A Stability Estimate for Robin Boundary Coefficients in Stokes Fluid Flows

Date Received: Feb 18, 2019

Date Published: Mar 18, 2019

DOI: https://doi.org/10.31817/vjas.2018.1.4.05

Views

2134

Download

426

Issue: Vol. 1 No. 4 (2018): Vietnam Journal of Agricultural Sciences

Section:

ENGINEERING AND TECHNOLOGY

How to Cite:

Sang, P., & Dung, N. (2019). A Stability Estimate for Robin Boundary Coefficients in Stokes Fluid Flows. Vietnam Journal of Agricultural Sciences, 1(4), 289–304. https://doi.org/10.31817/vjas.2018.1.4.05

A Stability Estimate for Robin Boundary Coefficients in Stokes Fluid Flows

Phan Quang Sang (*) 1   , Nguyen Thuy Dung 1

  • Corresponding author: pqsang@vnua.edu.vn
  • 1 Faculty of Information Technology, Vietnam National University of Agriculture, Hanoi 131000, Vietnam

    • Keywords

    inverse problems, Carleman inequality, Stokes equation, Stability estimate

    Abstract


    In this report, we examine the unsteady Stokes equations with non-homogeneous boundary conditions. As an application of a Carleman estimate, we first establish log type stabilities for the solution of the equations from either an interior measurement of the velocity, or a boundary observation depending on the trace of the velocity and of the Cauchy stress tensor measurements on a part of the boundary. We then consider the inverse problem of determining the time-independent Robin coefficient from a measurement of the solution and of Cauchy data on a sub-boundary.

    References

    Adams R. A. and Fournier J. (2003). Sobolev spaces. Pure and Applied Mathematics. New York - London: Acadmic Press. Vol 140.

    Badra M., Canbet F. and Dare J. (2016). Stability estimates for Navier - Stokes equations and application to inverse problems. Discrete and Continuous Dynamical Systems - Series B. Vol 21 (8). pp. 2379-2407.

    Baffico L., Grandmont C. and Maury B. (2010). Multiscale modeling of the respiratory tract. Mathematical Models and Methods in Applied Sciences. Vol 20 (1). pp. 59-93.

    Bellassoued M., Cheng J. and Choulli M. (2008). Stability estimate for an inverse boundary coefficient problem in thermal imaging. Journal of Mathematical Analysis and Applications. Vol 343 (1). pp. 328-336.

    Boulakia M., Egloffe A. C. and Grandmont C. (2013). Stability estimates for a Robin coefficient in the Two-dimensional Stokes system. Mathematical Control & Related Fields. Vol 3 (1). pp. 21-49.

    Bramble L. H. (2003). A proof of the inf-sup condition for the Stokes equations on Lipschitz domains. Mathematical Models and Methods in Applied Sciences. Vol 13 (3). pp. 361-371.

    Chaabane S., Fellah I., Jaoua M. and Leblond J. (2004). Logarithmic stability estimates for a Robin coefficient in two-dimensional Laplace inverse problems. Inverse Problems. Vol 20 (1). pp. 47-59.

    Cheng J., Choulli M. and Lin J. (2008). Stable determination of a boundary coefficient in an elliptic equation. Mathematical Models and Methods in Applied Sciences. Vol 18 (1). pp. 107-123.

    Dusenbery D. B. (2011). Living at Micro Scale. Harvard University Press. Cambridge, MA 02138, USA.

    Fabre C. and Lebeau G. (1996). Prolongement unique des solutions. Communications in Partial Differential Equations. Vol 21 (3-4). pp. 573-596.

    Imanuvilov O. Y. and Yamamoto M. (2003). Carleman Inequalities for Parabolic Equations in Sobolev Spaces of Negative Order and Exact Controllability for semilinear Parabolic Equations. Publications of the Research Institute for Mathematical Sciences. Vol 39 (2). pp. 227-274.

    Tucsnak M. and Weiss G. (2009). Observation and control for operator semigroups. Birkhauser Verlag: Birkhauser Advabced texts.

    Necas J. (2012). Direct Methods in the Theory of Elliptic Equations. New york: Springer-Verlag Berlin Heidelberg.

    Sincich E. (2007). Lipschitz stability for the inverse Robin problem. Inverse Problems. Vol 23 (3). pp. 1311-1326.

    Vignon-Clementel I. E., Figueroa C. A., Jansen K. E. and Taylor C. A. (2006). Outflow boundary conditions for three-dimensional finite element modeling of blood flow and pressure in arteries. Computer Methods in Applied Mechanics and Engineering. Vol 195 (29-32). pp. 3776-3796.