Perturbative Cauchy theory for a flux-incompressible Maxwell-Stefan system

<p style='text-indent:20px;'>Recently, the authors proved [<xref ref-type="bibr" rid="b2">2</xref>] that the Maxwell-Stefan system with an incompressibility-like condition on the total flux can be rigorously derived from the multi-species Boltzmann equation. Similar cross-diffusion models have been widely investigated, but the particular case of a perturbative incompressible setting around a non constant equilibrium state of the mixture (needed in [<xref ref-type="bibr" rid="b2">2</xref>]) seems absent of the literature. We thus establish a quantitative perturbative Cauchy theory in Sobolev spaces for it. More precisely, by reducing the analysis of the Maxwell-Stefan system to the study of a quasilinear parabolic equation on the sole concentrations and with the use of a suitable anisotropic norm, we prove global existence and uniqueness of strong solutions and their exponential trend to equilibrium in a perturbative regime around any macroscopic equilibrium state of the mixture. As a by-product, we show that the equimolar diffusion condition naturally appears from this perturbative incompressible setting.</p>

On the well-posedness of the anisotropically-reduced two-dimensional Kuramoto-Sivashinsky Equation

<p style='text-indent:20px;'>We address the global existence and uniqueness of solutions for the anisotropically reduced 2D Kuramoto-Sivashinsky equations in a periodic domain with initial data <inline-formula><tex-math id="M1">\begin{document}$ u_{01} \in L^2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ u_{02} \in H^{-1 + \eta} $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M3">\begin{document}$ \eta &gt; 0 $\end{document}</tex-math></inline-formula>.</p>

Asymptotic Separation of Solutions to Fractional Stochastic Multi-Term Differential Equations

In this paper, we study the exact asymptotic separation rate of two distinct solutions of Caputo stochastic multi-term differential equations (Caputo SMTDEs). Our goal in this paper is to establish results of the global existence and uniqueness and continuity dependence of the initial values of the solutions to Caputo SMTDEs with non-permutable matrices of order α∈(12,1) and β∈(0,1) whose coefficients satisfy a standard Lipschitz condition. For this class of systems, we then show the asymptotic separation property between two different solutions of Caputo SMTDEs with a more general condition based on λ. Furthermore, the asymptotic separation rate for the two distinct mild solutions reveals that our asymptotic results are general.

Global Existence and Uniqueness of Solution of Atangana–Baleanu Caputo Fractional Differential Equation with Nonlinear Term and Approximate Solutions

In this paper, a class of fractional order differential equation expressed with Atangana–Baleanu Caputo derivative with nonlinear term is discussed. The existence and uniqueness of the solution of the general fractional differential equation are expressed. To present numerical results, we construct approximate scheme to be used for producing numerical solutions of the considered fractional differential equation. As an illustrative numerical example, we consider two Riccati fractional differential equations with different derivatives: Atangana–Baleanu Caputo and Caputo derivatives. Finally, the study of those examples verifies the theoretical results of global existence and uniqueness of solution. Moreover, numerical results underline the difference between solutions of both examples.

Final Value Problem for Parabolic Equation with Fractional Laplacian and Kirchhoff’s Term

In this paper, we study a diffusion equation of the Kirchhoff type with a conformable fractional derivative. The global existence and uniqueness of mild solutions are established. Some regularity results for the mild solution are also derived. The main tools for analysis in this paper are the Banach fixed point theory and Sobolev embeddings. In addition, to investigate the regularity, we also further study the nonwell-posed and give the regularized methods to get the correct approximate solution. With reasonable and appropriate input conditions, we can prove that the error between the regularized solution and the search solution is towards zero when δ tends to zero.

Existence and Uniqueness of Weak Solution for chemotaxis model coupled with heat equation

Keller-Segel chemotaxis model is described by a system of nonlinear PDE : a convection diffusion equation for the cell density coupled with a reaction-diffusion equation for chemoattractant concentration. In this work, we study the phenomenon of Keller Segel model coupled with a heat equation, because The heat has an effect the density of the cells as well as the signal of chemical concentration, since the heat is a factor affecting the spread and attraction of cells as well in relation to the signal of chemical concentration, The main objectives of this work is the study of the global existence and uniqueness and boundedness of the weak solution for the problem defined in (8) for this we use the technical of Galerkin method.

The Cauchy Problem for the Incompressible 2D-MHD with Power Law-Type Nonlinear Viscous Fluid

We investigate a motion of the incompressible 2D-MHD with power law-type nonlinear viscous fluid. In this paper, we establish the global existence and uniqueness of a weak solution u , b depending on a number q in ℝ 2 . Moreover, the energy norm of the weak solutions to the fluid flows has decay rate 1 + t − 1 / 2 .

Applications of the Bielecki renorming technique

The renorming technique allows one to apply the Banach Contraction Principle for maps which are not contractions with respect to the original metric. This method was invented by Bielecki and manifested in an extremely elegant proof of the Global Existence and Uniqueness Theorem for ODEs. The present paper provides further extensions and applications of Bielecki’s method to problems stemming from functional analysis and from the theory of functional equations.