Global Null Controllability of the 1-Dimensional Nonlinear Slow Diffusion Equation
The authors prove the global null controllability for the 1-dimensional nonlinear slow diffusion equation by using both a boundary and an internal control. They assume that the internal control is only time dependent. The proof relies on the return method
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Abstract The authors prove the global null controllability for the 1-dimensional nonlinear slow diffusion equation by using both a boundary and an internal control. They assume that the internal control is only time dependent. The proof relies on the return method in combination with some local controllability results for nondegenerate equations and rescaling techniques. Keywords Nonlinear control · Nonlinear slow diffusion equation · Porous medium equation Mathematics Subject Classification 35L65 · 35L567
Project supported by the ITN FIRST of the Seventh Framework Programme of the European Community (No. 238702), the ERC advanced grant 266907 (CPDENL) of the 7th Research Framework Programme (FP7), DGISPI of Spain (ProjectMTM2011-26119) and the Research Group MOMAT (No. 910480) supported by UCM. J.-M. Coron (B) UMR 7598 Laboratoire Jacques-Louis Lions, Institut Universitaire de France and Université Pierre et Marie Curie (Paris 6), 4, place Jussieu, 75252 Paris cedex 5, France e-mail: [email protected] J.I. Díaz Instituto de Matemática Interdisiplinar and Dpto. de Matemática Aplicada, Universidad Complutense de Madrid, Plaza de las Ciencias 3, 28040 Madrid, Spain e-mail: [email protected] A. Drici UMR 7598 Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie (Paris 6), 4, place Jussieu, 75252 Paris cedex 5, France e-mail: [email protected] T. Mingazzini Dpto. de Matemática Aplicada, Universidad Complutense de Madrid, Plaza de las Ciencias 3, 28040 Madrid, Spain e-mail: [email protected] P.G. Ciarlet et al. (eds.), Partial Differential Equations: Theory, Control and Approximation, DOI 10.1007/978-3-642-41401-5_8, © Springer-Verlag Berlin Heidelberg 2014
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1 Introduction We study the null controllability of the 1-dimensional nonlinear slow diffusion equation, sometimes referred to as the Porous Media Equation (or PME for short), using both internal and boundary controls. The methods we used need such a combination of controls due to the degenerate nature of this quasilinear parabolic equation. The PME belongs to the more general family of nonlinear diffusion equations of the form yt − φ(y) = f,
(1.1)
where φ is a continuous nondecreasing function with φ(0) = 0. For the PME, the constitutive law is precisely given by φ(y) = |y|m−1 y
(1.2)
with m ≥ 1. This family of equations arises in many different frameworks and, depending on the nature of φ, it models different diffusion processes, mainly grouped into three categories: “slow diffusion”, “fast diffusion” and linear processes. The “slow diffusion” case is characterized by a finite speed of propagation and the formation of free boundaries, while the “fast diffusion” one is characterized by a finite extinction time, which means that the solution becomes identically zero after a finite time. If one neglects the source term, i.e., f ≡ 0, and imposes the constraint of nonnegativeness to the solutions (which is fundamental in all the applications where y represents for example a density), then one can prec
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