained by the current observations.

Lubrication and shallow-water systems Bernis-Friedman and BD entropies

This paper concerns the results recently announced by the authors, in C.R. Acad. Sciences Maths volume 357, Issue 1, 1-6 (2019), which make the link between the BD entropy introduced by D. Bresch and B. Desjardins for the viscous shallow-water equations and the Bernis-Friedman (called BF in our paper) dissipative entropy introduced to study the lubrication equations. More precisely different dissipative BF entropies are obtained from the BD entropies playing with drag terms and capillarity formula for viscous shallow water type equations. This is the main idea in the paper which makes the link between two communities. The limit processes employ the standard compactness arguments taking care of the control in the drag terms. It allows in one dimension for instance to prove global existence of nonnegative weak solutions for lubrication equations starting from the global existence of nonnegative weak solutions for appropriate viscous shallow-water equations (for which we refer to appropriate references). It also allows to prove global existence of nonnegative weak solutions for fourth-order equation including the Derrida-Lebowitz-Speer-Spohn equation starting from compressible Navier-Stokes type equations.