We study non-negative solutions to the chemotaxis system [Formula: see text] under no-flux boundary conditions in a bounded planar convex domain with smooth boundary, where f and S are given parameter functions on Ω × [0, ∞) 2 values ℝ 2×2 , respectively, which assumed satisfy certain regularity assumptions growth restrictions. Systems of type (⋆), special case reducing version standard Keller–Segel signal consumption, have recently been proposed as model for swimming bacteria near surface,...

In literature, it is usually very difficult to investigate the analytical and numerical solutions of fractional integro-differential equations (FIDEs). current work, linear non-linear FIDEs their systems have been analyzed by using Aboodh transform decomposition method (ATDM). The transformation first utilized simplify given problem, then implemented obtain required results. Adomian polynomials Daftardar-Jafari are used control term in system. obtained results compared with both different...

We study an initial boundary value problem for the 3D magnetohydrodynamics (MHD) equations of compressible fluids in $\mathbb{R}^3$. establish a blow-up criterion local strong solutions terms density and magnetic field. Namely, if is away from vacuum ($\rho= 0$) concentration mass ($\rho=\infty$) field bounded above $L^\infty$-norm, then solution can be continued globally time.

We study the three-dimensional active scalar equation called magneto-geostropic equation, which was proposed by Moffatt and Loper as a model for geodynamo processes in Earth's fluid core. When viscosity of is positive, constitutive law that relates drift velocity u(x, t) temperature produces two orders smoothing. implications this property. For example, we prove case non-diffusive () initial data implies existence unique, global weak solutions. If with s > 0, then solution all time. In...

<abstract><p>This article implements an efficient analytical technique within three different operators to investigate the solutions of some fractional partial differential equations and their systems. The generalized schemes proposed method are derived for every targeted problem under influence each derivative operator. numerical examples non-homogeneous Cauchy equation three-dimensional Navier-Stokes obtained using new iterative transform method. results found be identical. 2D...

In this article, we explore the effectiveness of two polynomial methods in solving non-linear time and space fractional partial differential equations. We first outline general methodology then apply it to five distinct experiments. The proposed method, noted for its simplicity, demonstrates a high degree accuracy. Comparative analysis with existing techniques reveals that our approach yields more precise solutions. results, presented through graphs tables, indicate He's Daftardar-Jafari...

We study the low-energy solutions to 3D compressible Navier-Stokes-Poisson equations. first obtain existence of smooth with small $ L^2 $-norm and essentially bounded densities. No smallness assumption is imposed on H^4 initial data. Using a compactness argument, we further weak which may have discontinuities across some hypersurfaces in \mathbb R^3 $. also provide blow-up criterion terms L^\infty density.

We study the 3-D compressible Navier-Stokes equations with an external potential force and a general non-decreasing pressure. prove global-in-time existence of weak solutions small-energy initial data densities being non-negative essentially bounded. A solution may have large oscillations contain vacuum states. No smallness assumption is made on nor perturbation in L∞ for density. Initial velocity u0 taken to be bounded Lq some q > 6 no further regularity imposed u0. Finally, we...

Abstract We study the 3‐D compressible Navier–Stokes equations with an external potential force and a general pressure. prove global‐in‐time existence of weak solutions small‐energy initial data densities being positive essentially bounded. No smallness assumption is made on force. Copyright © 2013 John Wiley & Sons, Ltd.

We prove the global-in-time existence of intermediate weak solutions equations chemotaxis system in a bounded domain $\mathbb{R}^2$ or $\mathbb{R}^3$ with initial chemical concentration small $H^1$. No smallness assumption is imposed on cell density which $L^2$. first show that when $c_0$ only $H^1$ and $(n_0-n_\infty,c_0)$ smooth, classical solution exists for all time. Then we construct as limits smooth corresponding to mollified data. Finally determine asymptotic behavior global solutions.

We study the local Morrey spaces with variable exponents.We show that block space exponents are pre-duals of exponents.Using this duality, we establish extrapolation theory for exponents.The gives mapping properties sharp maximal functions, geometric functions and rough function on exponents.

<p style='text-indent:20px;'>We address the compressible magnetohydrodynamics (MHD) equations in <inline-formula><tex-math id="M1">\begin{document}$ \mathbb R^3 $\end{document}</tex-math></inline-formula> and establish a blow-up criterion for local-in-time smooth solutions terms of density only. Namely, if is away from vacuum (<inline-formula><tex-math id="M2">\begin{document}$ \rho = 0 $\end{document}</tex-math></inline-formula>)...