In this mini-course we will analyze highly nonlinear parabolic higher-order equations, like the thin-film and the Derrida-Lebowitz-Speer-Spohn equation. The aims are to develop an existence analysis and to study some properties of the solutions, e.g. positivity or their long-time behavior. The analysis strongly depends on the derivation of appropriate a priori estimates. Recently, we have developed a method which allows to derive these estimates in an algorithmic way. The idea is to perform the needed integration by parts, leading to the desired estimates, in a systematic way and to identify the integrations by parts by a decision problem for polynomial systems. This problem can be solved, at least in principle, by quantifier elimination. We will show how this method works, which results can be obtained, and how to make some of these estimates rigorous, using exponential variable transformations. Finally, we will mention some open challenging problems.

• Event: Working group on "The quantum drift diffusion equation and similiar PDE's: gradient flow structure and entropy methods"; Speakers: Giuseppe Savaré (Univ. Pavia) - Ansgar Juengel (TU Vienna); (2007)

In this mini-course we will analyze highly nonlinear parabolic higher-order equations, like the thin-film and the Derrida-Lebowitz-Speer-Spohn equation. The aims are to develop an existence analysis and to study some properties of the solutions, e.g. positivity or their long-time behavior. The analysis strongly depends on the derivation of appropriate a priori estimates. Recently, we have developed a method which allows to derive these estimates in an algorithmic way. The idea is to perform the needed integration by parts, leading to the desired estimates, in a systematic way and to identify the integrations by parts by a decision problem for polynomial systems. This problem can be solved, at least in principle, by quantifier elimination. We will show how this method works, which results can be obtained, and how to make some of these estimates rigorous, using exponential variable transformations. Finally, we will mention some open challenging problems.

• Event: Working group on "The quantum drift diffusion equation and similiar PDE's: gradient flow structure and entropy methods"; Speakers: Giuseppe Savaré (Univ. Pavia) - Ansgar Juengel (TU Vienna); (2007)

After a short introduction to gradient flows in (metric) spaces of probability measures, we discuss the application of this point of view to study the global existence of non negative solutions to the fourth-order ``quantum-drift diffusion'' equation under variational boundary conditions. Despite the lack of a maximum principle for fourth order equations, non negative solutions can be obtained as a limit of a variational approximation scheme by exploiting the particular structure of this equation, which is the gradient flow of the Fisher Information functional with respect to the Kantorovich-Rubinstein-Wasserstein distance. We will also devote some time to study a new family of "second order logarithmic Sobolev" inequalities, which play a crucial role in the derivation of a priori estimates for the solutions.

• Event: Working group on "The quantum drift diffusion equation and similiar PDE's: gradient flow structure and entropy methods"; Speakers: Giuseppe Savaré (Univ. Pavia) - Ansgar Juengel (TU Vienna); (2007)

In this mini-course we will analyze highly nonlinear parabolic higher-order equations, like the thin-film and the Derrida-Lebowitz-Speer-Spohn equation. The aims are to develop an existence analysis and to study some properties of the solutions, e.g. positivity or their long-time behavior. The analysis strongly depends on the derivation of appropriate a priori estimates. Recently, we have developed a method which allows to derive these estimates in an algorithmic way. The idea is to perform the needed integration by parts, leading to the desired estimates, in a systematic way and to identify the integrations by parts by a decision problem for polynomial systems. This problem can be solved, at least in principle, by quantifier elimination. We will show how this method works, which results can be obtained, and how to make some of these estimates rigorous, using exponential variable transformations. Finally, we will mention some open challenging problems.

• Event: Working group on "The quantum drift diffusion equation and similiar PDE's: gradient flow structure and entropy methods"; Speakers: Giuseppe Savaré (Univ. Pavia) - Ansgar Juengel (TU Vienna); (2007)

After a short introduction to gradient flows in (metric) spaces of probability measures, we discuss the application of this point of view to study the global existence of non negative solutions to the fourth-order ``quantum-drift diffusion'' equation under variational boundary conditions. Despite the lack of a maximum principle for fourth order equations, non negative solutions can be obtained as a limit of a variational approximation scheme by exploiting the particular structure of this equation, which is the gradient flow of the Fisher Information functional with respect to the Kantorovich-Rubinstein-Wasserstein distance. We will also devote some time to study a new family of "second order logarithmic Sobolev" inequalities, which play a crucial role in the derivation of a priori estimates for the solutions.

• Event: Working group on "The quantum drift diffusion equation and similiar PDE's: gradient flow structure and entropy methods"; Speakers: Giuseppe Savaré (Univ. Pavia) - Ansgar Juengel (TU Vienna); (2007)

After a short introduction to gradient flows in (metric) spaces of probability measures, we discuss the application of this point of view to study the global existence of non negative solutions to the fourth-order ``quantum-drift diffusion'' equation under variational boundary conditions. Despite the lack of a maximum principle for fourth order equations, non negative solutions can be obtained as a limit of a variational approximation scheme by exploiting the particular structure of this equation, which is the gradient flow of the Fisher Information functional with respect to the Kantorovich-Rubinstein-Wasserstein distance. We will also devote some time to study a new family of "second order logarithmic Sobolev" inequalities, which play a crucial role in the derivation of a priori estimates for the solutions.

• Event: Working group on "The quantum drift diffusion equation and similiar PDE's: gradient flow structure and entropy methods"; Speakers: Giuseppe Savaré (Univ. Pavia) - Ansgar Juengel (TU Vienna); (2007)