arXiv subjects

Qingtian Zhang

Publications and source records attributed to Qingtian Zhang.

18 recordsLinked to original sources

arXiv2025

Poiseuille flow of hyperbolic Ericksen-Leslie system in dimension two

In this paper, we study the Poiseuille laminar flow in a tube for the full Ericksen-Leslie system. It is a parabolic-hyperbolic coupled system which may develop singularity in finite time. We will prove the global existence of energy weak solution, and the partial regularity of solution to system. We first construct global weak finite energy solutions by the Ginzburg- Landau approximation and the fixed-point arguments. Then we obtain the enhanced regularity of solution. Different from the solution in one space dimension, the finite energy solution of Poiseuille laminar flow in a tube may still form a discontinuity at the origin. We show that at the first possible blowup time, there are blowup sequences which converge to a non-constant time-independent (axisymmetric) harmonic map.

math.AP↗

arXiv2023

Global solutions of quasi-linear Hamiltonian mKdV equation

We study the initial value problem of quasi-linear Hamiltonian mKdV equations. Our goal is to prove the global-in-time existence of a solution given sufficiently smooth, localized, and small initial data. To achieve this, we utilize the bootstrap argument, Sobolev energy estimates, and the dispersive estimate. This proof relies on the space-time resonance method, as well as a bilinear estimate developed by Germain, Pusateri, and Rousset.

math.AP↗

arXiv2023

Initial-boundary value problems for Poiseuille flow of nematic liquid crystal via full Ericksen-Leslie model

In this paper, we study the initial-boundary value problem for the Poiseuille flow of hyperbolic-parabolic Ericksen-Leslie model of nematic liquid crystals in one space dimension. Due to the quasilinearity, the solution of this model in general forms cusp singularity. We prove the global existence of Hölder continuous solution, which may include cusp singularity, for initial-boundary value problems with different types of boundary conditions.

math.AP↗

arXiv2022

Large-Scale Integrated Flexible Tactile Sensor Array for Sensitive Smart Robotic Touch

In the long pursuit of smart robotics, it has been envisioned to empower robots with human-like senses, especially vision and touch. While tremendous progress has been made in image sensors and computer vision over the past decades, the tactile sense abilities are lagging behind due to the lack of large-scale flexible tactile sensor array with high sensitivity, high spatial resolution, and fast response. In this work, we have demonstrated a 64x64 flexible tactile sensor array with a record-high spatial resolution of 0.9 mm (equivalently 28.2 pixels per inch), by integrating a high-performance piezoresistive film (PRF) with a large-area active matrix of carbon nanotube thin-film transistors. PRF with self-formed microstructures exhibited high pressure-sensitivity of ~385 kPa-1 for MWCNTs concentration of 6%, while the 14% one exhibited fast response time of ~3 ms, good linearity, broad detection range beyond 1400 kPa, and excellent cyclability over 3000 cycles. Using this fully integrated tactile sensor array, the footprint maps of an artificial honeybee were clearly identified. Furthermore, we hardware-implemented a smart tactile system by integrating the PRF-based sensor array with a memristor-based computing-in-memory chip to record and recognize handwritten digits and Chinese calligraphy, achieving high classification accuracies of 98.8% and 97.3% in hardware, respectively. The integration of sensor networks with deep learning hardware may enable edge or near-sensor computing with significantly reduced power consumption and latency. Our work could pave the road to building large-scale intelligent sensor networks for next-generation smart robotics.

cond-mat.mtrl-sci↗

arXiv2022

Global solutions of quasi-geostrophic shallow-water fronts

In this paper, we consider a family of piecewise constant solutions of the quasi-geostrophic shallow-water (QGSW) equation. We derive the contour dynamics equation of the QGSW front, which is a nonlinear, nonlocal dispersive equation, and prove the global existence of the solutions when the initial data is sufficiently close to a flat front.

math.AP↗

arXiv2021

On the approximation of vorticity fronts by the Burgers-Hilbert equation

This paper proves that the motion of small-slope vorticity fronts in the two-dimensional incompressible Euler equations is approximated on cubically nonlinear timescales by a Burgers-Hilbert equation derived by Biello and Hunter (2010) using formal asymptotic expansions. The proof uses a modified energy method to show that the contour dynamics equations for vorticity fronts in the Euler equations and the Burgers-Hilbert equation are both approximated by the same cubically nonlinear asymptotic equation. The contour dynamics equations for Euler vorticity fronts are also derived.

math.AP↗

arXiv2020

Global Solutions of a Surface Quasi-Geostrophic Front Equation

We consider a nonlinear, spatially-nonlocal initial value problem in one space dimension on $\mathbb{R}$ that describes the motion of surface quasi-geostrophic (SQG) fronts. We prove that the initial value problem has a unique local smooth solution under a convergence condition on the multilinear expansion of the nonlinear term in the equation, and, for sufficiently smooth and small initial data, we prove that the solution is global.

math.AP↗

arXiv2020

Two-Front Solutions of the SQG Equation and its Generalizations

The generalized surface quasi-geostrophic (GSQG) equations are transport equations for an active scalar that depend on a parameter $0<α\le 2$. Special cases are the two-dimensional incompressible Euler equations ($α= 2$) and the surface quasi-geostrophic (SQG) equations ($α= 1$). We derive contour-dynamics equations for a class of two-front solutions of the GSQG equations when the fronts are a graph. Scalar reductions of these equations include ones that describe a single front in the presence of a rigid, flat boundary. We use the contour dynamics equations to determine the linearized stability of the GSQG shear flows that correspond to two flat fronts. We also prove local-in-time existence and uniqueness for large, smooth solutions of the two-front equations in the parameter regime $1<α\le 2$, and small, smooth solutions in the parameter regime $0<α\le 1$.

math.AP↗

arXiv2020

Global solutions for a family of GSQG front equations

We prove the global existence of solutions with small and smooth initial data of a nonlinear dispersive equation for the motion of generalized surface quasi-geostrophic (GSQG) fronts in a parameter regime $1<α<2$, where $α=1$ corresponds to the SQG equation and $α=2$ corresponds to the incompressible Euler equations. This result completes previous global well-posedness results for $0<α\le 1$. We also use contour dynamics to derive the GSQG front equations for $1<α<2$.

math.AP↗

arXiv2019

Contour Dynamics for Surface Quasi-Geostrophic Fronts

We use contour dynamics to derive equations of motion for infinite planar surface quasi-geostrophic (SQG) fronts, and show that it leads to the same result as a regularization procedure introduced previously by Hunter and Shu (2018).

math.AP↗

arXiv2018

Local wellposedness of an approximate equation for SQG fronts

We prove local well-posedness in the Sobolev spaces $\dot H^s(\mathbb{T})$, with $s>7/2$, for an initial value problem for a nonlocal, cubically nonlinear, dispersive equation that provides an approximate description of the evolution of surface quasi-geostrophic (SQG) fronts with small slopes.

math.AP↗

arXiv2017

On finite time BV blow-up for the p-system

The paper studies the possible blowup of the total variation for entropy weak solutions of the p-system, modeling isentropic gas dynamics. It is assumed that the density remains uniformly positive, while the initial data can have arbitrarily large total variation (measured in terms of Riemann invariants). Two main results are proved. (I) If the total variation blows up in finite time, then the solution must contain an infinite number of large shocks in a neighborhood of some point in the $t$-$x$ plane. (II) Piecewise smooth approximate solutions can be constructed whose total variation blows up in finite time. For these solutions the strength of waves emerging from each interaction is exact, while rarefaction waves satisfy the natural decay estimates stemming from the assumption of genuine nonlinearity.

math.AP↗

arXiv2015

No BV bounds for approximate solutions to p-system with general pressure law

For the p-system with large BV initial data, an assumption introduced in [3] by Bakhvalov guarantees the global existence of entropy weak solutions with uniformly bounded total variation. The present paper provides a partial converse to this result. Whenever Bakhvalov's condition does not hold, we show that there exist front tracking approximate solutions, with uniformly positive density, whose total variation becomes arbitrarily large. The construction extends the arguments in [4] to a general class of pressure laws.

math.AP↗

arXiv2014

Unique Conservative Solutions to a Variational Wave Equation

Relying on the analysis of characteristics, we prove the uniqueness of conservative solutions to the variational wave equation $u_{tt}-c(u) (c(u)u_x)_x=0$. Given a solution $u(t,x)$, even if the wave speed $c(u)$ is only Hölder continuous in the $t$-$x$ plane, one can still define forward and backward characteristics in a unique way. Using a new set of independent variables $X,Y$, constant along characteristics, we prove that $t,x,u$, together with other variables, satisfy a semilinear system with smooth coefficients. From the uniqueness of the solution to this semilinear system, one obtains the uniqueness of conservative solutions to the Cauchy problem for the wave equation with general initial data $u(0,\cdot)\in H^1(\mathbb{R})$, $u_t(0,\cdot)\in L^2(\mathbb{R})$.

math.AP↗

arXiv2014

Uniqueness of Conservative Solutions to the Camassa-Holm Equation via Characteristics

The paper provides a direct proof the uniqueness of solutions to the Camassa-Holm equation, based on characteristics. Given a conservative solution $u=u(t,x)$, an equation is introduced which singles out a unique characteristic curve through each initial point. By studying the evolution of the quantities $u$ and $v= 2\arctan u_x$ along each characteristic, it is proved that the Cauchy problem with general initial data $u_0\in H^1(\mathbb{R})$ has a unique solution, globally in time.

math.AP↗

arXiv2013

Lack of BV Bounds for Approximate Solutions to the p-system with Large Data

We consider front tracking approximate solutions to the p-system of isentropic gas dynamics. At interaction times, the outgoing wave fronts have the same strength as in the exact solution of the Riemann problem, but some error is allowed in their speed. For large BV initial data, we construct examples showing that the total variation of these approximate solutions can become arbitrarily large, or even blow up in finite time. This happens even if the density of the gas remains uniformly positive.

math.AP↗

arXiv2012

Shock formation in the compressible Euler equations and related systems

We prove shock formation results for the compressible Euler equations and related systems of conservation laws in one space dimension, or three dimensions with spherical symmetry. We establish an $L^\infty$ bound for $C^1$ solutions of the one-D Euler equations, and use this to improve recent shock formation results of the authors. We prove analogous shock formation results for one-D MHD with orthogonal magnetic field, and for compressible flow in a variable area duct, which has as a special case spherically symmetric three dimensional flow on the exterior of a ball

math.AP↗

arXiv2012

Singularities of solutions to compressible Euler equations with vacuum

Presented are two results on the formation of finite time singularities of solutions to the compressible Euler equations in two and three space dimensions for isentropic, polytropic, ideal fluid flows. The initial velocity is assumed to be symmetric and the initial sound speed is required to vanish at the origin. They are smooth in Sobolev space $H^3$, but not required to have a compact support. It is shown that the $H^3$ norm of the velocity field and the sound speed will blow up in a finite time.

math.AP↗