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In this paper, we treat the corner singularity expansion and its convergence result regarding the penalized system obtained by eliminating the pressure variable in the Stokes problem of incompressible flow. The penalized problem is a kind of the Lam\'{e} system, so we first discuss the corner singularity theory of the Lam\'{e} system with inhomogeneous Dirichlet boundary condition on a non-convex polygon. Considering the inhomogeneous condition, we show the decomposition of its solution, composed of singular parts and a smoother remainder near a re-entrant corner, and furthermore, we provide the explicit formulae of coefficients in singular parts. In particular, these formulae can be used in the development of highly accurate numerical scheme. In addition, we formulate coefficients in singular parts regarding the Stokes equations with inhomogeneous boundary condition and non-divergence-free property of velocity field, and thus we show the convergence results of coefficients in singular parts and remainder regarding the concerned penalized problem., Comment: 34 pages, 1 figure

We study the (generalized) one-dimensional population model developed by Anguige \& Schmeiser [1], which reflects cell-cell adhesion and volume filling under no-flux boundary condition. In this generalized model, depending on the adhesion and volume filling parameters $\alpha,\beta\in[0,1],$ the resulting equation is classified into six types. Among these, we focus on the type exhibiting strong effects of both adhesion and volume filling, which results in a class of advection-diffusion equations of the forward-backward-forward type. For five distinct cases of initial maximum, minimum and average population densities, we derive the corresponding patterns for the global behavior of weak solutions to the initial and no-flux boundary value problem. Due to the presence of a negative diffusion regime, we indeed prove that the problem is ill-posed and admits infinitely many global-in-time weak solutions, with the exception of one specific case of the initial datum. This nonuniqueness is inherent in the method of convex integration that we use to solve the Dirichlet problem of a partial differential inclusion arising from the ill-posed problem., Comment: 29 pages, 3 figures. arXiv admin note: substantial text overlap with arXiv:2409.04689

We generalize the one-dimensional population model of Anguige \& Schmeiser [1] reflecting the cell-to-cell adhesion and volume filling and classify the resulting equation into the six types. Among these types, we fix one that yields a class of advection-diffusion equations of forward-backward-forward type and prove the existence of infinitely many global-in-time weak solutions to the initial-Dirichlet boundary value problem when the maximum value of an initial population density exceeds a certain threshold. Such solutions are extracted from the method of convex integration by M\"uller \& \v Sver\'ak [12]; they exhibit fine-scale density mixtures over a finite time interval, then become smooth and identical, and decay exponentially and uniformly to zero as time approaches infinity., Comment: 40 pages, 4 figures

In this work, we establish the linear convergence estimate for the gradient descent involving the delay $\tau\in\mathbb{N}$ when the cost function is $\mu$-strongly convex and $L$-smooth. This result improves upon the well-known estimates in Arjevani et al. \cite{ASS} and Stich-Karmireddy \cite{SK} in the sense that it is non-ergodic and is still established in spite of weaker constraint of cost function. Also, the range of learning rate $\eta$ can be extended from $\eta\leq 1/(10L\tau)$ to $\eta\leq 1/(4L\tau)$ for $\tau =1$ and $\eta\leq 3/(10L\tau)$ for $\tau \geq 2$, where $L >0$ is the Lipschitz continuity constant of the gradient of cost function. In a further research, we show the linear convergence of cost function under the Polyak-{\L}ojasiewicz\,(PL) condition, for which the available choice of learning rate is further improved as $\eta\leq 9/(10L\tau)$ for the large delay $\tau$. The framework of the proof for this result is also extended to the stochastic gradient descent with time-varying delay under the PL condition. Finally, some numerical experiments are provided in order to confirm the reliability of the analyzed results., Comment: A result for the delayed SGD was added. accepted in Analysis and Applications

We establish convergence results related to the operator splitting scheme on the Cauchy problem for the nonlinear Schr\"odinger equation with rough initial data in $L^2$, $$ \left\{ \begin{array}{ll} i\partial_t u +\Delta u = \lambda |u|^{p} u, & (x,t) \in \mathbb{R}^d \times \mathbb{R}_+, u (x,0) =\phi (x), & x\in\mathbb{R}^d, \end{array} \right. $$ where $\lambda \in \{-1,1\}$ and $p >0$. While the Lie approximation $Z_L$ is known to converge to the solution $u$ when the initial datum $\phi$ is sufficiently smooth, the convergence result for rough initial data is open to question. In this paper, for rough initial data $\phi\in L^2 (\mathbb{R}^d)$, we prove the convergence of the filtered Lie approximation $Z_{flt}$ to the solution $u$ in the mass-subcritical range, $\max\left\{1,\frac{2}{d}\right\} \leq p < \frac{4}{d}$. Furthermore, we provide a precise convergence result for radial initial data $\phi\in L^2 (\mathbb{R}^d)$, Comment: typos are fixed

In this paper, we analyze an operator splitting scheme of the nonlinear heat equation in $\Omega\subset\mathbb{R}^d$ ($d\geq 1$): $\partial_t u = \Delta u + \lambda |u|^{p-1} u$ in $\Omega\times(0,\infty)$, $u=0$ in $\partial\Omega\times(0,\infty)$, $u ({\bf x},0) =\phi ({\bf x})$ in $\Omega$. where $\lambda\in\{-1,1\}$ and $\phi \in W^{1,q}(\Omega)\cap L^{\infty} (\Omega)$ with $2\leq p < \infty$ and $d(p-1)/20$. Finally, we give some numerical examples to confirm the reliability of the analyzed result.