A stability estimate in the problem of determining the dispersion index and relaxation for the transport equation
The transport equation \[ L(\sigma,K)V \equiv\nabla U\cdot V(\theta)+ \sigma U+SU=0, \quad (x,\theta)\in G\equiv D\times [0,2\pi], \] where \(\sigma= \sigma(x)\), \(\nabla= ({\partial\over \partial x_1}, {\partial\over \partial x_2})\), \(V(\theta)= (\cos\theta, \sin \theta)\), \(\nabla U\cdot V\) is the inner product of the vectors \(\nabla U\) and \(V(\theta)\), and \(SU\) is the operator: \[ SU\equiv \int^{2\pi}_0 K\bigl(x,V (\theta)\cdot V(\theta')\bigr) \cdot U(x, \theta') d\theta' \] describing the scattering has been the object of serious study for long. The present author has contributed already much to this field of studies. The inverse problem for the transport equation, consisting in finding the relaxation \(\sigma(x)\), \(x\in\mathbb{R}^2\) and the dispersion index \(K(x,V(\theta), V(\theta'))\) is considered here. After presenting the statement of the problem and the main results, two theorems and ten lemmas are given in four sections. Details of most of these are also written down alongwith certain stability estimates as well.
scientific article; zbMATH DE number 1023969