Uniqueness of solutions of the Derrida-Lebowitz-Speer-Spohn equation and quantum drift diffusion models

J.L. Fischer, Communications in Partial Differential Equations 38 (2013) 2004–2047.

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Abstract
We prove uniqueness of solutions of the DLSS equation in a class of sufficiently regular functions. The global weak solutions of the DLSS equation constructed by Jüngel and Matthes belong to this class of uniqueness. We also show uniqueness of solutions for the quantum drift-diffusion equation, which contains additional drift and second-order diffusion terms. The results hold in case of periodic or Dirichlet-Neumann boundary conditions. Our proof is based on a monotonicity property of the DLSS operator and sophisticated approximation arguments; we derive a PDE satisfied by the pointwise square root of the solution, which enables us to exploit the monotonicity property of the operator.
Publishing Year
Date Published
2013-11-01
Journal Title
Communications in Partial Differential Equations
Volume
38
Issue
11
Page
2004 - 2047
IST-REx-ID

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Fischer JL. Uniqueness of solutions of the Derrida-Lebowitz-Speer-Spohn equation and quantum drift diffusion models. Communications in Partial Differential Equations. 2013;38(11):2004-2047. doi:10.1080/03605302.2013.823548
Fischer, J. L. (2013). Uniqueness of solutions of the Derrida-Lebowitz-Speer-Spohn equation and quantum drift diffusion models. Communications in Partial Differential Equations, 38(11), 2004–2047. https://doi.org/10.1080/03605302.2013.823548
Fischer, Julian L. “Uniqueness of Solutions of the Derrida-Lebowitz-Speer-Spohn Equation and Quantum Drift Diffusion Models.” Communications in Partial Differential Equations 38, no. 11 (2013): 2004–47. https://doi.org/10.1080/03605302.2013.823548.
J. L. Fischer, “Uniqueness of solutions of the Derrida-Lebowitz-Speer-Spohn equation and quantum drift diffusion models,” Communications in Partial Differential Equations, vol. 38, no. 11, pp. 2004–2047, 2013.
Fischer JL. 2013. Uniqueness of solutions of the Derrida-Lebowitz-Speer-Spohn equation and quantum drift diffusion models. Communications in Partial Differential Equations. 38(11), 2004–2047.
Fischer, Julian L. “Uniqueness of Solutions of the Derrida-Lebowitz-Speer-Spohn Equation and Quantum Drift Diffusion Models.” Communications in Partial Differential Equations, vol. 38, no. 11, Taylor & Francis, 2013, pp. 2004–47, doi:10.1080/03605302.2013.823548.

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