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We establish the well/ill-posedness theories for the inviscid $\alpha$-surface quasi-geostrophic ($\alpha$-SQG) equations in H\"older spaces, where $\alpha = 0$ and $\alpha = 1$ correspond to the two-dimensional Euler equation in the vorticity formulation and SQG equation of geophysical significance, respectively. We first prove the local-in-time well-posedness of $\alpha$-SQG equations in $C([0,T);C^{0,\beta}(\mathbb{R}^2))$ with $\beta \in (\alpha,1)$ for some $T>0$. We then analyze the strong ill-posedness in $C^{0,\alpha}(\mathbb{R}^2)$ constructing smooth solutions to the $\alpha$-SQG equations that exhibit $C^{0,\alpha}$--norm growth in a short time. In particular, we develop the nonexistence theory for $\alpha$-SQG equations in $C^{0,\alpha}(\mathbb{R}^2)$.

In this paper, we derive decay rates of the solutions to the incompressible Navier-Stokes equations and Hall-magnetohydrodynamic equations. We first improve the decay rate of weak solutions of these equations by refining the Fourier splitting method with initial data in the space of pseudo-measures. We also deal with these equations with initial data in Lei-Lin spaces and find decay rates of solutions in Lei-Lin spaces., Comment: 17 pages

We completely resolve the global Cauchy problem for the multi-dimensional Euler-Riesz equations, where the interaction forcing is given by $\nabla (-\Delta)^{-\sigma/2}\rho$ for some $\sigma \in (0,2)$. We construct the global-in-time unique solution to the Euler-Riesz system in a $H^s$ Sobolev space under a smallness assumption on the initial density and a dispersive spectral condition on the initial velocity. Moreover, we investigate the algebraic time decay of convergences for the constructed solutions. Our results cover the both attractive and repulsive cases as well as the whole regime $\sigma \in (0,2)$., Comment: 40 pages

We consider the barotropic Euler equations with pairwise attractive Riesz interactions and linear velocity damping in the periodic domain. We establish the global-in-time well-posedness theory for the system near an equilibrium state. We also analyze the large-time behavior of solutions showing the exponential rate of convergence toward the equilibrium state as time goes to infinity., Comment: 24 pages

We propose a systematic framework to conduct design-technology pathfinding for PPAC in advanced nodes. Our goal is to provide configurable, scalable generation of process design kit (PDK) and standard-cell library, spanning key scaling boosters (backside PDN and buried power rail), to explore PPAC across given technology and design parameters. We build on PROBE2.0, which addressed only area and cost (AC), to include power and performance (PP) evaluations through automated generation of full design enablements. We also improve the use of artificial designs in the PPAC assessment of technology and design configurations. We generate more realistic artificial designs by applying a machine learning-based parameter tuning flow. We further employ clustering-based cell width-regularized placements at the core of routability assessment, enabling more realistic placement utilization and improved experimental efficiency. We demonstrate PPAC evaluation across scaling boosters and artificial designs in a predictive technology node., Comment: 14 pages, 17 figures, submitted to IEEE Trans. on CAD

Sprays are complex flows constituted of dispersed particles in an underlying gas. In this paper, we are interested in the equations for moderately thick sprays consisting of the compressible Navier-Stokes equations and Boltzmann BGK equation. Here the coupling of two equations is through a friction (or drag) force which depends on the density of compressible fluid and the relative velocity between particles and fluid. For the Navier-Stokes-BGK system, we establish the existence and uniqueness of solutions in Sobolev spaces with exponential weight.

In this paper, we present a refined framework for the global-in-time well-posedness theory for the pressureless Euler--Navier--Stokes system and the optimal temporal decay rates of certain norms of solutions. Here the coupling of two systems, pressureless Euler system and incompressible Navier--Stoke system, is through the drag force. We construct the global-in-time existence and uniqueness of regular solutions for the pressureless Euler--Navier--Stokes system without using a priori large time behavior estimates. Moreover, we seek for the optimal Sobolev regularity for the solutions. Concerning the temporal decay for solutions, we show that the fluid velocities exhibit the same decay rate as that of the heat equations. In particular, our result provides that the temporal decay rate of difference between two velocities, which is faster than the fluid velocities themselves, are at least the same as the second-order derivatives of fluid velocities.

In this paper, we provide modulated interaction energy estimates for the kernel $K(x) = |x|^{-\alpha}$ with $\alpha \in (0,d)$, and its applications to quantified asymptotic analyses for kinetic equations. The proof relies on a dimension extension argument for an elliptic operator and its commutator estimates. For the applications, we first discuss the quantified small inertia limit of kinetic equation with singular nonlocal interactions. The aggregation equations with singular interaction kernels are rigorously derived. We also study the rigorous quantified hydrodynamic limit of the kinetic equation to derive the isothermal Euler or pressureless Euler system with the nonlocal singular interactions forces.

We analyze the Vlasov equation coupled with the compressible Navier--Stokes equations with degenerate viscosities and vacuum. These two equations are coupled through the drag force which depends on the fluid density and the relative velocity between particle and fluid. We first establish the existence and uniqueness of local-in-time regular solutions with arbitrarily large initial data and a vacuum. We then present sufficient conditions on the initial data leading to the finite-time blowup of regular solutions. In particular, our study makes the result on the finite-time singularity formation for Vlasov/Navier-Stokes equations discussed by Choi [J. Math. Pures Appl., 108, (2017), 991-1021] completely rigorous.

In this paper, we analyze the pressureless damped Euler-Riesz equations posed in either $\mathbb{R}^d$ or $\mathbb{T}^d$. We construct the global-in-time existence and uniqueness of classical solutions for the system around a constant background state. We also establish large-time behaviors of classical solutions showing the solutions towards the equilibrium as time goes to infinity. For the whole space case, we first show the algebraic decay rate of solutions under additional assumptions on the initial data compared to the existence theory. We then refine the argument to have the exponential decay rate of convergence even in the whole space. In the case of the periodic domain, without any further regularity assumptions on the initial data, we provide the exponential convergence of solutions.

In this work, we study the global existence of strong solutions and large-time behavior of a two-phase fluid model in a bounded domain. The model consists of the isothermal Euler equations and the isentropic compressible Navier--Stokes equations, coupled via the drag force. It was derived in [13[ from a kinetic-fluid model describing the dynamics of particles subject to local alignment force and Brownian noises immersed in a compressible viscous fluid. For this system, we extend the local existence theory for strong solutions developed in [13] to obtain the global existence of strong solutions to the system. Moreover, we use the Lyapunov functional associated with the system to get large-time behavior estimates for global classical solutions.

In this paper, we are interested in the dynamics of charged particles interacting with the incompressible viscous flow. More precisely, we consider the Vlasov-Poisson or Vlasov-Poisson-Fokker-Planck equation coupled with the incompressible Navier-Stokes system through the drag force. For the proposed kinetic-fluid model, we study the asymptotic regime corresponding to strong local alignment and diffusion forces. Under suitable assumptions on the initial data, we rigorously derive a coupled isothermal/pressureless Euler-Poisson system and incompressible Navier-Stokes system(in short, EPNS system). For this hydrodynamic limit, we employ the modulated kinetic energy estimate together with the relative entropy method and the bounded Lipschitz distance. We also construct a global-in-time strong solvability for the isothermal/pressureless EPNS system. In particular, this global-in-time solvability gives the estimates of hydrodynamic limit hold for all time., Comment: The paper is reorganized; the introduction is updated and the large-time behavior estimate for the EPNS system is removed. The large-time behavior estimate will be addressed in another paper

In this paper, we study the global Cauchy problem for a two-phase fluid model consisting of the pressureless Euler equations and the incompressible Navier-Stokes equations where the coupling of two equations is through the drag force. We establish the global-in-time existence and uniqueness of classical solutions for that system when the initial data are sufficiently small and regular. Main difficulties arise in the absence of pressure in the Euler equations. In order to resolve it, we properly combine the large-time behavior of classical solutions and the bootstrapping argument to construct the global-in-time unique classical solutions.

In this paper, we quantify the asymptotic limit of collective behavior kinetic equations arising in mathematical biology modeled by Vlasov-type equations with nonlocal interaction forces and alignment. More precisely, we investigate the hydrodynamic limit of a kinetic Cucker--Smale flocking model with confinement, nonlocal interaction, and local alignment forces, linear damping and diffusion in velocity. We first discuss the hydrodynamic limit of our main equation under strong local alignment and diffusion regime, and we rigorously derive the isothermal Euler equations with nonlocal forces. We also analyze the hydrodynamic limit corresponding to strong local alignment without diffusion. In this case, the limiting system is pressureless Euler-type equations. Our analysis includes the Coulombian interaction potential for both cases and explicit estimates on the distance towards the limiting hydrodynamic equations. The relative entropy method is the crucial technology in our main results, however, for the case without diffusion, we combine a modulated macroscopic kinetic energy with the bounded Lipschitz distance to deal with the nonlocality in the interaction forces. For the sake of completeness, the existence of weak and strong solutions to the kinetic and fluid equations are also established.

We study an asymptotic analysis of a coupled system of kinetic and fluid equations. More precisely, we deal with the nonlinear Vlasov-Fokker-Planck equation coupled with the compressible isentropic Navier-Stokes system through a drag force in a bounded domain with the specular reflection boundary condition for the kinetic equation and homogeneous Dirichlet boundary condition for the fluid system. We establish a rigorous hydrodynamic limit corresponding to strong noise and local alignment force. The limiting system is a type of two-phase fluid model consisting of the isothermal Euler system and the compressible Navier-Stokes system. Our main strategy relies on the relative entropy argument based on the weak-strong uniqueness principle. For this, we provide a global-in-time existence of weak solutions for the coupled kinetic-fluid system. We also show the existence and uniqueness of strong solutions to the limiting system in a bounded domain with the kinematic boundary condition for the Euler system and Dirichlet boundary condition for the Navier-Stokes system.

We study emergent dynamics of the discrete Cucker-Smale (in short, DCS) model with randomly switching network topologies. For this, we provide a sufficient framework leading to the stochastic flocking with probability one. Our sufficient framework is formulated in terms of an admissible set of network topologies realized by digraphs and probability density function for random switching times. As examples for the law of switching times, we use the Poisson process and the geometric process and show that these two processes satisfy the required conditions in a given framework so that we have a stochastic flocking with probability one. As a corollary of our flocking analysis, we improve the earlier result [J.-G. Dong, S.-Y. Ha, J. Jung and D. Kim: On the stochastic flocking of the Cucker-Smale flock with randomly switching topologies. arXiv:1911.07390.] on the continuous C-S model.

We present an emergent stochastic flocking dynamics of the Cucker-Smale (CS) ensemble under randomly switching topologies. The evolution of the CS ensemble with randomly switching topologies involves two random components (switching times and choices of network topologies at switching instant). First, we allow switching times for the network topology to be random so that the successive increments are i.i.d. processes following the common probability distribution. Second, at each switching instant, we choose a network topology randomly from a finite set of admissible network topologies whose union contains a spanning tree. Even for the fixed deterministic network topology, the CS ensemble may not exhibit a mono-cluster flocking depending on the initial data and the decay mode of the communication weight functions measuring the degree of interactions between particles. For the flocking dynamics of the CS ensemble with these two random components, we first use a priori conditions on the network topologies and uniform boundedness of position diameter, and derive the flocking estimates via matrix theory together with a priori conditions, and then replace the a priori condition for the position diameter by some suitable condition on the system parameters and communication weight. The a priori condition on the network topology will be guaranteed by the suitable spanning tree time-blocks with probability one., Comment: 18 pages

We present collective dynamics of the Cucker-Smale (C-S) ensemble under random communications. As an effective modeling of the C-S ensemble, we introduce a stochastic kinetic C-S equation with a multiplicative white noise. For the proposed stochastic kinetic model with a multiplicative noise, we present a global well-posedness of strong solutions and their asymptotic flocking dynamics, when initial datum is sufficiently regular, and communication weight function has a positive lower bound.

We study a hydrodynamic limit of a system of coupled kinetic and fluid equations under a strong local alignment force and a strong Brownian motion. More precisely, we consider the Vlasov-Fokker-Planck type equation and compressible Navier-Stokes equations with a density-dependent viscosity. Based on a relative entropy argument, by assuming the existence of weak solutions to that kinetic-fluid system, we rigorously derive a two-phase fluid model consisting of isothermal Euler equations and compressible Navier-Stokes equations with a density-dependent viscosity.

In this paper, we present DATC Robust Design Flow (RDF) from logic synthesis to detailed routing. Our goals are 1) to provide an open-source academic design flow from logic synthesis to detailed routing based on existing contest results, 2) to construct a database for design benchmarks and point tool libraries, and 3) to interact with industrial designs by using industrial standard design input/output formats. We also demonstrate RDF in a scalable cloud infrastructure. Design methodology and cross-stage optimization research can be conducted via RDF.