Suman Kumar Sahoo, Manmohan Vashisth
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
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.
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 $2/3< γ\leq 1$.
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。
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。
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。
An inverse problem for a quasilinear convection--diffusion equation
针对拟线性对流-扩散方程，利用边界通量数据，通过线性化奇异解和高阶线性化方法，完全恢复非线性扩散项与对流项。
Recovery of time-dependent coefficients for the convection-diffusion equation on conformally transversally anisotropic manifolds from partial data
该研究针对共形横截各向异性流形上的对流扩散方程反问题，利用部分输入-输出算子，在1-形式与函数的衰减测地射线变换单射的条件下，唯一确定时变对流项和密度系数。
In this paper we reconstruct convection terms from boundary measurements.We reduce the Beals and Coifman inverse scattering scattering formalism from a first order system to a formalism for the $\bar{\partial}$ equation.
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.
Parabolic problems with slightly superlinear convection terms
本文研究了具有略微超线性对流项的抛物型问题，证明了有界和无界弱解的唯一性。
Recovery of coefficients for a convection-diffusion equation from partial data
研究从部分边界测量中恢复抛物型方程中依赖时空变量的零阶和一阶系数的反问题，通过开发特殊解、引入非线性权重卡尔曼估计并结合X射线变换部分数据结果来实现，还建立了新的卡尔曼估计。
The numerical approximation of an inverse problem subject to the convection--diffusion equation when diffusion dominates is studied. We derive Carleman estimates that are on a form suitable for use in numerical analysis and with explicit dependence on the Péclet number. A stabilized finite element method is then proposed and analysed. An upper bound on the condition number is first derived. Combining the stability estimates on the continuous problem with the numerical stability of the method, we then obtain error estimates in local $H^1$- or $L^2$-norms that are optimal with respect to the app…
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…
Hierarchical matrix approximability of inverse of convection dominated finite element matrices
本文提出基于
Inverse Optimal Cardano-Lyapunov Feedback for PDEs with Convection
本文针对控制非仿射的带对流和扩散 PDE，利用 CLF 导数的降三次或二次结构构造逆最优代价泛函，提出卡尔达诺-李雅普诺夫控制器及双根切换策略以降低控制代价。
We consider the inverse problem of recovering the spherically symmetric sound speed, density and attenuation in the Sun from the observations of the acoustic field randomly excited by turbulent convection. We show that observations at two heights above the photosphere and at two frequencies above the acoustic cutoff frequency uniquely determine the solar parameters. We also present numerical simulations which confirm this theoretical result.
In this note we present $L^\infty$-results for problems of the form $A(x,u,Du)=B(x,u,Du)$ in $Ω$, $u>0$ in $Ω$, $u=0$ on $\partialΩ$, where the growth condition for the function $B\colon Ω\times \mathbb{R}\times \mathbb{R}^N\to \mathbb{R}$ contains both a singular and a convection term. We use ideas from the works of Giacomoni-Schindler-Takac (2007) and the authors (2019) to prove the boundedness of weak solutions for such general problem by applying appropriate bootstrap arguments.
Let \begin{equation*} L=\sum_{i,j=1}^da_{i,j}\frac{\partial^2}{\partial x_i\partial x_j}-\sum_{i=1}^db_i\frac{\partial}{\partial x_i} \end{equation*} be a second order elliptic operator and consider the reaction-diffusion equation with Neumann boundary condition, \begin{equation*} \begin{aligned} &Lu=Λu^p\ \text{in}\ \mathbb{R}^d-D;\\ &\nabla u\cdot \bar n=-h\ \text{on}\ \partial D;\\ &u\ge0 \ \text{is minimal}, \end{aligned} \end{equation*} where $p\in(0,1)$, $d\ge2$, $h$ and $Λ$ are continuous positive functions, $D\subset R^d$ is bounded, and $\bar n$ is the unit inward normal to the domain…
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 the highly nonlinear and ill-posed inverse problem of determining some general expression $F(x,t,u,\nabla_xu)$ appearing in the diffusion equation $\partial_tu-Δ_x u+F(x,t,u,\nabla_xu)=0$ on $Ω\times(0,T)$, with $T>0$ and $Ω$ a bounded open subset of $\mathbb R^n$, $n\geq2$, from measurements of solutions on the lateral boundary $\partialΩ\times(0,T)$. We consider both linear and nonlinear expression of $F(x,t,u,\nabla_xu)$. In the linear case, the equation can be seen as a convection-diffusion equation and our inverse problem corresponds to the unique recovery, in some suitable se…
We consider a stochastic Camassa-Holm equation driven by a one-dimensional Wiener process with a first order differential operator as diffusion coefficient. We prove the existence and uniqueness of local strong solutions of this equation. In order to do so, we transform it into a random quasi-linear partial differential equation and apply Kato's operator theory methods. Some of the results have potential to find applications to other nonlinear stochastic partial differential equations.
We consider the quasilinear 1D inverse heat convection problem (IHCP) of determining the enthalpy-dependent heat fluxes from noisy internal enthalpy measurements. This problem arises in the Accelerated Cooling (ACC) process of producing thermomechanically controlled processed (TMCP) heavy plates made of steel. In order to adjust the complex microstructure of the underlying material, the Leidenfrost behavior of the hot surfaces with respect to the application of the cooling fluid has to be studied. Since the heat fluxes depend on the enthalpy and hence on the solution of the underlying initial …
We classify ``arithmetic convection equations'' on modular curves, and describe their space of solutions. Certain of these solutions involve the Fourier expansions of the Eisenstein modular forms of weight 4 and 6, while others involve the Serre-Tate expansions of the same modular forms; in this sense, our arithmetic convection equations can be seen as "unifying" the two types of expansions. The theory can be generalized to one of ``arithmetic heat equations'' on modular curves, but we prove that modular curves do not carry ``arithmetic wave equations.'' Finally, we prove an instability result…
We are concerned with singular elliptic problems of the form $-Δu\pm p(d(x))g(u)=\la f(x,u)+μ|\nabla u|^a$ in $Ω,$ where $Ω$ is a smooth bounded domain in $\RR^N$, $d(x)={\rm dist}(x,\partialΩ),$ $\la>0,$ $μ\in\RR$, $0<a\leq 2$, and $f,k$ are nonnegative and nondecreasing functions. We assume that $p(d(x))$ is a positive weight with possible singular behavior on the boundary of $Ω$ and that the nonlinearity $g$ is unbounded around the origin. Taking into account the competition between the anisotropic potential $p(d(x))$, the convection term $|\nabla u|^a$, and the singular nonlinearity $g$, w…
Regularization operators for identifying the unknown source in the time-fractional convection-diffusion-reaction equation
本文研究了基于噪声时间依赖测量在任意位置确定时间依赖源项的问题，采用傅里叶方法解析求解，发现解不稳定，提出三种正则化算子以克服不稳定性，并引入新的正则化参数选择规则和误差界。
In this contribution we prove the existence of weak solutions to degenerate parabolic systems arising from the coupled moisture movement, transport of dissolved species and heat transfer through partially saturated porous materials. Physically motivated mixed Dirichlet-Neumann boundary conditions and initial conditions are considered. Existence of a global weak solution of the problem is proved by means of semidiscretization in time and by passing to the limit from discrete approximations. Degeneration occurs in the nonlinear transport coefficients which are not assumed to be bounded below and…
In this work we establish log-type stability estimates for the inverse potential and conductivity problems with partial Dirichlet-to-Neumann map, where the Dirichlet data is homogeneous on the inaccessible part. This result, to some extent, improves our former result on the partial data problem in which log-log-type estimates were derived.
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。
An inverse boundary value problem for the Maxwell's equations with partial data
本文研究动态麦克斯韦方程组的逆边界问题，在存在严格凸叶状结构的条件下，证明了介电常数、电导率和磁导率的局部唯一确定性。
An Inverse Problem with Partial Neumann Data and $L^{n/2}$ Potentials
本文研究具有无界位势的部分数据反问题，通过构造显式Green函数和复几何光学解，证明了Schrödinger方程在部分Neumann数据下$L^{n/2}$位势的唯一可确定性。
We describe a simple way of constructing exponentially growing solutions of the second order systems with the Laplacian as the principal term.
We show that the linear span of the set of scalar products of gradients of harmonic functions on a bounded smooth domain $Ω\subset \mathbb{R}^n$ which vanish on a closed proper subset of the boundary is dense in $L^1(Ω)$. We apply this density result to solve some partial data inverse boundary problems for a class of semilinear elliptic PDE with quadratic gradient terms.
We study the inverse problem of identifying a periodic potential perturbation of the Dirichlet Laplacian acting in an infinite cylindrical domain, whose cross section is assumed to be bounded. We prove log-log stable determination of the potential with respect to the partial Dirichlet-to-Neumann map, where the Neumann data is taken on slightly more than half of the boundary of the domain.
We show that the knowledge of the Dirichlet-to-Neumann map on an arbitrary open portion of the boundary of a domain in $\mathbb{R}^n$, $n\ge 2$, for a class of semilinear elliptic equations, determines the nonlinearity uniquely.
We consider two inverse boundary value problems for the time-harmonic Maxwell equations in an infinite slab. Assuming that tangential boundary data for the electric and magnetic fields at a fixed frequency is available either on subsets of one boundary hyperplane, or on subsets of different boundary hyperplanes, we show that the electromagnetic material parameters, the conductivity, electric permittivity, and magnetic permeability, are uniquely determined by these partial measurements.
In this paper, we study the stability of two inverse boundary value problems in an infinite slab with partial data. These problems have been studied by Li and Uhlmann for the case of the Schrodinger equation and by Krupchyk, Lassas and Uhlmann for the case of the magnetic Schrodinger equation. Here we quantify the method of uniqueness proposed by Li and Uhlmann and prove a log-log stability estimate for the inverse problems associated to the Schrodinger equation. The boundary measurements considered in these problems are modelled by partial knowledge of the Dirichlet-to-Neumann map: in the fir…
The initial value problem for the conservation law $\partial_t u+(-Δ)^{α/2}u+\nabla \cdot f(u)=0$ is studied for $α\in (1,2)$ and under natural polynomial growth conditions imposed on the nonlinearity. We find the asymptotic expansion as $|x|\to \infty$ of solutions to this equation corresponding to initial conditions, decaying sufficiently fast at infinity.
In this article, high frequency stability estimates for the determination of the potential in the Schrödinger equation are studied when the boundary measurements are made on slightly more than half the boundary. The estimates reflect the increasing stability property with growing frequency.
We prove that the scattering matrix at all energies restricted to an open subset of the boundary determines an asymptotically hyperbolic manifold modulo isometries that are equal to the identity on the open subset where the scattering matrix is known.
We consider Calderón's inverse boundary value problems for a class of nonlinear Helmholtz Schrödinger equations and Maxwell's equations in a bounded domain in $\R^n$. The main method is the higher-order linearization of the Dirichlet-to-Neumann map of the corresponding equations. The local uniqueness of the linearized partial data Calderón's inverse problem is obtained following \cite{DKSU}. The Runge approximation properties and unique continuation principle allow us to extend to global situations. Simultaneous recovery of some unknown cavity$/$boundary and coefficients are given as some appl…
Notes on unique continuation properties for Partial Differential Equations -- Introduction to the stability estimates for inverse problems
本文为熟悉测度论与PDE的学生介绍反问题稳定性估计及唯一延拓定量估计，自包含论述Cauchy问题与唯一延拓性质。
In this article we consider an inverse boundary value problem for the time-harmonic Maxwell equations. We show that the electromagnetic material parameters are determined by boundary measurements where part of the boundary data is measured on a possibly very small set. This is an extension of earlier scalar results of Bukhgeim-Uhlmann and Kenig-Sjöstrand-Uhlmann to the Maxwell system. The main contribution is to show that the Carleman estimate approach to scalar partial data inverse problems introduced in those works can be carried over to the Maxwell system.
We consider the inverse boundary value problem for operators of the form $-\triangle+q$ in an infinite domain $Ω=\mathbb{R}\timesω\subset\mathbb{R}^{1+n}$, $n\geq3$, with a periodic potential $q$. For Dirichlet-to-Neumann data localized on a portion of the boundary of the form $Γ_1=\mathbb{R}\timesγ_1$, with $γ_1$ being the complement either of a flat or spherical portion of $\partialω$, we prove that a log-type stability estimate holds.
In this paper, we consider a mixed boundary value problem with a double phase partial differential operator, an obstacle effect and a multivalued reaction convection term. Under very general assumptions, an existence theorem for the mixed boundary value problem under consideration is proved by using a surjectivity theorem for multivalued pseudomonotone operators together with the approximation method of Moreau-Yosida. Then, we introduce a family of the approximating problems without constraints corresponding to the mixed boundary value problem. Denoting by $\mathcal S$ the solution set of the …
We extend results of Dos Santos Ferreira-Kenig-Sjoestrand-Uhlmann (math.AP/0601466) to less smooth coefficients, and we show that measurements on part of the boundary for the magnetic Schroedinger operator determine uniquely the magnetic field related to a Hoelder continuous potential. We give a similar result for determining a convection term. The proofs involve Carleman estimates, a smoothing procedure, and an extension of the Nakamura-Uhlmann pseudodifferential conjugation method to logarithmic Carleman weights.
We prove that the knowledge of the Dirichlet-to-Neumann map, measured on a part of the boundary of a bounded domain in $\mathbb{R}^n, n\geq2$, can uniquely determine, in a nonlinear magnetic Schrödinger equation, the vector-valued magnetic potential and the scalar electric potential, both being nonlinear in the solution.
Using a new method and additional (conditional and partial) equivalence transformations, we performed group classification in a class of variable coefficient $(1+1)$-dimensional nonlinear diffusion-convection equations of the general form $f(x)u_t=(D(u)u_x)_x+K(u)u_x.$ We obtain new interesting cases of such equations with the density $f$ localized in space, which have large invariance algebra. Exact solutions of these equations are constructed. We also consider the problem of investigation of the possible local trasformations for an arbitrary pair of equations from the class under considerati…
In this paper we study a convection-diffusion equation on a star-shaped graph composed by $n$ incoming edges and $m$ outgoing edges with a nonlinearity $f\in C^1(\rr)$ satisfying some additional general conditions. First, we prove the global well-posedness of the solutions of the system under consideration. Next, in the particular case that the nonlinear convection is given by $\partial_x(f(u(t, x))$ with $f(s)=-a|s|^{q-1}s$ with $q\geq 2$ and $a\in \rr$ verifying $(n-m)a\geq 0$, we analyze the long time behavior of the solutions. For $q> 2$ we find that the asymptotic behavior of the solution…
We consider a nonlinear 4th-order degenerate parabolic partial differential equation that arises in modelling the dynamics of an incompressible thin liquid film on the outer surface of a rotating horizontal cylinder in the presence of gravity. The parameters involved determine a rich variety of qualitatively different flows. Depending on the initial data and the parameter values, we prove the existence of nonnegative periodic weak solutions. In addition, we prove that these solutions and their gradients cannot grow any faster than linearly in time; there cannot be a finite-time blow-up. Finall…
In this technical report, we consider a nonlinear 4th-order degenerate parabolic partial differential equation that arises in modelling the dynamics of an incompressible thin liquid film on the outer surface of a rotating horizontal cylinder in the presence of gravity. The parameters involved determine a rich variety of qualitatively different flows. Depending on the initial data and the parameter values, we prove the existence of nonnegative periodic weak solutions. In addition, we prove that these solutions and their gradients cannot grow any faster than linearly in time; there cannot be a f…
On the local solvability and stability of the partial inverse problems for the non-self-adjoint Sturm-Liouville operators with a discontinuity
本文研究带复系数和间断点的非自伴Sturm-Liouville算子部分逆问题，在已知部分势函数和参数的前提下从特征值恢复剩余信息，得到其局部可解性、稳定性及两个新唯一性定理。
For v\in R^n let K be a compact set in R^n containing a suitable smooth surface and such that the intersection {tv+x:t\in R}\cap K is a closed interval or a single point for all x\in K. We prove that every linear first order differential operator with constant coefficients in direction v on space of Whitney functions E(K) admits a continuous linear right inverse.
We establish a global uniqueness result for an inverse boundary problem with partial data for the magnetic Schrödinger operator with a magnetic potential of class $W^{1,n}\cap L^\infty$, and an electric potential of class $L^n$. Our result is an extension, in terms of the regularity of the potentials, of the results [16] and [25]. As a consequence, we also show global uniqueness for a partial data inverse boundary problem for the advection-diffusion operator with the advection term of class $W^{1,n}\cap L^\infty$.
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。
Let $Ω\subset \Bbb R^2$ be a bounded domain with $\partialΩ\in C^\infty$ and $L$ be a positive number. For a three dimensional cylindrical domain $Q=Ω\times (0,L)$, we obtain some uniqueness result of determining a complex-valued potential for the Schrödinger equation from partial Cauchy data when Dirichlet data vanish on a subboundary $(\partialΩ\setminus\widetildeΓ) \times [0,L]$ and the corresponding Neumann data are observed on $\widetildeΓ\times [0,L]$, where $\widetildeΓ$ is an arbitrary fixed open set of $\partialΩ.$
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。