arXiv subjects

Chen-Chih Lai

Publications and source records attributed to Chen-Chih Lai.

At least 19 recordsLinked to original sources

arXiv2026

Global Well-posedness of the 2D Stochastic Self-consistent Keller-Segel-Navier-Stokes System with Subcritical Cellular Mass

We consider a stochastic Keller-Segel-Navier-Stokes system in $R^2$ describing the collective motion of cells in an ambient stochastic fluid flow, where the cells are attracted by a chemical substance and transported by the ambient fluid velocity, and the fluid motion is self-consistently driven by forces induced by the cells. We prove the existence of a unique mild solution globally-in-time to the two-dimensional stochastic Keller-Segel-Navier-Stokes system with subcritical mass.

math.AP↗

arXiv2025

Joint inviscid, incompressible, and continuous phenotype limit in nonlocal models of tissue growth

We rigorously derive the joint limit of vanishing viscosity, singular pressure, and discrete-to-continuous phenotype structure in a class of tissue growth models. Starting from a viscoelastic (Brinkman-type) system describing multiple interacting phenotypes, where pressure depends nonlinearly on total cell density, we establish convergence to an incompressible Darcy-type model with a continuous phenotypic structure. Our analysis builds upon recent advances on the Brinkman-to-Darcy limit and incompressible transitions by David, Jacob, and Kim [arXiv:2503.18870], and on hydrodynamic limits for phenotype-structured populations by Debiec, Mandal, and Schmidtchen [J. Differential Equations 2025]. A key novelty lies in combining all three singular limits simultaneously, under uniform a priori estimates, compactness in space-phenotype-time, and a generalized entropy-dissipation structure. This provides a unified framework for modeling constrained tissue growth with mechanical feedback and phenotypic plasticity.

math.AP↗

arXiv2025

Moment Estimates and DeepRitz Methods on Learning Diffusion Systems with Non-gradient Drifts

Conservative-dissipative dynamics are ubiquitous across a variety of complex open systems. We propose a data-driven two-phase method, the Moment-DeepRitz Method, for learning drift decompositions in generalized diffusion systems involving conservative-dissipative dynamics. The method is robust to noisy data, adaptable to rough potentials and oscillatory rotations. We demonstrate its effectiveness through several numerical experiments.

cs.LG↗

arXiv2025

Moment Estimate and Variational Approach for Learning Generalized Diffusion with Non-gradient Structures

This paper proposes a data-driven learning framework for identifying governing laws of generalized diffusions with non-gradient components. By combining energy dissipation laws with a physically consistent penalty and first-moment evolution, we design a two-stage method to recover the pseudo-potential and rotation in the pointwise orthogonal decomposition of a class of non-gradient drifts in generalized diffusions. Our two-stage method is applied to complex generalized diffusion processes including dissipation-rotation dynamics, rough pseudo-potentials and noisy data. Representative numerical experiments demonstrate the effectiveness of our approach for learning physical laws in non-gradient generalized diffusions.

physics.comp-ph↗

arXiv2025

Nonspherically symmetric equilibrium bubbles in a steadily rotating incompressible fluid

This note presents two nontrivial, rotational equilibrium solutions to the spatial uniform gas pressure (isobaric) approximate model of Prosperetti in the inviscid case. Building on Gavrilov's work [GAFA 2019], we first establish the existence of equilibrium solutions with nontrivial (rotational) liquid flow. Second, we construct a nonspherically symmetric, horn-torus-shaped equilibrium bubble under mild spatial decay conditions of the liquid flow. In addition, we extend earlier results on the characterization of spherical equilibrium bubbles to the axisymmetric, purely azimuthal setting. Finally, we implement a numerical simulation of the equilibrium bubble shape using the Physics-Informed Neural Network (PINN) approximation.

math.AP↗

arXiv2025

Weak and mild solutions to the MHD equations and the viscoelastic Navier-Stokes equations with damping in Wiener amalgam spaces

We study the three-dimensional incompressible magnetohydrodynamic (MHD) equations and the incompressible viscoelastic Navier-Stokes equations with damping. Building on techniques developed by Bradshaw, Lai, and Tsai (Math. Ann. 2024), we prove the existence of mild solutions in Wiener amalgam spaces that satisfy the corresponding spacetime integral bounds. In addition, we construct global-in-time local energy weak solutions in these amalgam spaces using the framework introduced by Bradshaw and Tsai (SIAM J. Math. Anal. 2021). As part of this construction, we also establish several properties of local energy solutions with $L^2_{\rm uloc}$ initial data, including initial and eventual regularity as well as small-large uniqueness, extending analogous results obtained for the Navier-Stokes equations by Bradshaw and Tsai (Comm. Partial Differential Equations 2020).

math.AP↗

arXiv2024

Asymmetric deformations of a perturbed spherical bubble in an incompressible fluid

We study the dynamics of a gas bubble in a fluid with surface tension, initially near a spherical equilibrium. While there are many studies and applications of radial bubble dynamics, the theory of general deformations from a spherical equilibrium is less developed. We aim to understand how asymmetrically perturbed equilibrium bubbles evolve toward spherical equilibrium due to thermal or viscous dissipation in an incompressible liquid. We focus on the isobaric approximation [Prosperetti, JFM, 1991], under which the gas pressure within the bubble is spatially uniform and obeys the ideal gas law. The liquid outside the bubble is incompressible, irrotational, and has surface tension. We prove that any equilibrium gas bubble must be spherical by showing that the bubble boundary is a closed surface of constant mean curvature. We then study the initial value problem (IVP) for the coupled PDEs, constitutive laws and interface conditions of the isobaric approximation for general (asymmetric) small initial perturbations of the spherical bubble in the linearized approximation. Our first result, considering thermal damping without viscosity, proves that the linearized IVP is globally well-posed. The monopole (radial) component of the perturbation decays exponentially over time, while the multipole (non-radial) components undergo undamped oscillations. This indicates a limitation of the isobaric model for non-spherical dynamics. Our second result, incorporating viscous dissipation, shows that the IVP is linear and nonlinearly ill-posed due to an incompatibility of normal stress boundary conditions, for non-spherical solutions, and the irrotationality assumption. Our study concludes that to accurately capture the dynamics of general deformations of a gas bubble, the model must account for either vorticity generated at the bubble-fluid boundary, spatial non-uniformities in the gas pressure, or both.

math.AP↗

arXiv2024

Applications of the Green tensor estimates of the nonstationary Stokes system in the half space

In this paper, we present a series of applications of the pointwise estimates of the (unrestricted) Green tensor of the nonstationary Stokes system in the half space, established in our previous work [CMP 2023]. First, we show the $L^1$-$L^q$ estimates for the Stokes flow with possibly non-solenoidal $L^1$ initial data, generalizing the results of Giga-Matsui-Shimizu [Math. Z. 1999] and Desch-Hieber-Prüss [J. Evol. Equ. 2001]. Second, we construct mild solutions of the Navier-Stokes equations in the half space with mixed-type pointwise decay or with pointwise decay alongside boundary vanishing. Finally, we explore various coupled fluid systems in the half space including viscous resistive magnetohydrodynamics equations, a coupled system for the flow and the magnetic field of MHD type, and the nematic liquid crystal flow. For each of these systems, we construct mild solutions in $L^q$, pointwise decay, and uniformly local $L^q$ spaces.

math.AP↗

arXiv2023

Free boundary problem for a gas bubble in a liquid, and exponential stability of the manifold of spherically symmetric equilibria

We consider the dynamics of a gas bubble immersed in an incompressible fluid of fixed temperature, and focus on the relaxation of an expanding and contracting spherically symmetric bubble due to thermal effects. We study two models, both systems of PDEs with an evolving free boundary: the full mathematical model as well as an approximate model, arising for example in the study of sonoluminescence. For fixed physical parameters (surface tension of the gas-liquid interface, liquid viscosity, thermal conductivity of the gas, etc.), both models share a family of spherically symmetric equilibria, smoothly parametrized by the mass of the gas bubble. Our main result concerns the approximate model. We prove the nonlinear asymptotic stability of the manifold of equilibria with respect to small spherically symmetric perturbations. The rate of convergence is exponential in time. To prove this result we first prove a weak form of nonlinear asymptotic stability -- with no explicit rate of time-decay -- using the energy dissipation law, and then, via a center manifold analysis, bootstrap the weak time-decay to exponential time-decay. We also study the uniqueness of the family of spherically symmetric equilibria within each model. The family of spherically symmetric equilibria captures all regular spherically symmetric equilibria of the approximate system. However within the full model, this family is embedded in a larger family of spherically symmetric solutions. For the approximate system, we prove that all equilibrium bubbles are spherically symmetric, by an application of Alexandrov's theorem on closed surfaces of constant mean curvature.

math.AP↗

arXiv2023

Global Existence and Aggregation of Chemotaxis-fluid Systems in Dimension Two

To describe the cellular self-aggregation phenomenon, some strongly coupled PDEs named as Keller-Segel (KS) and Patlak-Keller-Segel (PKS) systems were proposed in 1970s. Since KS and PKS systems possess relatively simple structures but admit rich dynamics, plenty of scholars have studied them and obtained many significant results. However, the cells in general direct their movement in liquid. As a consequence, it seems more realistic to consider the influence of ambient fluid flow on the chemotactic mechanism. Motivated by this, He et al. (SIAM J. Math. Anal., Vol. 53, No. 3, 2021) proposed a coupled Patlak-Keller-Segel-Navier-Stokes system that features the effect of the friction induced by the cells on the ambient fluid flow. In their pioneer work, the global existence of solutions of such system in 2D was established when the initial mass is strictly less than a threshold, which is referred to as the subcritical case. The last two authors and Zhou (Indiana Univ. Math. J., Vol. 72, No. 1, 2023) extended their result to the critical case. To our best knowledge, this system has only been studied in either the whole space or periodic domains. In this paper, we consider the chemotaxis-fluid system in two-dimensional bounded domains, in which the boundary conditions are Neumann conditions for the cell density and the chemical concentration, and the Navier slip boundary condition with zero friction for the fluid velocity. We prove that the solution of the system exists globally in time with the subcritical mass. Concerning the critical mass case, we construct the boundary spot equilibrium rigorously via the inner-outer gluing method. In particular, we develop the global $W^{2,p}$ theory for the 2D stationary Stokes system subject to Navier boundary conditions and further establish semigroup estimates of the nonstationary counterpart by analyzing the Stokes eigenvalue problem.

math.AP↗

arXiv2023

Thermal relaxation toward equilibrium and periodically pulsating spherical bubbles in an incompressible liquid

We study the radial relaxation dynamics toward equilibrium and time-periodic pulsating spherically symmetric gas bubbles in an incompressible liquid due to thermal effects. The asymptotic model ([A. Prosperetti, J. Fluid Mech., 1991] and [Z. Biro and J. J. L. Velazquez, SIAM J. Math. Anal., 2000]) is one where the pressure within the gas bubble is spatially uniform and satisfies an ideal gas law, relating the pressure, density and temperature of the gas. The temperature of the surrounding liquid is taken to be constant and the behavior of the liquid pressure at infinity is prescribed to be constant or periodic in time. In arXiv:2207.04079, for the case where the liquid pressure at infinity is a positive constant, we proved the existence of a one-parameter manifold of spherical equilibria, parameterized by the bubble mass, and further proved that it is a nonlinearly and exponentially asymptotically stable center manifold. In the present article, we first refine the exponential time-decay estimates, via a study of the linearized dynamics subject to the constraint of fixed mass. We obtain, in particular, estimates for the exponential decay rate constant, which highlight the interplay between the effects of thermal diffusivity and the liquid viscosity. We then study the nonlinear radial dynamics of the bubble-fluid system subject to a pressure field at infinity which is a small-amplitude and time-periodic perturbation about a positive constant. We prove that nonlinearly and exponentially asymptotically stable time-periodically pulsating solutions of the nonlinear (asymptotic) model exist for all sufficiently small forcing amplitudes. The existence of such states is formulated as a fixed point problem for the Poincaré return map, and the existence of a fixed point makes use of our (constant mass constrained) exponential time-decay estimates of the linearized problem.

math.AP↗

arXiv2022

The Green tensor of the nonstationary Stokes system in the half space

We prove the first ever pointwise estimates of the (unrestricted) Green tensor and the associated pressure tensor of the nonstationary Stokes system in the half-space, for every space dimension greater than one. The force field is not necessarily assumed to be solenoidal. The key is to find a suitable Green tensor formula which maximizes the tangential decay, showing in particular the integrability of Green tensor derivatives. With its pointwise estimates, we show the symmetry of the Green tensor, which in turn improves pointwise estimates. We also study how the solutions converge to the initial data, and the (infinitely many) restricted Green tensors acting on solenoidal vector fields. As applications, we give new proofs of existence of mild solutions of the Navier-Stokes equations in $L^q$, pointwise decay, and uniformly local $L^q$ spaces in the half-space.

math.AP↗

arXiv2022

Mild solutions and spacetime integral bounds for Stokes and Navier-Stokes flows in Wiener amalgam spaces

We first prove decay estimates and spacetime integral bounds for Stokes flows in amalgam spaces $E^r_q$ which connect the classical Lebesgue spaces to the spaces of uniformly locally $r$-integrable functions. Using these estimates, we construct mild solutions of the Navier-Stokes equations in the amalgam spaces satisfying the corresponding spacetime integral bounds. Time-global solutions are constructed for small data in $E^3_q$, $1\le q \le 3$. Our results provide new bounds for the strong solutions classically constructed by Kato and the more recent solutions in uniformly local spaces constructed by Maekawa and Terasawa. As an application we obtain a result on the stability of suitability for weak solutions to the perturbed Navier-Stokes equation where the drift velocity solves the Navier-Stokes equations and has small data in a local $L^3$ class. Extending an earlier result, we also construct global-in-time local energy weak solutions in $E^2_q$, $1\le q <2$.

math.AP↗

arXiv2021

Finite energy Navier-Stokes flows with unbounded gradients induced by localized flux in the half-space

For the Stokes system in the half space, Kang [Math.~Ann.~2005] showed that a solution generated by a compactly supported, Hölder continuous boundary flux may have unbounded normal derivatives near the boundary. In this paper we first prove explicit global pointwise estimates of the above solution, showing in particular that it has finite global energy and its derivatives blow up everywhere on the boundary away from the flux. We then use the above solution as a profile to construct solutions of the Navier-Stokes equations which also have finite global energy and unbounded normal derivatives due to the flux. Our main tool is the pointwise estimates of the Green tensor of the Stokes system proved by us in \cite{Green} (arXiv:2011.00134). We also examine the Stokes flows generated by dipole bumps boundary flux, and identify the regions where the normal derivatives of the solutions tend to positive or negative infinity near the boundary.

math.AP↗

arXiv2021

Global existence of free-energy solutions to the 2D Patlak--Keller--Segel--Navier--Stokes system with critical and subcritical mass

We consider a coupled Patlak-Keller-Segel-Navier-Stokes system in $\mathbb{R}^2$ that describes the collective motion of cells and fluid flow, where the cells are attracted by a chemical substance and transported by ambient fluid velocity, and the fluid flow is forced by the friction induced by the cells. The main result of the paper is to show the global existence of free-energy solutions to the 2D Patlak-Keller-Segel-Navier-Stokes system with critical and subcritical mass.

math.AP↗

arXiv2020

New type II Finite time blow-up for the energy supercritical heat equation

We consider the energy supercritical heat equation with the $(n-3)$-th Sobolev exponent \begin{equation*} \begin{cases} u_t=Δu+u^{3},~&\mbox{ in } Ω\times (0,T),\\ u(x,t)=u|_{\partialΩ},~&\mbox{ on } \partialΩ\times (0,T),\\ u(x,0)=u_0(x),~&\mbox{ in } Ω, \end{cases} \end{equation*} where $5\leq n\leq 7$, $Ω=\R^n$ or $Ω\subset \R^n$ is a smooth, bounded domain enjoying special symmetries. We construct type II finite time blow-up solution $u(x,t)$ with the singularity taking place along an $(n-4)$-dimensional {\em shrinking sphere} in $Ω$. More precisely, at leading order, the solution $u(x,t)$ is of the sharply scaled form $$u(x,t)\approx \la^{-1}(t)\frac{2\sqrt{2}}{1+\left|\frac{(r,z)-(ξ_r(t),ξ_z(t))}{\la(t)}\right|^2}$$ where $r=\sqrt{x_1^2+\cdots+x_{n-3}^2}$, $z=(x_{n-2},x_{n-1},x_n)$ with $x=(x_1,\cdots,x_n)\inΩ$. Moreover, the singularity location $$(ξ_r(t),ξ_z(t))\sim (\sqrt{2(n-4)(T-t)},z_0)~\mbox{ as }~t\nearrow T,$$ for some fixed $z_0$, and the blow-up rate $$\la(t)\sim \frac{T-t}{|\log(T-t)|^2}~\mbox{ as }~t\nearrow T.$$ This is a completely new phenomenon in the parabolic setting.

math.AP↗

arXiv2019

Finite time blow-up for the nematic liquid crystal flow in dimension two

We consider the initial-boundary value problem of a simplified nematic liquid crystal flow in a bounded, smooth domain $Ω\subset \mathbb R^2$. Given any $k$ distinct points in the domain, we develop a new {\em inner--outer gluing method} to construct solutions which blow up exactly at those $k$ points as $t$ goes to a finite time $T$. Moreover, we obtain a precise description of the blow-up.

math.AP↗

arXiv2019

Forward Discretely Self-Similar Solutions of the MHD Equations and the Viscoelastic Navier-Stokes Equations with Damping

In this paper, we prove the existence of forward discretely self-similar solutions to the MHD equations and the viscoelastic Navier-Stokes equations with damping with large weak $L^3$ initial data. The same proving techniques are also applied to construct self-similar solutions to the MHD equations and the viscoelastic Navier-Stokes equations with damping with large weak $L^3$ initial data. This approach is based on [Z. Bradshaw and T.-P. Tsai, Ann. Henri Poincar'{e}, vol. 18, no. 3, 1095-1119, 2017].

math.AP↗