arXiv subjects

Sagbo Marcel Zodji

Publications and source records attributed to Sagbo Marcel Zodji.

3 recordsLinked to original sources

arXiv2025

A well-posedness result for the compressible two-fluid model with density-dependent viscosity

In this paper, we study a system of PDEs describing the motion of two compressible viscous fluids occupying the whole space $\mathbb R^d\;(d\in \{2,3\}$). The two phases of the mixture are separated by a $\mathscr{C}^{1+α}$-regular sharp interface $\mathcal{C}$ across which the density can experience jumps. We prove the existence of a unique local-in-time solution assuming that the initial density is $α$-Hölder continuous on both sides of $\mathcal{C}$. The initial velocity belongs to the Sobolev space $H^1(\mathbb R^d)$, and the divergence of the initial stress tensor belongs to $L^2(\mathbb R^d)$. The later assumption expresses somehow the continuity of the stress tensor. This result is more general than the one by Tani [32], as it allows for less regular initial data and furthermore it can serve as a building block for the construction of global-in-time solutions.

math.AP↗

arXiv2024

Discontinuous solutions for the Navier-Stokes equations with density-dependent viscosity

We prove existence of a unique global-in-time weak solutions of the Navier-Stokes equations that govern the motion of a compressible viscous fluid with density-dependent viscosity in two-dimensional space. The initial velocity belongs to the Sobolev space $H^1(\mathbb{R}^2)$, and the initial fluid density is $α$-Hölder continuous on both sides of a $\mathscr{C}^{1+α}$-regular interface with some geometrical assumption. We prove that this configuration persists over time: the initial interface is transported by the flow to an interface that maintains the same regularity as the initial one. Our result generalizes previous known of Hoff [21], Hoff and Santos [22] concerning the propagation of regularity for discontinuity surfaces by allowing more general nonlinear pressure law and density-dependent viscosity. Moreover, it supplements the work by Danchin, Fanelli and Paicu [6] with global-in-time well-posedness, even for density-dependent viscosity and we achieve uniqueness in a large space.

math.AP↗

arXiv2024

Global-in-time well-posedness of the compressible Navier-Stokes equations with striated density

We first show local-in-time well-posedness of the compressible Navier-Stokes equations, assuming striated regularity while no other smoothness or smallness conditions on the initial density. With these local-in-time solutions served as blocks, for \textit{less} regular initial data where the vacuum is permitted, the global-in-time well-posedness follows from the energy estimates and the propagated striated regularity of the density function, if the bulk viscosity coefficient is large enough in the two dimensional case. The global-in-time well-posedness holds also true in the three dimensional case, provided with large bulk viscosity coefficient together with small initial energy. This solves the density-patch problem in the exterior domain for the compressible model with $W^{2,p}$-Interfaces. Finally, the singular incompressible limit toward the inhomogenous incompressible model when the bulk viscosity coefficient tends to infinity is obtained.

math.AP↗