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## Time discretisation of monotone nonlinear evolution problems by the discontinuous Galerkin method

### BIT Numerical Mathematics (2011-09-01) 51: 581-607 , September 01, 2011

A class of discontinuous Galerkin methods is studied for the time discretisation of the initial-value problem for a nonlinear first-order evolution equation that is governed by a monotone, coercive, and hemicontinuous operator. The numerical solution is shown to converge towards the weak solution of the original problem. Furthermore, well-posedness of the time-discrete problem as well as a priori error estimates for sufficiently smooth exact solutions are studied.

## Short-time existence theory toward stability for nonlinear parabolic systems

### Journal of Evolution Equations (2015-06-01) 15: 403-456 , June 01, 2015

We establish existence of classical solutions for nonlinear parabolic systems in divergence form on $${\mathbb{R}^n}$$ , under mild regularity assumptions on coefficients in the problem, and under the assumption of Hölder continuous initial conditions. Our analysis is motivated by the study of stability for stationary and traveling wave solutions arising in such systems. In this setting, large time bounds obtained by pointwise semigroup techniques are often coupled with appropriate short time bounds in order to close an iteration based on Duhamel-type integral equations, and our analysis gives precisely the required short time bounds. This development both clarifies previous applications of this idea (by Zumbrun and Howard) and establishes a general result that covers many additional cases.

## Operator Splitting Methods with Error Estimator and Adaptive Time-Stepping. Application to the Simulation of Combustion Phenomena

### Splitting Methods in Communication, Imaging, Science, and Engineering (2016-01-01): 627-641 , January 01, 2016

Operator splitting techniques were originally introduced with the main objective of saving computational costs. A multi-physics problem is thus split in subproblems of different nature with a significant reduction of the algorithmic complexity and computational requirements of the numerical solvers. Nevertheless, splitting errors are introduced in the numerical approximations due to the separate evolution of the split subproblems and can compromise a reliable simulation of the coupled dynamics. In this chapter we present a numerical technique to estimate such splitting errors on the fly and dynamically adapt the splitting time steps according to a user-defined accuracy tolerance. The method applies to the numerical solution of time-dependent stiff PDEs, illustrated here by propagating laminar flames investigated in combustion applications.

## The dynamics of elastic closed curves under uniform high pressure

### Calculus of Variations and Partial Differential Equations (2008-12-01) 33: 493-521 , December 01, 2008

We consider the dynamics of an inextensible elastic closed wire in the plane under uniform high pressure. In 1967, Tadjbakhsh and Odeh (J. Math. Anal. Appl. 18:59–74, 1967) posed a variational problem to determine the shape of a buckled elastic ring under uniform pressure. In order to comprehend a dynamics of the wire, we consider the following two mathematical questions: (i) can we construct a gradient flow for the Tadjbakhsh–Odeh functional under the inextensibility condition?; (ii) what is a behavior of the wire governed by the gradient flow near every critical point of the Tadjbakhsh–Odeh variational problem? For (i), first we derive a system of equations which governs the gradient flow, and then, give an affirmative answer to (i) by solving the system involving fourth order parabolic equations. For (ii), we first prove a stability and instability of each critical point by considering the second variation formula of the Tadjbakhsh–Odeh functional. Moreover, we give a lower bound of its Morse index. Finally we prove a dynamical aspects of the wire near each equilibrium state.

## Decay estimates for “anisotropic” viscous Hamilton-Jacobi equations in ℝ N

### Nonlinear Evolution Equations and Related Topics (2004-01-01): 27-37 , January 01, 2004

The large time behaviour of the *L*^{q}-norm of nonnegative solutions to the “anisotropic” viscous Hamilton-Jacobi equation
$$
{u_{t}} - \Delta u + {\sum\limits_{{i = 1}}^{m} {|{u_{{xi}}}|} ^{{Pi}}} = 0 in {\mathbb{R}_{ + }} x {\mathbb{R}^{N}},
$$
is studied for *q* = 1 and *q* = ∞, where *m* ∈ {1,...,*N*} and *p*_{i} for *i* ∈ {1,...,*m*}. The limit of the*L*^{1}-norm is identified, and temporal decay estimates for the *L*^{∞}-norm are obtained, according to the values of the *p*_{i}’s. The main tool in our approach is the derivation of L^{∞}-decay estimates for
$$
\nabla ({u^{\alpha }}),\alpha \in (0,1]
$$
, by a Bernstein technique inspired by the ones developed by Bénilan for the porous medium equation.

## A one-dimensional model of cell diffusion and aggregation, incorporating volume filling and cell-to-cell adhesion

### Journal of Mathematical Biology (2008-06-18) 58: 395-427 , June 18, 2008

We develop and analyse a discrete model of cell motility in one dimension which incorporates the effects of volume filling and cell-to-cell adhesion. The formal continuum limit of the model is a nonlinear diffusion equation with a diffusivity which can become negative if the adhesion coefficient is sufficiently large. This appears to be related to the presence of spatial oscillations and the development of plateaus (pattern formation) in numerical solutions of the discrete model. A combination of stability analysis of the discrete equations and steady-state analysis of the limiting PDE (and a higher-order correction thereof) can be used to shed light on these and other qualitative predictions of the model.

## Nonlinear degenerate parabolic equations with time-dependent singular potentials for Baouendi-Grushin vector fields

### Acta Mathematica Sinica, English Series (2015-01-01) 31: 123-139 , January 01, 2015

In this paper, we are concerned with the following three types of nonlinear degenerate parabolic equations with time-dependent singular potentials:
$$\begin{array}{*{20}c}
{\frac{{\partial u^q }}
{{\partial t}} = \nabla _\alpha \cdot \left( {\left\| z \right\|^{ - p\gamma } \left| {\nabla _\alpha u} \right|^{p - 2} \nabla _\alpha u} \right) + V(z,t)u^{p - 1} ,} \\
{\frac{{\partial u^q }}
{{\partial t}} = \nabla _\alpha \cdot \left( {\left\| z \right\|^{ - 2\gamma } \nabla _\alpha u^m } \right) + V(z,t)u^m ,} \\
{\frac{{\partial u^q }}
{{\partial t}} = u^\mu \nabla _\alpha \cdot \left( {u^\tau \left| {\nabla _\alpha u} \right|^{p - 2} \nabla _\alpha u} \right) + V(z,t)u^{p - 1 + \mu + \tau } } \\
\end{array}$$
in a cylinder Ω × (0, *T*) with initial condition *u* (*z*, 0) = *u*_{0} (*z*) ≥ 0 and vanishing on the boundary *∂*Ω × (0, T), where Ω is a Carnot-Carathéodory metric ball in ℝ^{d+k} and the time-dependent singular potential function is *V* (*z, t*) ∈ *L*_{loc}^{1}
(Ω × (0, *T*)). We investigate the nonexistence of positive solutions of these three problems and present our results on nonexistence.

## Finite element approximation of spatially extended predator–prey interactions with the Holling type II functional response

### Numerische Mathematik (2007-10-01) 107: 641-667 , October 01, 2007

We study the numerical approximation of the solutions of a class of nonlinear reaction–diffusion systems modelling predator–prey interactions, where the local growth of prey is logistic and the predator displays the Holling type II functional response. The fully discrete scheme results from a finite element discretisation in space (with lumped mass) and a semi-implicit discretisation in time. We establish a priori estimates and error bounds for the semi discrete and fully discrete finite element approximations. Numerical results illustrating the theoretical results and spatiotemporal phenomena are presented in one and two space dimensions. The class of problems studied in this paper are real experimental systems where the parameters are associated with real kinetics, expressed in nondimensional form. The theoretical techniques were adapted from a previous study of an idealised reaction–diffusion system (Garvie and Blowey in Eur J Appl Math 16(5):621–646, 2005).

## Existence and regularity of mean curvature flow with transport term in higher dimensions

### Mathematische Annalen (2016-04-01) 364: 857-935 , April 01, 2016

Given an initial $$C^1$$ hypersurface and a time-dependent vector field in a Sobolev space, we prove a time-global existence of a family of hypersurfaces which start from the given hypersurface and which move by the velocity equal to the mean curvature plus the given vector field. We show that the hypersurfaces are $$C^1$$ for a short time and, even after some singularities occur, almost everywhere $$C^1$$ away from the higher multiplicity region.

## On a Convection Diffusion Equation with Absorption Term

### Bulletin of the Malaysian Mathematical Sciences Society (2017-04-01) 40: 523-544 , April 01, 2017

The paper studies the posedness of the convection diffusion equation $$\begin{aligned} u_{t}=\text {div}\left( \left| \nabla u^{m}\right| ^{p-2}\nabla u^{m}\right) +\sum _{i=1}^{N}\frac{\partial b_{i}\left( u^{m}\right) }{\partial x_{i}}-u^{mr}. \end{aligned}$$ with homogeneous boundary condition and with the initial value $$u_0(x)\in L^{q-1+\frac{1}{m}}(\Omega )$$ . By considering its regularized problem, using Moser iteration technique, the local bounded properties of the $$L^{\infty }$$ -norm of $$u_{k}$$ and that of the $$L^{p}$$ -norm of the gradient $$\nabla u_{k}$$ are got, where $$u_{k}$$ is the solution of the regularized problem of the equation. By the compactness theorem, the existence of the solution of the equation itself is obtained. By using some techniques in Zhao and Yuan (Chin Ann Math A 16(2):179–194, 1995), the stability of the solutions is obtained too.