Related papers: Inverse problem on conservation laws
We study hyperbolic systems of one-dimensional partial differential equations under general, possibly non-local boundary conditions. A large class of evolution equations, either on individual 1-dimensional intervals or on general networks,…
We are concerned with the well-posedness of an inverse problem for determining the wedge boundary and associated two-dimensional steady supersonic Euler flow past the wedge, provided that the pressure distribution on the boundary surface of…
The solution of time dependent differential equations with neural networks has attracted a lot of attention recently. The central idea is to learn the laws that govern the evolution of the solution from data, which might be polluted with…
We consider an inverse boundary value problem for diffusion equations with multiple fractional time derivatives. We prove the uniqueness in determining a number of fractional time-derivative terms, the orders of the derivatives and…
We describe a new method which allows us to obtain a result of exact controllability to trajectories of multidimensional conservation laws in the context of entropy solutions and under a mere non-degeneracy assumption on the flux and a…
Despite the large number of publications on symmetry analysis of the geopotential forecast equation, its group foliations laws have not been considered previously. The present publication aims to address this shortcoming. First, group…
We prove the conservation of energy for weak and statistical solutions of the two-dimensional Euler equations, generated as strong (in an appropriate topology) limits of the underlying Navier-Stokes equations and a Monte Carlo-Spectral…
We show that, for first-order systems of conservation laws with a strictly convex entropy,in particular for the very simple so-called "inviscid" Burgers equation,it is possible to address the Cauchy problem by a suitable convex…
This book aims to provide a brief overview of recent advancements in the theory of inverse problems for stochastic partial differential equations. In order to keep the content concise, we will only discuss the inverse problems of two…
Conservation laws are formulated for systems of differential equations by using symmetries and adjoint symmetries, and an application to systems of evolution equations is made, together with illustrative examples. The formulation does not…
We deal with direct and inverse problems of the calculus of variations on arbitrary time scales. Firstly, using the Euler-Lagrange equation and the strengthened Legendre condition, we give a general form for a variational functional to…
In this article, we will study unbounded solutions of the 2D incompressible Euler equations. One of the motivating factors for this is that the usual functional framework for the Euler equations (e.g. based on finite energy conditions, such…
The divergence theorem of Gauss plays a central role in the derivation of the governing differential equations in fluid dynamics, electrodynamics, gravitational fields, and optics. One is often interested in an evolution equation for the…
This study derives conservative and skew-symmetric formulations of the incompressible flow equations in a terrain-following sigma-coordinate system that preserve key structural properties of the Cartesian formulation. Unlike conventional…
We consider a hyperbolic conservation law posed on an (N+1)-dimensional spacetime, whose flux is a field of differential forms of degree N. Generalizing the classical Kuznetsov's method, we derive an L1 error estimate which applies to a…
As it is known various dynamical processes can be modeled through the systems of time-fractional order pseudo-differential equations. In the modeling process one frequently faces with determining the adequate orders of time-fractional…
The initial inverse problem of finding solutions and their initial values ($t = 0$) appearing in a general class of fractional reaction-diffusion equations from the knowledge of solutions at the final time ($t = T$). Our work focuses on the…
We introduce in this document a direct method allowing to solve numerically inverse type problems for linear parabolic equations. We consider the reconstruction of the full solution of the parabolic equation posed in $\Omega\times (0,T)$ -…
We prove global existence of solutions to the initial value problem for a third order dispersive flow into compact locally Hermitian symmetric spaces. The equation we consider generalizes two-sphere-valued completely integrable systems…
We provide a clear and concise introduction to the subjects of inverse problems and data assimilation, and their inter-relations. The first part of our notes covers inverse problems; this refers to the study of how to estimate unknown model…