Nonlinear PDE's and Applications: C.I.M.E. Summer School, by Stefano Bianchini, Eric A. Carlen, Alexander Mielke, Cédric

By Stefano Bianchini, Eric A. Carlen, Alexander Mielke, Cédric Villani

This quantity collects the notes of the CIME direction "Nonlinear PDE’s and purposes" held in Cetraro (Italy) on June 23–28, 2008. It includes 4 sequence of lectures, introduced by way of Stefano Bianchini (SISSA, Trieste), Eric A. Carlen (Rutgers University), Alexander Mielke (WIAS, Berlin), and Cédric Villani (Ecole Normale Superieure de Lyon). They provided a vast evaluation of far-reaching findings and intriguing new advancements referring to, specifically, optimum shipping conception, nonlinear evolution equations, practical inequalities, and differential geometry. A sampling of the most subject matters thought of the following comprises optimum delivery, Hamilton-Jacobi equations, Riemannian geometry, and their hyperlinks with sharp geometric/functional inequalities, variational equipment for learning nonlinear evolution equations and their scaling homes, and the metric/energetic concept of gradient flows and of rate-independent evolution difficulties. The booklet explores the basic connections among all of those themes and issues to new study instructions in contributions through prime specialists in those fields.

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2 involve evolution equations (in the form of partial differential equations), while the example in Sect. 3 involves a discrete-time dynamical system. In Sect. 2 we shall discuss certain applications of the inequalities (1)–(4) to the study of the evolution equations introduced there, while in Sect. 3 we shall apply the discrete-time dynamics introduced there to prove the inequalities themselves. The remaining sections further develop the themes introduce in the first three, but in the context of other examples.

1 19 The Sharp Hardy-Littlewood-Sobolev Inequality For two non-negative measurable functions f and g on Rd , d ≥ 1, and a number λ with 0 ≤ λ < d, define the functional Iλ (f, g) by Iλ (f, g) = and for 1 ≤ p < ∞, let f p Rd Rd f (x) 1 g(y)dxdy, |x − y|λ (1) denote the Lp norm of f defined by 1/p f p = Rd f p dx . (2) The HLS inequality provides an upper bound on Iλ (f, f ) in terms of p(λ) where 2d p(λ) = . (3) 2d − λ 2d is determined by the scale invariance of the The value p(λ) = 2d − λ functional Iλ (f, f ): For s > 0, define f (s) by f f (s) (x) = f (x/s).

The bacteria themselves also diffuse across the surface, but with a drift: The bacteria tend to move towards higher concentrations of the attractant, and this induces a drift term tending to concentrate the population, and countering the spreading effects of the diffusion. • The key point of mathematical interest in this model is the competition between the concentrating effects of the drift induced by the chemical attractant and the spreading effects of the diffusion. If ρ(x) denotes the population density on R2 , and c(x) denotes the concentration of the chemical attractant, also on R2 of course, the system of equations is Functional Inequalities and Dynamics 41 ⎧ ∂ρ ⎪ ⎪ (t, x) = div [∇ρ(t, x) − ρ(t, x)∇c(t, x)] ⎨ ∂t −Δc(t, x) = ρ(t, x) , ⎪ ⎪ ⎩ ρ(0, x) = ρ0 (x) ≥ 0 t > 0 , x ∈ R2 , t > 0 , x ∈ R2 , x ∈ R2 .

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