arXiv subjects

Jian-Guo Liu

Publications and source records attributed to Jian-Guo Liu.

At least 127 records · Page 7Linked to original sources

arXiv2017

On the best constant for Gagliardo-Nirenberg interpolation inequalities

In this paper we derive the best constant for the following Gagliardo-Nirenberg interpolation inequality \begin{eqnarray*} \|u\|_{L^{m+1}}\leq C_{q,m,p} \|u\|^{1-θ}_{L^{q+1}}\|\nabla u\|^θ_{L^p},\quad θ=\frac{pd(m-q)}{(m+1)[d(p-q-1)+p(q+1)]}, \end{eqnarray*} where parameters $q,m,p$ respectively belong to the following two ranges: (i) $p>d\geq 1$, $q\geq0$ and $m=\infty$. That shows $L^{\infty}$-type Gagliardo-Nirenberg interpolation inequality. (ii) $p>\max\{1,\frac{2d}{d+2}\}$, $0\leq q<σ-1$, and $q 0$, $λ>0$ and $x_{0}\in \mathbb{R}^d$. In particular, for the case $m=+\infty$, the generalized Lane-Emden equation becomes a Thomas-Fermi type equation. For $q=0,~m=\infty$ or $d=1$, $u_{c,m}$ are closed form solutions expressed in term of the incomplete Beta functions. Moreover, we show that $u_{c,m}\to u_{c,\infty}$ and $C_{q,m,p}\to C_{q,\infty,p}$ as $m\to +\infty$ for $d=1$.

math.AP↗

arXiv2017

$p$-Euler equations and $p$-Navier-Stokes equations

We propose in this work new systems of equations which we call $p$-Euler equations and $p$-Navier-Stokes equations. $p$-Euler equations are derived as the Euler-Lagrange equations for the action represented by the Benamou-Brenier characterization of Wasserstein-$p$ distances, with incompressibility constraint. $p$-Euler equations have similar structures with the usual Euler equations but the `momentum' is the signed ($p-1$)-th power of the velocity. In the 2D case, the $p$-Euler equations have streamfunction-vorticity formulation, where the vorticity is given by the $p$-Laplacian of the streamfunction. By adding diffusion presented by $γ$-Laplacian of the velocity, we obtain what we call $p$-Navier-Stokes equations. If $γ=p$, the {\it a priori} energy estimates for the velocity and momentum have dual symmetries. Using these energy estimates and a time-shift estimate, we show the global existence of weak solutions for the $p$-Navier-Stokes equations in $\mathbb{R}^d$ for $γ=p$ and $p\ge d\ge 2$ through a compactness criterion.

math.AP↗

arXiv2017

On Runge-Kutta methods for the water wave equation and its simplified nonlocal hyperbolic model

There is a growing interest in investigating numerical approximations of the water wave equation in recent years, whereas the lack of rigorous analysis of its time discretization inhibits the design of more efficient algorithms. In this work, we focus on a nonlocal hyperbolic model, which essentially inherits the features of the water wave equation, and is simplified from the latter. For the constant coefficient case, we carry out systematical stability studies of the fully discrete approximation of such systems with the Fourier spectral approximation in space and general Runge-Kutta method in time. In particular, we discover the optimal time step constraints, in the form of a modified CFL condition, when certain explicit Runge-Kutta method are applied. Besides, the convergence of the semi-discrete approximation of variable coefficient case is shown, which naturally connects to the water wave equation. Extensive numerical tests have been performed to verify the stability conditions and simulations of the simplified hyperbolic model in the high frequency regime and the water wave equation are also provided.

math.NA↗

arXiv2017

Asymmetry in crystal facet dynamics of homoepitaxy by a continuum model

In the absence of external material deposition, crystal surfaces usually relax to become flat by decreasing their free energy. We study an asymmetry in the relaxation of macroscopic plateaus, facets, of a periodic surface corrugation in 1+1 dimensions via a continuum model below the roughening transition temperature. The model invokes a highly degenerate parabolic partial differential equation (PDE) for surface diffusion, which is related to the weighted-$H^{-1}$ (nonlinear) gradient flow of a convex, singular surface free energy in homoepitaxy. The PDE is motivated both by an atomistic broken-bond model and a mesoscale model for steps. By constructing an explicit solution to the PDE, we demonstrate the lack of symmetry in the evolution of top and bottom facets in periodic surface profiles. Our explicit, analytical solution is compared to numerical simulations of the PDE via a regularized surface free energy.

cond-mat.stat-mech↗

arXiv2017

An accurate front capturing scheme for tumor growth models with a free boundary limit

We consider a class of tumor growth models under the combined effects of density-dependent pressure and cell multiplication, with a free boundary model as its singular limit when the pressure-density relationship becomes highly nonlinear. In particular, the constitutive law connecting pressure $p$ and density $ρ$ is $p(ρ)=\frac{m}{m-1} ρ^{m-1}$, and when $m \gg 1$, the cell density $ρ$ may evolve its support due to a pressure-driven geometric motion with sharp interface along the boundary of its support. The nonlinearity and degeneracy in the diffusion bring great challenges in numerical simulations, let alone the capturing of the singular free boundary limit. Prior to the present paper, there is lack of standard mechanism to numerically capture the front propagation speed as $m\gg 1$. In this paper, we develope a numerical scheme based on a novel prediction-correction reformulation that can accurately approximate the front propagation even when the nonlinearity is extremely strong. We show that the semi-discrete scheme naturally connects to the free boundary limit equation as $m \rightarrow \infty$, and with proper spacial discretization, the fully discrete scheme has improved stability, preserves positivity, and implements without nonlinear solvers. Finally, extensive numerical examples in both one and two dimensions are provided to verify the claimed properties and showcase good performance in various applications.

math.NA↗

arXiv2017

Fractional stochastic differential equations satisfying fluctuation-dissipation theorem

We propose in this work a fractional stochastic differential equation (FSDE) model consistent with the over-damped limit of the generalized Langevin equation model. As a result of the `fluctuation-dissipation theorem', the differential equations driven by fractional Brownian noise to model memory effects should be paired with Caputo derivatives, and this FSDE model should be understood in an integral form. We establish the existence of strong solutions for such equations and discuss the ergodicity and convergence to Gibbs measure. In the linear forcing regime, we show rigorously the algebraic convergence to Gibbs measure when the `fluctuation-dissipation theorem' is satisfied, and this verifies that satisfying `fluctuation-dissipation theorem' indeed leads to the correct physical behavior. We further discuss possible approaches to analyze the ergodicity and convergence to Gibbs measure in the nonlinear forcing regime, while leave the rigorous analysis for future works. The FSDE model proposed is suitable for systems in contact with heat bath with power-law kernel and subdiffusion behaviors.

math.PR↗

arXiv2017

Continuous and discrete one dimensional autonomous fractional ODEs

In this paper, we study 1D autonomous fractional ODEs $D_c^γu=f(u), 0< γ<1$, where $u: [0,\infty)\mapsto\mathbb{R}$ is the unknown function and $D_c^γ$ is the generalized Caputo derivative introduced by Li and Liu ( arXiv:1612.05103). Based on the existence and uniqueness theorem and regularity results in previous work, we show the monotonicity of solutions to the autonomous fractional ODEs and several versions of comparison principles. We also perform a detailed discussion of the asymptotic behavior for $f(u)=Au^p$. In particular, based on an Osgood type blow-up criteria, we find relatively sharp bounds of the blow-up time in the case $A>0, p>1$. These bounds indicate that as the memory effect becomes stronger ($γ\to 0$), if the initial value is big, the blow-up time tends to zero while if the initial value is small, the blow-up time tends to infinity. In the case $A<0, p>1$, we show that the solution decays to zero more slowly compared with the usual derivative. Lastly, we show several comparison principles and Grönwall inequalities for discretized equations, and perform some numerical simulations to confirm our analysis.

math.CA↗

arXiv2017

A dispersive regularization for the modified Camassa-Holm equation

In this paper, we present a dispersive regularization for the modified Camassa-Holm equation (mCH) in one dimension, which is achieved through a double mollification for the system of ODEs describing trajectories of $N$-peakon solutions. From this regularized system of ODEs, we obtain approximated $N$-peakon solutions with no collision between peakons. Then, a global $N$-peakon solution for the mCH equation is obtained, whose trajectories are global Lipschitz functions and do not cross each other. When $N=2$, the limiting solution is a sticky peakon weak solution. By a limiting process, we also derive a system of ODEs to describe $N$-peakon solutions. At last, using the $N$-peakon solutions and through a mean field limit process, we obtain global weak solutions for general initial data $m_0$ in Radon measure space.

math.AP↗

arXiv2017

Analytical validation of a continuum model for the evolution of a crystal surface in multiple space dimensions

In this paper we are concerned with the existence of a weak solution to the initial boundary value problem for the equation $\frac{\partial u}{\partial t} = Δ\left(Δu\right)^{-3}$. This problem arises in the mathematical modeling of the evolution of a crystal surface. Existence of a weak solution $u$ with $Δu\geq 0$ is obtained via a suitable substitution. Our investigations reveal the close connection between this problem and the equation $\partial_tρ+ρ^2Δ^2ρ^3=0$, another crystal surface model first proposed by H. Al Hajj Shehadeh, R. V. Kohn and J. Weare in Physica D: Nonlinear Phenomena, bf 240 (2011), no. 21, 1771-1784.

math.AP↗

arXiv2017

The modified Camassa-Holm equation in Lagrangian coordinates

In this paper, we study the modified Camassa-Holm (mCH) equation in Lagrangian coordinates. For some initial data $m_0$, we show that classical solutions to this equation blow up in finite time $T_{max}$. Before $T_{max}$, existence and uniqueness of classical solutions are established. Lifespan for classical solutions is obtained: $T_{max}\geq \frac{1}{||m_0||_{L^\infty}||m_0||_{L^1}}.$ And there is a unique solution $X(ξ,t)$ to the Lagrange dynamics which is a strictly monotonic function of $ξ$ for any $t\in[0,T_{max})$: $X_ξ(\cdot,t)>0$. As $t$ approaching $T_{max}$, we prove that classical solution $m(\cdot ,t)$ in Eulerian coordinate has a unique limit $m(\cdot,T_{max})$ in Radon measure space and there is a point $ξ_0$ such that $X_ξ(ξ_0,T_{max})=0$ which means $T_{max}$ is an onset time of collision of characteristics. We also show that in some cases peakons are formed at $T_{max}$. After $T_{max}$, we regularize the Lagrange dynamics to prove global existence of weak solutions $m$ in Radon measure space.

math.AP↗

arXiv2017

Continuum limit of a mesoscopic model with elasticity of step motion on vicinal surfaces

This work considers the rigorous derivation of continuum models of step motion starting from a mesoscopic Burton-Cabrera-Frank (BCF) type model following the work [Xiang, SIAM J. Appl. Math. 2002]. We prove that as the lattice parameter goes to zero, for a finite time interval, a modified discrete model converges to the strong solution of the limiting PDE with first order convergence rate.

math.AP↗

arXiv2017

Patched peakon weak solutions of the modified Camassa-Holm equation

In this paper, we study traveling wave solutions and peakon weak solutions of the modified Camassa-Holm (mCH) equation with dispersive term $2ku_x$ for $k\in\mathbb{R}$. We study traveling wave solutions through a Hamiltonian system obtained from the mCH equation by using a nonlinear transformation. The typical traveling wave solutions given by this Hamiltonian system are unbounded or multi-valued. We provide a method, called patching technic, to truncate these traveling wave solutions and patch different segments to obtain patched bounded single-valued peakon weak solutions which satisfy jump conditions at peakons. Then, we study some special peakon weak solutions constructed by the fundamental solution of the Helmholtz operator $1-\partial_{xx}$, which can also be obtained by the patching technic. At last, we study some length and total signed area preserving closed planar curve flows that can be described by the mCH equation when $k=1$, for which we give a Hamiltonian structure and use the patched periodic peakon weak solutions to investigate loops with cusps.

math-ph↗

arXiv2017

Global existence of solutions to a tear film model with locally elevated evaporation rates

Motivated by a model proposed by Peng et al. [Advances in Coll. and Interf. Sci. 206 (2014)] for break-up of tear films on human eyes, we study the dynamics of a generalized thin film model. The governing equations form a fourth-order coupled system of nonlinear parabolic PDE for the film thickness and salt concentration subject to non-conservative effects representing evaporation. We analytically prove the global existence of solutions to this model with mobility exponents in several different ranges and the results are then validated against PDE simulations. We also numerically capture other interesting dynamics of the model, including finite-time rupture-shock phenomenon due to the instabilities caused by locally elevated evaporation rates, convergence to equilibrium and infinite-time thinning.

math.AP↗

arXiv2016

Weak solution of a continuum model for vicinal surface in the attachment-detachment-limited regime

We study in this work a continuum model derived from 1D attachment-detachment-limited (ADL) type step flow on vicinal surface, $ u_t=-u^2(u^3)_{hhhh}$, where $u$, considered as a function of step height $h$, is the step slope of the surface. We formulate a notion of weak solution to this continuum model and prove the existence of a global weak solution, which is positive almost everywhere. We also study the long time behavior of weak solution and prove it converges to a constant solution as time goes to infinity. The space-time Hölder continuity of the weak solution is also discussed as a byproduct.

math.AP↗

arXiv2016

Positivity-preserving and asymptotic preserving method for 2D Keller-Segal equations

We propose a semi-discrete scheme for 2D Keller-Segel equations based on a symmetrization reformation, which is equivalent to the convex splitting method and is free of any nonlinear solver. We show that, this new scheme is unconditionally stable as long as the initial condition does not exceed certain threshold, and it asymptotically preserves the quasi-static limit in the transient regime. Furthermore, we prove that the fully discrete scheme is conservative and positivity preserving, which makes it ideal for simulations. The analogical schemes for the radial symmetric cases and the subcritical degenerate cases are also presented and analyzed. With extensive numerical tests, we verify the claimed properties of the methods and demonstrate their superiority in various challenging applications.

math.NA↗

arXiv2016

Explicit and implicit TVD schemes for conservation laws with Caputo derivatives

In this paper, we investigate numerical approximations of the scalar conservation law with the Caputo derivative, which introduces the memory effect. We construct the first order and the second order explicit upwind schemes for such equations, which are shown to be conditionally $\ell^1$ contracting and TVD. However, the Caputo derivative leads to the modified CFL-type stability condition, $ (Δt)^α = O(Δx)$, where $α\in (0,1]$ is the fractional exponent in the derivative. When $α$ small, such strong constraint makes the numerical implementation extremely impractical. We have then proposed the implicit upwind scheme to overcome this issue, which is proved to be unconditionally $\ell^1$ contracting and TVD. Various numerical tests are presented to validate the properties of the methods and provide more numerical evidence in interpreting the memory effect in conservation laws.

math.NA↗

arXiv2016

Continuum dynamics of the intention field under weakly cohesive social interactions

We investigate the long-time dynamics of an opinion formation model inspired by a work by Borghesi, Bouchaud and Jensen. Firstly, we derive a Fokker-Planck type equation under the assumption that interactions between individuals produce little consensus of opinion (grazing collision approximation). Secondly, we study conditions under which the Fokker-Planck equation has non-trivial equilibria and derive the macroscopic limit (corresponding to the long-time dynamics and spatially localized interactions) for the evolution of the mean opinion. Finally, we compare two different types of interaction rates: the original one given in the work of Borghesi, Bouchaud and Jensen (symmetric binary interactions) and one inspired from works by Motsch and Tadmor (non-symmetric binary interactions). We show that the first case leads to a conservative model for the density of the mean opinion whereas the second case leads to a non-conservative equation. We also show that the speed at which consensus is reached asymptotically for these two rates has fairly different density dependence.

math.AP↗

arXiv2015

Convergence of Stochastic Interacting Particle Systems in Probability under a Sobolev Norm

In this paper, we consider particle systems with interaction and Brownian motion. We prove that when the initial data is from the sampling of Chorin's method, i.e., the initial vertices are on lattice points $hi\in \mathbb{R}^d$ with mass $ρ_0(hi) h^d$, where $ρ_0$ is some initial density function, then the regularized empirical measure of the interacting particle system converges in probability to the corresponding mean-field partial differential equation with initial density $ρ_0$, under the Sobolev norm of $L^\infty(L^2)\cap L^2(H^1)$. Our result is true for all those systems when the interacting function is bounded, Lipschitz continuous and satisfies certain regular condition. And if we further regularize the interacting particle system, it also holds for some of the most important systems of which the interacting functions are not. For systems with repulsive Coulomb interaction, this convergence holds globally on any interval $[0,t]$. And for systems with attractive Newton force as interacting function, we have convergence within the largest existence time of the regular solution of the corresponding Keller-Segel equation.

math.PR↗