Approximations of 2D and 3D Stochastic Convective Brinkman-Forchheimer Extended Darcy EquationsMohan, Manil T.
doi: 10.1007/s10440-024-00680-zpmid: N/A
In this article, we consider two- and three- dimensional stochastic convective Brinkman-Forchheimer extended Darcy (CBFeD) equations ∂u∂t−μΔu+(u⋅∇)u+α|u|q−1u+β|u|r−1u+∇p=f,∇⋅u=0,\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ \frac{\partial \boldsymbol{u}}{\partial t}-\mu \Delta \boldsymbol{u}+( \boldsymbol{u}\cdot \nabla )\boldsymbol{u}+\alpha |\boldsymbol{u}|^{q-1} \boldsymbol{u}+\beta |\boldsymbol{u}|^{r-1}\boldsymbol{u}+\nabla p= \boldsymbol{f},\ \nabla \cdot \boldsymbol{u}=0, $$\end{document} on a torus, where μ,β>0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\mu ,\beta >0$\end{document}, α∈R\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\alpha \in \mathbb{R}$\end{document}, r∈[1,∞)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$r\in [1,\infty )$\end{document} and q∈[1,r)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$q\in [1,r)$\end{document}. The goal is to show that the solutions of 2D and 3D stochastic CBFeD equations driven by Brownian motion can be approximated by 2D and 3D stochastic CBFeD equations forced by pure jump noise/random kicks on the state space D([0,T];H)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\mathrm{D}([0,T];\mathbb{H})$\end{document}. For the cases d=2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$d=2$\end{document}, r∈[1,∞)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$r\in [1,\infty )$\end{document} and d=3\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$d=3$\end{document}, r∈(3,∞)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$r\in (3,\infty )$\end{document}, by using minimal regularity assumptions on the noise coefficient, the results are established for any μ,β>0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\mu ,\beta >0$\end{document}. For the case d=r=3\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$d=r=3$\end{document}, the same results are obtained for 2βμ≥1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$2\beta \mu \geq 1$\end{document}.
Pseudorandomness of the Schrödinger Map EquationKumar, Sandeep
doi: 10.1007/s10440-024-00687-6pmid: N/A
A unique behaviour of the Schrödinger map equation, a geometric partial differential equation, is presented by considering its evolution for regular polygonal curves in both Euclidean and hyperbolic spaces. The results are consistent with those for the vortex filament equation, an equivalent form of the Schrödinger map equation in the Euclidean space. Thus, with all possible choices of regular polygons in a given setting, our analysis not only provides a novel extension to its usefulness as a pseudorandom number generator but also complements the existing results.
Wong-Zakai Approximations for the Stochastic Landau-Lifshitz-Bloch Equation with HelicityGokhale, Soham Sanjay
doi: 10.1007/s10440-024-00681-ypmid: N/A
For temperatures below and beyond the Curie temperature, the stochastic Landau-Lifshitz-Bloch equation describes the evolution of spins in ferromagnetic materials. In this work, we consider the stochastic Landau-Lifshitz-Bloch equation driven by a real valued Wiener process and show Wong-Zakai type approximations for the same. We consider non-zero contribution from the helicity term in the energy. First, using a Doss-Sussmann type transform, we convert the stochastic partial differential equation into a deterministic equation with random coefficients. We then show that the solution of the transformed equation depends continuously on the driving Wiener process. We then use this result, along with the properties of the said transform to show that the solution of the originally considered equation depends continuously on the driving Wiener process.
Unveiling the Interplay Between Degree-Based Graph Invariants of a Graph and Its Random SubgraphsHosseinzadeh, Mohammad Ali
doi: 10.1007/s10440-024-00688-5pmid: N/A
This paper investigates the significance of employing random subgraphs and analyzing the expected values of Zagreb ndices within chemical graphs. By examining smaller, representative subsets, we uncover valuable insights into the properties and characteristics of complex networks. The expected values of Zagreb indices serve as critical mathematical measures for quantifying the structural complexity of chemical graphs, providing essential information about connectivity and branching patterns within molecules. Our primary contribution includes deriving theoretical expressions for these indices and validating them through extensive computational experiments on fullerene graphs C20\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$C_{20}$\end{document} and C60\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$C_{60}$\end{document}. The results demonstrate that our theoretical predictions closely align with experimental findings, affirming the robustness of Zagreb indices in characterizing molecular structures. Additionally, our analysis of specific cases, such as complete graphs and complete bipartite graphs, is consistent with previous studies, further reinforcing our methodology. This research emphasizes the relevance of random subgraphs and expected values of Zagreb indices in advancing our understanding of molecular behavior and stability, with important implications for materials science and drug design.
Global Bounded Solutions and Large Time Behavior of a Chemotaxis System with Flux LimitationWu, Chun
doi: 10.1007/s10440-024-00690-xpmid: N/A
In this paper, the following cross-diffusion system is investigated {ut=∇⋅((u+1)m∇u)−∇⋅(u(u+1)β−1∇v(1+|∇v|2)α)+a−bur,x∈Ω,t>0,0=Δv−v+u,x∈Ω,t>0,\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ \textstyle\begin{cases} u_{t}=\nabla \cdot \big((u+1)^{m}\nabla u\big)-\nabla \cdot \Bigg( \frac{u(u+1)^{\beta -1}\nabla v}{(1+|\nabla v|^{2})^{\alpha }}\Bigg)+a-bu^{r}, \,\,& x\in \Omega ,\,\,t>0, \\ 0=\Delta v-v+u, & x\in \Omega ,\,\,t>0, \end{cases} $$\end{document} in a bounded domain Ω⊂Rn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\Omega \subset \mathbb{R}^{n}$\end{document} (n≥2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$n\ge 2$\end{document}) with smooth boundary ∂Ω\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\partial \Omega $\end{document}. Under the condition that α>2n−mn−22(n−1)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\alpha >\frac{2n-mn-2}{2(n-1)}$\end{document}, m≥1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$m\geq 1$\end{document}, and β≤m+22\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\beta \leq \frac{m+2}{2}$\end{document}, it is shown that the problem possesses a unique global bounded classical solution. Moreover, it is obtained that the corresponding solution exponentially converge to a constant stationary solution when the initial data u0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$u_{0}$\end{document} is sufficiently small.
Partially Dissipative Viscous System of Balance Laws and Application to Kuznetsov–Westervelt EquationPeralta, Gilbert
doi: 10.1007/s10440-024-00686-7pmid: N/A
We provide the well-posedness for a partially dissipative viscous system of balance laws in smooth Sobolev spaces under the same assumptions as in the case of inviscid balance laws. A priori estimates for coupled hyperbolic-parabolic linear systems with coefficients having limited regularity are derived using Friedrichs regularization and Moser-type estimates. Local existence for nonlinear systems will be established using the results of the linear theory and a suitable iteration scheme. The local existence theory is then applied to the Kuznetsov–Westervelt equation with damping for nonlinear wave acoustic propagation. Existence of global solutions for small data and their asymptotic stability are established.
Asymptotically Linear Euclidean Bosonic EquationsLong, Cuicui; Tan, Jinggang; Xia, Aliang
doi: 10.1007/s10440-024-00693-8pmid: N/A
We investigate the following nonlinear bosonic equation on Euclidean space arising in string theory and cosmology: P−Δe−cΔu+mu=f(x,u),x∈Rn,\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ -\Delta e^{-c\Delta }u+mu=f(x,u),\quad x\in {\mathbb{R}}^{n}, $$\end{document} where n≥3\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$n\ge 3$\end{document}, m>0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$m>0$\end{document}, c>0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$c>0$\end{document} and f(x,u)u\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\frac{f(x,u)}{u}$\end{document} tends to a positive function h(x)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$h(x)$\end{document} independent of u\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$u$\end{document} as u→+∞\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$u\rightarrow +\infty $\end{document}, e−cΔ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$e^{-c\Delta }$\end{document} is given by a power series with Δ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\Delta $\end{document} is the Euclidean Laplace operator. Here, the nonlinear term f(x,u)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$f(x,u)$\end{document} does not satisfy the usual condition: AR0≤F(x,u):=∫0uf(x,t)dt≤12+θf(x,u)u,\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ 0\le F(x,u):=\int _{0}^{u}f(x,t)\,dt\le \frac{1}{2+\theta }f(x,u)u, $$\end{document} for θ>0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\theta >0$\end{document} and |u|\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$|u|$\end{document} is large, which is important in using the mountain pass theorem, see Alves et al. (J. Differ. Equ. 323:229-252, 2022) and Corrêa et al. (J. Differ. Equ. 363:491-517, 2023). This paper is devoted to discuss how to use the mountain pass theorem to obtain the existence of nontrivial solution to problem (P) without the (AR) condition.
Stability of the Ionic Parameters of a Nonlocal FitzHugh-Nagumo Model of Cardiac ElectrophysiologyBen Abid, Narjess; Bendahmane, Mostafa; Mahjoub, Moncef
doi: 10.1007/s10440-024-00682-xpmid: N/A
This paper presents an inverse problem of identifying two ionic parameters of a nonlocal reaction-diffusion system in cardiac electrophysiology modelling. We used a nonlocal FitzHugh-Nagumo monodomain model which describes the electrical activity in cardiac tissue with the diffusion rate assumed to depend on the total electrical potential in the heart. We established at first, the global Carleman estimate adapted to nonlocal diffusion to obtain our main result which is the uniqueness and the Lipschitz stability estimate for two ionic parameters (k,γ)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$(k,\gamma )$\end{document}.