More Like this

1. Gradient flow approach to an exponential thin film equation: global existence and latent singularity

   https://doi.org/10.1051/cocv/2018037  

   Gao, Yuan; Liu, Jian-Guo; Lu, Xin Yang (January 2019, ESAIM: Control, Optimisation and Calculus of Variations)

   null (Ed.)

   In this work, we study a fourth order exponential equation, u t = Δ e −Δ u derived from thin film growth on crystal surface in multiple space dimensions. We use the gradient flow method in metric space to characterize the latent singularity in global strong solution, which is intrinsic due to high degeneration. We define a suitable functional, which reveals where the singularity happens, and then prove the variational inequality solution under very weak assumptions for initial data. Moreover, the existence of global strong solution is established with regular initial data. 

   more » « less
2. Global well‐posedness for the one‐phase Muskat problem

   https://doi.org/10.1002/cpa.22124  

   Dong, Hongjie; Gancedo, Francisco; Nguyen, Huy Q. (December 2023, Communications on Pure and Applied Mathematics)

   Abstract The free boundary problem for a two‐dimensional fluid permeating a porous medium is studied. This is known as the one‐phase Muskat problem and is mathematically equivalent to the vertical Hele‐Shaw problem driven by gravity force. We prove that if the initial free boundary is the graph of a periodic Lipschitz function, then there exists a global‐in‐time Lipschitz solution in the strong sense and it is the unique viscosity solution. The proof requires quantitative estimates for layer potentials and pointwise elliptic regularity in Lipschitz domains. This is the first construction of unique global strong solutions for the Muskat problem with initial data of arbitrary size. 

   more » « less
3. Stability of a Subcritical Fluid Model for Fair Bandwidth Sharing with General File Size Distributions

   https://doi.org/10.1287/stsy.2019.0058  

   Fu, Yingjia; Williams, Ruth J. (September 2020, Stochastic Systems)

   null (Ed.)

   This work concerns the asymptotic behavior of solutions to a (strictly) subcritical fluid model for a data communication network, where file sizes are generally distributed and the network operates under a fair bandwidth-sharing policy. Here we consider fair bandwidth-sharing policies that are a slight generalization of the [Formula: see text]-fair policies introduced by Mo and Walrand [Mo J, Walrand J (2000) Fair end-to-end window-based congestion control. IEEE/ACM Trans. Networks 8(5):556–567.]. Since the year 2000, it has been a standing problem to prove stability of the data communications network model of Massoulié and Roberts [Massoulié L, Roberts J (2000) Bandwidth sharing and admission control for elastic traffic. Telecommunication Systems 15(1):185–201.], with general file sizes and operating under fair bandwidth sharing policies, when the offered load is less than capacity (subcritical conditions). A crucial step in an approach to this problem is to prove stability of subcritical fluid model solutions. In 2012, Paganini et al. [Paganini F, Tang A, Ferragut A, Andrew LLH (2012) Network stability under alpha fair bandwidth allocation with general file size distribution. IEEE Trans. Automatic Control 57(3):579–591.] introduced a Lyapunov function for this purpose and gave an argument, assuming that fluid model solutions are sufficiently smooth in time and space that they are strong solutions of a partial differential equation and assuming that no fluid level on any route touches zero before all route levels reach zero. The aim of the current paper is to prove stability of the subcritical fluid model without these strong assumptions. Starting with a slight generalization of the Lyapunov function proposed by Paganini et al., assuming that each component of the initial state of a measure-valued fluid model solution, as well as the file size distributions, have no atoms and have finite first moments, we prove absolute continuity in time of the composition of the Lyapunov function with any subcritical fluid model solution and describe the associated density. We use this to prove that the Lyapunov function composed with such a subcritical fluid model solution converges to zero as time goes to infinity. This implies that each component of the measure-valued fluid model solution converges vaguely on [Formula: see text] to the zero measure as time goes to infinity. Under the further assumption that the file size distributions have finite pth moments for some p > 1 and that each component of the initial state of the fluid model solution has finite pth moment, it is proved that the fluid model solution reaches the measure with all components equal to the zero measure in finite time and that the time to reach this zero state has a uniform bound for all fluid model solutions having a uniform bound on the initial total mass and the pth moment of each component of the initial state. In contrast to the analysis of Paganini et al., we do not need their strong smoothness assumptions on fluid model solutions and we rigorously treat the realistic, but singular situation, where the fluid level on some routes becomes zero, whereas other route levels remain positive. 

   more » « less
4. Self-Improving inequalities for bounded weak solutions to nonlocal double phase equations

   https://doi.org/10.3934/cpaa.2021174  

   Scott, James M.; Mengesha, Tadele (January 2021, Communications on Pure & Applied Analysis)

   We prove higher Sobolev regularity for bounded weak solutions to a class of nonlinear nonlocal integro-differential equations. The leading operator exhibits nonuniform growth, switching between two different fractional elliptic "phases" that are determined by the zero set of a modulating coefficient. Solutions are shown to improve both in integrability and differentiability. These results apply to operators with rough kernels and modulating coefficients. To obtain these results we adapt a particular fractional version of the Gehring lemma developed by Kuusi, Mingione, and Sire in their work "Nonlocal self-improving properties" Analysis & PDE, 8(1):57–114 for the specific nonlinear setting under investigation in this manuscript. 

   more » « less
5. A Note on Estimates of Level Sets and Their Role in Demonstrating Regularity of Solutions to Nonlocal Double-Phase Equations

   https://doi.org/10.1007/978-3-031-34089-5_7  

   Mengesha, Tadele; Scott, J.M. (May 2023, The IMA volumes in mathematics and its applications)

   Mengesha, Tadele; Salgado, Abner J. (Ed.)

   In this note, we prove an estimate on the level sets of a function with (𝑝,𝑞) growth that depends on the difference quotient of a bounded weak solution to a nonlocal double-phase equation. This estimate is related to a self-improving property of these solutions. 

   more » « less