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Hind Al Baba

Publications and source records attributed to Hind Al Baba.

7 recordsLinked to original sources
arXiv2025

Remark on the local well-posedness of compressible non-Newtonian fluids with initial vacuum

We discuss in this short note the local-in-time strong well-posedness of the compressible Navier-Stokes system for non-Newtonian fluids on the three dimensional torus. We show that the result established recently by Kalousek, Mácha, and Nečasova in \doi{10.1007/s00208-021-02301-8} can be extended to the case where vanishing density is allowed initially. Our proof builds on the framework developed by Cho, Choe, and Kim in \doi{10.1016/j.matpur.2003.11.004} for compressible Navier-Stokes equations in the case of Newtonian fluids. To adapt their method, special attention is given to the elliptic regularity of a challenging nonlinear elliptic system. We show particular results in this direction, however, the main result of this paper is proven in the general case when elliptic $W^{2,p}$-regularity is imposed as an assumption. Also, we give a finite time blow-up criterion.

math.AP↗
arXiv2021

Lp-strong solution to fluid-rigid body interaction system with Navier slip boundary condition

We study a fluid-structure interaction problem describing movement of a rigid body inside a bounded domain filled by a viscous fluid. The fluid is modelled by the generalized incompressible Naiver-Stokes equations which include cases of Newtonian and non-Newtonian fluids. The fluid and the rigid body are coupled via the Navier slip boundary conditions and balance of forces at the fluid-rigid body interface. Our analysis also includes the case of the nonlinear slip condition. The main results assert the existence of strong solutions, in an $L^p-L^q$ setting, globally in time, for small data in the Newtonian case, while existence of strong solutions in $L^p$-spaces, locally in time, is obtained for non-Newtonian case. The proof for the Newtonian fluid essentially uses the maximal regularity property of the associated linear system which is obtained by proving the $\mathcal{R}$-sectoriality of the corresponding operator. The existence and regularity result for the general non-Newtonian fluid-solid system then relies upon the previous case. Moreover, we also prove the exponential stability of the system in the Newtonian case.

math.AP↗
arXiv2019

Non-uniqueness of admissible weak solution to the Riemann problem for the full Euler system in 2D

The question of well- and ill-posedness of entropy admissible solutions to the multi-dimensional systems of conservation laws has been studied recently in the case of isentropic Euler equations. In this context special initial data were considered, namely the 1D Riemann problem which is extended trivially to a second space dimension. It was shown that there exist infinitely many bounded entropy admissible weak solutions to such a 2D Riemann problem for isentropic Euler equations, if the initial data give rise to a 1D self-similar solution containing a shock. In this work we study such a 2D Riemann problem for the full Euler system in two space dimensions and prove the existence of infinitely many bounded entropy admissible weak solutions in the case that the Riemann initial data give rise to the 1D self-similar solution consisting of two shocks and possibly a contact discontinuity.

math.AP↗
arXiv2016

Maximal $L^p-L^q$ regularity to the Stokes Problem with Navier boundary conditions

We prove in this paper some results on the complex and fractional powers of the Stokes operator with slip frictionless boundary conditions involving the stress tensor. This is fundamental and plays an important role in the associated parabolic problem and will be used to prove maximal $L^{p}-L^{q}$ regularity results for the non-homogeneous Stokes problem.

math.AP↗
arXiv2016

Semi-group theory for the Stokes operator with Navier-type boundary conditions on $L^{p}$-spaces

In this article we consider the Stokes problem with Navier-type boundary conditions on a domain $Ω$, not necessarily simply connected. Since under these conditions the Stokes problem has a non trivial kernel, we also study the solutions lying in the orthogonal of that kernel. We prove the analyticity of several semigroups generated by the Stokes operator considered in different functional spaces. We obtain strong, weak and very weak solutions for the time dependent Stokes problem with the Navier-type boundary condition under different hypothesis on the initial data $\boldsymbol{u}_0$ and external force $\boldsymbol{f}$. Then, we study the fractional and pure imaginary powers of several operators related with our Stokes operators. Using the fractional powers, we prove maximal regularity results for the homogeneous Stokes problem. On the other hand, using the boundedness of the pure imaginary powers we deduce maximal $L^{p}-L^{q}$ regularity for the inhomogeneous Stokes problem.

math.AP↗