Mehdi Nadjafikhah, Saeed Dodangeh
symmetry, group classification, differential invariants, Lie-classical method,infinitesimal criterion method, RDC equation, KPP equation, similarity solutions.
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We consider a reaction-diffusion equation with a convection term in one space variable, where the diffusion changes sign from the positive to the negative and the reaction term is bistable. We study the existence of wavefront solutions, their uniqueness and regularity. The presence of convection reveals several new features of wavefronts: according to the mutual positions of the diffusivity and reaction, profiles can occur either for a single value of the speed or for a bounded interval of such values; uniqueness (up to shifts) is lost; moreover, plateaus of arbitrary length can appear; profil…
We investigate blow-up phenomena for positive solutions of nonlinear reaction-diffusion equations including a nonlinear convection term $\partial_t u = Δu - g(u) \cdot \nabla u + f(u)$ in a bounded domain of $\mathbb{R}^N$ under the dissipative dynamical boundary conditions $σ\partial_t u + \partial_νu =0$. Some conditions on $g$ and $f$ are discussed to state if the positive solutions blow up in finite time or not. Moreover, for certain classes of nonlinearities, an upper-bound for the blow-up time can be derived and the blow-up rate can be determinated.
We consider in this paper a diffusion-convection reaction equation in one space dimension. The main assumptions are about the reaction term, which is monostable, and the diffusivity, which changes sign once or twice; then, we deal with a forward-backward parabolic equation. Our main results concern the existence of globally defined traveling waves, which connect two equilibria and cross both regions where the diffusivity is positive and regions where it is negative. We also investigate the monotony of the profiles and show the appearance of sharp behaviours at the points where the diffusivity …
We study the global stability issue of the reaction-convection-diffusion cholera epidemic PDE model and show that the basic reproduction number serves as a threshold parameter that predicts whether cholera will persist or become globally extinct. Specifically, when the basic reproduction number is beneath one, we show that the disease-free-equilibrium is globally attractive. On the other hand, when the basic reproduction number exceeds one, if the infectious hosts or the concentration of bacteria in the contaminated water are not initially identically zero, we prove the uniform persistence res…
Traveling waves for bistable reaction-diffusion-convection equations with discontinuous density-dependent coefficients
本文研究了具有不连续密度依赖系数的双稳反应-扩散-对流方程的行波解存在性，扩展了之前在更高正则性假设下的结果到弱正则性设置中。
In this paper we establish the uniqueness of a solution to a stationary convection-diffusion equation in divergence form with an exponentially summable generalized divergence-free drift.
In this article, we study the unique determination of convection term and the time-dependent density coefficient appearing in a convection-diffusion equation from partial Dirichlet to Neumann map measured on boundary.
This work investigates how we can extend the invariant subspace method to two-dimensional time-fractional non-linear PDEs. More precisely, the systematic study has been provided for constructing the various dimensions of the invariant subspaces for the two-dimensional time-fractional generalized convection-reaction-diffusion-wave equation along with the initial conditions for the first time. Additionally, the special types of the above-mentioned equation are discussed through this method separately such as reaction-diffusion-wave equation, convection-diffusion-wave equation and diffusion-wave …
Traveling waves for monostable reaction-diffusion-convection equations with discontinuous density-dependent coefficients
本文研究了具有不连续密度依赖系数的单稳反应-扩散-对流方程中行波的存在性与影响因素，通过非光滑剖面概念和渐近分析探讨了对流速度对最小波速的作用。
This paper is devoted to the interfacial behaviors of a class of backward forward diffusion convection equations. Under the assumption that the equations have classical solutions in one dimension, we prove that the backward region shrinks and the forward region expands with a positive rate.
In this article, we study the stability in the inverse problem of determining the time-dependent convection term and density coefficient appearing in the convection-diffusion equation, from partial boundary measurements. For dimension $n\geq 2$, we show the convection term (modulo the gauge term) admits log-log stability, whereas log-log-log stability estimate is obtained for the density coefficient.
Analytic regularity for a singularly perturbed reaction-convection-diffusion boundary value problem with two small parameters
本文研究含两个小参数的奇异摄动反应-对流-扩散边值问题，建立了显式依赖于微分阶数和摄动参数的正则性界，并证明解可分解为光滑部分、端点边界层和可忽略余项。
Poincaré compactification for semiflows of reaction-diffusion equations with large diffusion and convection heating at the boundary
本文对具有大扩散和边界对流加热的耦合PDE-ODE系统极限半流进行庞加莱紧化，利用不变流形定理降维，并证明紧化向量场在C^1范数下接近。
Determining time-dependent convection and density terms in convection-diffusion equation using partial data
该研究针对时变对流扩散方程的逆边值问题，提出利用非线性Carleman权函数的方法，仅需在边界极小开子集上测量Dirichlet-to-Neumann映射，即可在自然规范下唯一恢复时变对流项与密度系数，证明了该逆问题的唯一性。
We analyze upwind difference methods for strongly degenerate convection-diffusion equations in several spatial dimensions. We prove that the local $L^1$-error between the exact and numerical solutions is $\mathcal{O}(Δx^{2/(19+d)})$, where $d$ is the spatial dimension and $Δx$ is the grid size. The error estimate is robust with respect to vanishing diffusion effects. The proof makes effective use of specific kinetic formulations of the difference method and the convection-diffusion equation.
We consider a scalar parabolic equation in one spatial dimension. The equation is constituted by a convective term, a reaction term with one or two equilibria, and a positive diffusivity which can however vanish. We prove the existence and several properties of traveling-wave solutions to such an equation. In particular, we provide a sharp estimate for the minimal speed of the profiles and improve previous results about the regularity of wavefronts. Moreover, we show the existence of an infinite number of semi-wavefronts with the same speed.
The aim of this paper is to study the $L^1$-stability of periodic stationary solutions of scalar convection-diffusion equations. We obtain dispersion in $L^2$ for all space dimensions using Kružkov type entropy. And when the space dimension is one, we estimate the number of sign changes of a solution to obtain $L^1$-stability.
In this paper we consider a model that involves nonlocal diffusion and a classical convective term. Using a scaling argument and a new compactness argument we obtain the first term in the asymptotic behavior of the solutions.
We consider the inverse boundary value problem for the steady state convection diffusion equation. We prove that a velocity field $V$, is uniquely determined by the Dirichlet-to-Neumann map, when $V \in C^{0,γ} (Ω)$, $2/3< γ\leq 1$, i.e. when $V$ is a Hölder continuous vector field with $2/3< γ\leq 1$.
We consider a convection-diffusion model with linear fractional diffusion in the sub-critical range. We prove that the large time asymptotic behavior of the solution is given by the unique entropy solution of the convective part of the equation. The proof is based on suitable a-priori estimates, among which proving an Oleinik type inequality plays a key role.
Existence and uniqueness of the solutions to convection-diffusion equations
该研究针对有界漂移和势梯度诱导漂移的对流扩散方程，提出新的解的概念并证明其存在唯一性，同时验证了质量守恒、初值收敛性与强极大值原理，丰富了对流扩散方程的解理论。
We analyze monotone difference schemes for strongly degenerate convection-diffusion equations in one spatial dimension. These nonlinear equations are well-posed within a class of (discontinuous) entropy solutions. We prove that the L1 difference between the approximate solutions and the unique entropy solution is bounded above by a constant times the cube root of the spatial discretization parameter.
We consider a viscous approximation for a nonlinear degenerate convection-diffusion equations in two space dimensions, and prove an $L^1$ error estimate. Precisely, we show that the $L^1_{\mathrm{loc}}$ difference between the approximate solution and the unique entropy solution converges at a rate $\mathcal{O}(\eps^{1/2})$, where $\eps$ is the viscous parameter.
We study a Robin boundary problem for degenerate parabolic equation. We suggest a notion of entropy solution and propose a result of existence and uniqueness. Numerical simulations illustrate some aspects of solution behaviour. Monodimensional experiments are presented.
We prove that some non-self-adjoint differential operator admits factorization and apply this new representation of the operator to construct explicitly its domain. We also show that this operator is J-self-adjoint in some Krein space.
Higher order asymptotic expansions for the convection-diffusion equation in the Fujita-subcritical case
本文针对Fujita次临界情形下的对流扩散方程，改进已有结果，建立带余项衰减估计的高阶渐近展开，并探讨余项衰减率的最优性。
We study the existence and properties of Lipschitz continuous weak solutions to the Neumann boundary value problem for a class of one-dimensional quasilinear forward-backward diffusion equations with linear convection and reaction. The diffusion flux function is assumed to be of a forward-backward type that contains two forward-diffusion phases. We prove that, for all smooth initial data, there exists at least one weak solution whose spatial derivative stays in the two forward phases. Also, for all smooth initial data that have a derivative value lying in a certain phase transition range, we s…
We consider an evolution equation generalising the viscous Burgers equation supplemented by an initial condition which is a homogeneous random field. We develop a non-linear framework enabling us to show the existence and regularity of solutions as well as their long time behaviour.
We study a reaction-diffusion-convection problem with nonlinear drift posed in a domain with periodically arranged obstacles. The non-linearity in the drift is linked to the hydrodynamic limit of a totally asymmetric simple exclusion process (TASEP) governing a population of interacting particles crossing a domain with obstacle. Because of the imposed large drift scaling, this nonlinearity is expected to explode in the limit of a vanishing scaling parameter. As main working techniques, we employ two-scale formal homogenization asymptotics with drift to derive the corresponding upscaled model e…
The large time behavior of zero mass solutions to the Cauchy problem for a convection-diffusion equation. We provide conditions on the size and shape of the initial datum such that the large time asymptotics of solutions is given either by the derivative of the Guass-Weierstrass kernel or by a self-similar solution or by a hyperbolic N-wave
We consider semi-discrete first-order finite difference schemes for a nonlinear degenerate convection-diffusion equations in one space dimension, and prove an L1 error estimate. Precisely, we show that the L1 loc difference between the approximate solution and the unique entropy solution converges at a rate O(\Deltax 1/11), where \Deltax is the spatial mesh size. If the diffusion is linear, we get the convergence rate O(\Deltax 1/2), the point being that the O is independent of the size of the diffusion
Large-time behaviour for anisotropic stable nonlocal diffusion problems with convection
研究带对流项的非局部热方程非负解的大时间行为，发现其大时间极限由对流与扩散相对强度决定的极限方程的源型解给出，并确定了各向异性扩散算子在正交子空间上的投影。
Inverse problem for a time-dependent Convection-diffusion equation in admissible geometries
本文首次在黎曼几何背景下研究时间依赖对流-扩散方程的部分数据反问题，证明在已知规范不变性下可唯一恢复时间依赖的对流项和密度。
One-dimensional free boundary problem for a nonlinear diffusion - convection equation with a Dirichlet condition at fixed face $x=0$, variable in time, is considered. Throught several transformations the problem is reduced to a free boundary problem for a diffusion equation and the integral formulation is obtained. By using fixed point theorems, the existence of at least a solution, for small time, to a system of coupled nonlinear integral equations is obtained.
We study the existence and uniqueness of a solution to a linear stationary convection-diffusion equation stated in an infinite cylinder, Neumann boundary condition being imposed on the boundary. We assume that the cylinder is a junction of two semi-infinite cylinders with two different periodic regimes. Depending on the direction of the effective convection in the two semi-infinite cylinders, we either get a unique solution, or one-parameter family of solutions, or even non-existence in the general case. In the latter case we provide necessary and sufficient conditions for the existence of a s…
We study in this paper the periodic homogenization problem related to a strongly nonlinear reaction-diffusion equation. Owing to the large reaction term, the homogenized equation has a rather quite different form which puts together both the reaction and convection effects. We show in a special case that, the homogenized equation is exactly of a convection-diffusion type. The study relies on a suitable version of the well-known two-scale convergence method.
Stable determination of time-dependent coefficients in a reaction-diffusion-convection system
本文针对R^{1+n}（n≥2）有界域内反应-扩散-对流系统的逆边值问题，结合Carleman估计与几何光学解，通过部分边界测量得到时变对流系数的双对数稳定性估计，进而得到矩阵值势的三对数稳定性估计。
We consider a nonlinear degenerate convection-diffusion equation with inhomogeneous convection and prove that its entropy solutions in the sense of Kružkov are obtained as the - a posteriori unique - limit points of the JKO variational approximation scheme for an associated gradient flow in the $L^2$-Wasserstein space. The equation lacks the necessary convexity properties which would allow to deduce well-posedness of the initial value problem by the abstract theory of metric gradient flows. Instead, we prove the entropy inequality directly by variational methods and conclude uniqueness by doub…
Analysis of a three-dimensional rapidly rotating convection model without thermal diffusion
本文针对无热扩散的三维快速旋转对流模型，通过先研究带热扩散的正则化模型并建立与扩散系数无关的精细估计，证明了弱解的全局存在唯一性和强解的Hadamard适定性。
On the optimal decay rate of global solutions to the convection-diffusion equation with zero-mass initial data
研究具有零质量初始数据的对流扩散方程全局解的最优衰减率，建立零质量条件下全局解的改进衰减估计并推导自相似渐近轮廓，给出达到改进衰减率的充要条件，表明衰减率是最优的。
Traveling waves in reaction-diffusion-convection equations with combustion nonlinearity
本文研究实直线上具有燃烧非线性的反应-扩散-对流方程的行波解，将$p=2$的结果推广到奇异和退化扩散及非Lipschitz反应情形，并分析平衡点附近行波剖面的形状。
Free boundary problem governed by a non-linear diffusion-convection equation with Neumann condition
该研究针对带时变Neumann条件和类Stefan对流自由边界条件的一维非线性扩散-对流方程自由边界问题，通过变换得到耦合积分方程组的积分表示，并用不动点定理证明了解的全局存在性。
Traveling waves for combustion reaction-diffusion-convection equations: the full range of wave speeds
研究燃烧型反应-扩散-对流方程的行波解，通过覆盖整个波速范围 $c \ge H^*$ 填补了已有结果中 $c \ge -\min h(u)$ 限制下的存在性与不存在性之间的空白。
Inverse boundary value problem for the Convection-Diffusion equation with local data
针对部分边界不可达的非稳态CDE，本文利用其余边界的局部测量数据唯一恢复时变对流项和密度项，并指出自然规范变换是唯一性的唯一障碍。
Weak Reproductive Solutions for a Convection-Diffusion Model Describing a Binary Alloy Solidification Processes
本文研究在 $\mathbb{R}^3$ 有界正则区域中具有混合边界条件的二元合金凝固模型，证明了该系统弱生殖解的存在性。
This comment is devoted to the paper "Exact solutions of nonlinear diffusion-convection-reaction equation: A Lie symmetry approach" (CNSNS, vol. 67 (2019), 253-263) in which several results are not new because were derived much earlier. Moreover, some results in the paper are incorrect or incomplete.
We develop a general framework for finding error estimates for convection-diffusion equations with nonlocal, nonlinear, and possibly degenerate diffusion terms. The equations are nonlocal because they involve fractional diffusion operators that are generators of pure jump Levy processes (e.g. the fractional Laplacian). As an application, we derive continuous dependence estimates on the nonlinearities and on the Levy measure of the diffusion term. Estimates of the rates of convergence for general nonlinear nonlocal vanishing viscosity approximations of scalar conservation laws then follow as a …
We study a one dimensional metastable dynamics of internal interfaces for the initial boundary value problem for the following convection-reaction-diffusion equation \begin{equation*} \partial_t u = \varepsilon \partial_x^2 u -\partial_x f(u)+ f'(u). \end{equation*} A metastable behavior appears when the time-dependent solution develops into a layered function in a relatively short time, and subsequent approaches its steady state in a very long time interval. A rigorous analysis is used to study such behavior, by means of the construction of a one-parameter family $\{ U^\varepsilon(x;ξ)\}_ξ$ o…
Using the asymmetric fractional calculus of variations, we derive a fractional Lagrangian variational formulation of the convection-diffusion equation in the special case of constant coefficients.
Leak localisation with a measure source convection-diffusion model
本文研究利用输运-扩散模型与测度源定位气体泄漏，通过稀疏正则化恢复点源位置与强度，联合估计风场与扩散参数，理论分析模型稳定性，数值实验验证方法在实际气体泄漏检测中的有效性。
A class of nonlinear reaction-diffusion-convection equations describing various processes in physics, biology, chemistry etc. is under study in the case of time and two space variables. The group of equivalence transformations is constructed, which is applied for deriving a Lie symmetry classification for the class of such equations by the well-known algorithm. It is proved that the algorithm leads to 32 reaction-diffusion-convection equations admitting nontrivial Lie symmetries. Furthermore a set of form-preserving transformations for this class is constructed in order to reduce this number o…
A nonlinear stochastic diffusion-convection equation with reflection
本文研究了任意空间维度下具有反射的非线性伪单调随机扩散-对流演化问题，证明了带非负初值的方程解及防止解取负值的反射测度的存在性，并通过研究抛物容度下解的拟处处定义代表元证明了测度的极小性条件。
A steady scalar convection-diffusion problem has been studied for one and two dimensional cases. The major problem of unrealistic oscillations of the convection dominated problems is relaxed thanks to the wide range of the elements FEMLAB 3.1 benefits. The FEMLAB 3.1 solution has been presented for the problems, unique features and illustrations of the software have been used and results have been tested against analytic solution.
A finite volume method on general meshes for a degenerate parabolic convection-reaction-diffusion equation
针对退化抛物对流-反应-扩散方程，提出一种基于一般网格的有限体积法，采用混合有限体积格式离散扩散项和部分迎风格式处理对流项，证明了收敛性。
A unified spatiotemporal formulation with physics-preserving structure for time-dependent convection-diffusion problems
本文提出了一种统一的四维时空 formulation，用于时间依赖的对流-扩散问题，通过引入时间作为额外坐标并利用外微积分，实现了对物理结构的保持，并证明了其收敛性。
The role of convection in the limit shape of the critical front profile for Born-Infeld diffusion models
本文研究Born-Infeld扩散模型中引入对流项对临界前沿极限形状的影响，通过估计临界速度和分析等价一阶问题，证明不同对流项可导致极限剖面完全锐化或完全正则化，并辅以数值模拟。
A New Splitting Method for Time-dependent Convection-dominated Diffusion Problems
提出一种将对流扩散系统分裂为纯对流和扩散子系统的显式-隐式分裂方法，通过多步技术扩大稳定区域，并推广至Navier-Stokes方程，无需调节稳定化参数。
We study a problem involving thermosolutal convection in a fluid when the solute concentration is subject to a chemical reaction in which the solubility of the dissolved component is a function of temperature. When the spatial domain is a bounded one in $\mathbb{R}^{2}$ we show that the solution depends continuously on the reaction rate using true \emph{a priori} bounds for the solution when the chemical equilibrium function is an arbitrary function of temperature.
Existence and stability of steady states of a reaction convection diffusion equation modeling microtubule formation
本文推广了微管动力学的Dogterom-Leibler模型，考虑GTP-微管蛋白浓度对伸长速率及伸长/缩短相寿命的影响，并引入阻尼项形式的成核速率，研究边界条件下稳态的存在性与稳定性，首次将Evans函数数值与解析计算作为微管动力学研究的新数学工具。
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