4 research outputs found
Coefficient identification in parabolic equations with final data
In this work we determine the second-order coefficient in a parabolic
equation from the knowledge of a single final data. Under assumptions on the
concentration of eigenvalues of the associated elliptic operator, and the
initial state, we show the uniqueness of solution, and we derive a Lipschitz
stability estimate for the inversion when the final time is large enough. The
Lipschitz stability constant grows exponentially with respect to the final
time, which makes the inversion ill-posed. The proof of the stability estimate
is based on a spectral decomposition of the solution to the parabolic equation
in terms of the eigenfunctions of the associated elliptic operator, and an ad
hoc method to solve a nonlinear stationary transport equation that is itself of
interest
Inverse moving source problems for parabolic equations
This paper is concerned with the inverse moving source problems for parabolic
equations. Given the temporal function, we prove the uniqueness of the
nonlinear inverse problem of determining the orbit function by final data
measured in a bounded domain. On the other hand, given the orbit function we
also show that the profile function can be uniquely determined by final data
measured in a bounded domain away from the domain enclosing the moving orbit.
The proofs adopt the Fourier approach and results from complex analysis
Direct Problem of Gas Diffusion in Polar Firn
Simultaneous use of partial differential equations in conjunction with data
analysis has proven to be an efficient way to obtain the main parameters of
various phenomena in different areas, such as medical, biological, and
ecological. In the ecological field, the study of climate change (including
global warming) over the past centuries requires estimating different gas
concentrations in the atmosphere, mainly CO2. Antarctic and Greenland Polar
snow and ice constitute a unique archive of past climates and atmospheres.
The mathematical model of gas trapping in deep polar ice (firns) has been
derived in [8, 11, 12, 13]. In this paper, we study the theoretical aspects of
existence and uniqueness for the obtained, almost singular, parabolic partial
differential equations
Coefficient identification in parabolic equations with final data
International audienceIn this work we determine the second-order coefficient in a parabolic equation from the knowledge of a single final data. Under assumptions on the concentration of eigenvalues of the associated elliptic operator, and the initial state, we show the uniqueness of solution, and we derive a Lipschitz stability estimate for the inversion when the final time is large enough. The Lipschitz stability constant grows exponentially with respect to the final time, which makes the inversion ill-posed. The proof of the stability estimate is based on a spectral decomposition of the solution to the parabolic equation in terms of the eigenfunctions of the associated elliptic operator, and an ad hoc method to solve a nonlinear stationary transport equation that is itself of interest