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Über dieses Buch

The topic of the 2010 Abel Symposium, hosted at the Norwegian Academy of Science and Letters, Oslo, was Nonlinear Partial Differential Equations, the study of which is of fundamental importance in mathematics and in almost all of natural sciences, economics, and engineering.

This area of mathematics is currently in the midst of an unprecedented development worldwide. Differential equations are used to model phenomena of increasing complexity, and in areas that have traditionally been outside the realm of mathematics. New analytical tools and numerical methods are dramatically improving our understanding of nonlinear models. Nonlinearity gives rise to novel effects reflected in the appearance of shock waves, turbulence, material defects, etc., and offers challenging mathematical problems. On the other hand, new mathematical developments provide new insight in many applications.

These proceedings present a selection of the latest exciting results by world leading researchers.



Convergence of Wigner Transforms in a Semiclassical Limit

We prove convergence of the Wigner transforms of solutions to the Schrödinger equation, in a semiclassical limit, to solutions to the Liouville equation. We are able to include in our convergence result rough or singular potentials (with Coulomb repulsive singularities), provided convergence is understood for “almost all” initial data. The rigorous statement involves a suitable extension of the DiPerna–Lions theory to the infinite-dimensional space of probability measure, where both the Wigner and the Liouville dynamics can be read.
Luigi Ambrosio

Contractive Metrics for Nonsmooth Evolutions

Given an evolution equation, a standard way to prove the well posedness of the Cauchy problem is to establish a Gronwall type estimate, bounding the distance between any two trajectories. There are important cases, however, where such estimates cannot hold, in the usual distance determined by the Euclidean norm or by a Banach space norm.
In alternative, one can construct different distance functions, related to a Riemannian structure or to an optimal transportation problem. This paper reviews various cases where this approach can be implemented, in connection with discontinuous ODEs on ℝ n , nonlinear wave equations, and systems of conservation laws. For all the evolution equations considered here, a metric can be constructed such that the distance between any two solutions satisfies a Gronwall type estimate. This yields the uniqueness of solutions, and estimates on their continuous dependence on the initial data.
Alberto Bressan

Non-local Diffusions, Drifts and Games

This is a brief discussion of the properties of solutions to several non-linear elliptic equations involving diffusive processes of non-local nature. These equation arise in several contexts: from continuum mechanics and phase transition, from population dynamics, from optimal control and game theory. The equations coming from continuum mechanics exhibit a variational structure and a theory parallel to the De Giorgi–Nash–Moser was necessary to show existence of regular solutions. Population dynamics suggests “porous media like equations” with a non-local pressure, and from optimal control we obtain fully non-linear equations that require methods of the type of the Krylov–Safonov–Evans theory. Finally, we discuss some non-local p and infinite Laplacian models coming from game theory.
Luis Caffarelli

Characteristic Discontinuities and Free Boundary Problems for Hyperbolic Conservation Laws

We are concerned with entropy solutions of hyperbolic systems of conservation laws in several space variables. The Euler equations of gas dynamics and magnetohydrodynamics (MHD) are prototypes of hyperbolic conservation laws. In general, there are two types of discontinuities in the entropy solutions: shock waves and characteristic discontinuities, in which characteristic discontinuities can be either vortex sheets or entropy waves. In gas dynamics and MHD, across a vortex sheet, the tangential velocity field has a jump while the normal velocity is continuous; across an entropy wave, the entropy has a jump while the velocity field is continuous. A vortex sheet or entropy wave front is a part of the unknowns, which is a free boundary. Compressible vortex sheets and entropy waves, along with shock and rarefaction waves, occur ubiquitously in nature and are fundamental waves in the entropy solutions to multidimensional hyperbolic conservation laws. The local stability of shock and rarefaction waves has been relatively better understood. In this paper we discuss the stability issues for vortex sheets/entropy waves and present some recent developments and further open problems in this direction. First we discuss vortex sheets and entropy waves for the Euler equations in gas dynamics and some recent developments for a rigorous mathematical theory on their nonlinear stability/instability. Then we review our recent study and present a supplement to the proof on the nonlinear stability of compressible vortex sheets under the magnetic effect in three-dimensional MHD. The compressible vortex sheets in three dimensions are unstable in the regime of pure gas dynamics. Our main concern is whether such vortex sheets can be nonlinearly stabilized under the magnetic fields. To achieve this, we first set up the current-vortex sheet problem as a free boundary problem; then we establish high-order energy estimates of the solutions to the linearized problem, which shows that the current-vortex sheets are linearly stable when the jump of the tangential velocity is dominated by the jump of the non-paralleled tangential magnetic fields; and finally we develop a suitable iteration scheme of the Nash–Moser–Hörmander type to obtain the existence and nonlinear stability of compressible current-vortex sheets, locally in time. Some further open problems and several related remarks are also presented.
Gui-Qiang Chen, Ya-Guang Wang

h-Principle and Rigidity for C 1,α Isometric Embeddings

In this paper we study the embedding of Riemannian manifolds in low codimension. The well-known result of Nash and Kuiper (Nash in Ann. Math. 60:383–396, 1954; Kuiper in Proc. Kon. Acad. Wet. Amsterdam A 58:545–556, 1955; Kuiper in Proc. Kon. Acad. Wet. Amsterdam A 58:683–689, 1955) says that any short embedding in codimension one can be uniformly approximated by C 1 isometric embeddings. This statement clearly cannot be true for C 2 embeddings in general, due to the classical rigidity in the Weyl problem. In fact Borisov extended the latter to embeddings of class C 1,α with α>2/3 in (Borisov in Vestn. Leningr. Univ. 14(13):20–26, 1959; Borisov in Vestn. Leningr. Univ. 15(19):127–129, 1960). On the other hand he announced in (Borisov in Doklady 163:869–871, 1965) that the Nash–Kuiper statement can be extended to local C 1,α embeddings with α<(1+n+n 2)−1, where n is the dimension of the manifold, provided the metric is analytic. Subsequently a proof of the 2-dimensional case appeared in (Borisov in Sib. Mat. Zh. 45(1):25–61, 2004). In this paper we provide analytic proofs of all these statements, for general dimension and general metric.
Sergio Conti, Camillo De Lellis, László Székelyhidi

About Existence, Symmetry and Symmetry Breaking for Extremal Functions of Some Interpolation Functional Inequalities

This chapter is devoted to a review of some recent results on existence, symmetry and symmetry breaking of optimal functions for Caffarelli–Kohn–Nirenberg (CKN) and weighted logarithmic Hardy (WLH) inequalities. These results have been obtained in a series of papers (Dolbeault et al. in Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 7(2):313–341, 2008; Dolbeault et al. in Adv. Nonlinear Stud. 9(4):713–726, 2009; Dolbeault, Esteban in Extremal functions for Caffarelli–Kohn–Nirenberg and logarithmic Hardy inequalities Calc. Var. Partial Differ. Equ., 2010; del Pino et al. in J. Funct. Anal. 259(8):2045–2072, 2010; Dolbeault et al. in Radial symmetry and symmetry breaking for some interpolation inequalities, preprint, 2010) in collaboration with M. del Pino, S. Filippas, M. Loss, G. Tarantello and A. Tertikas. Here we put the highlights on a symmetry breaking result: extremals of some inequalities are not radially symmetric in regions where the symmetric extremals are linearly stable. Special attention is paid to the study of the critical cases for (CKN) and (WLH).
Jean Dolbeault, Maria J. Esteban

On the Oberbeck–Boussinesq Approximation on Unbounded Domains

We study the Oberbeck–Boussinesq approximation describing the motion of an incompressible, heat-conducting fluid occupying a general unbounded domain in R 3. We provide a rigorous justification of the model by means of scale analysis of the full Navier–Stokes–Fourier system in the low Mach and Froude number regime on large domains, the diameter of which is proportional to the speed of sound. Finally, we show that the total energy of any solution of the resulting Oberbeck–Boussinesq system tends to zero with growing time.
Eduard Feireisl, Maria E. Schonbek

Universal Profiles and Rigidity Theorems for the Energy Critical Wave Equation

In this note we review recent joint works with F. Merle and with T. Duyckaerts and F. Merle on global existence, scattering and finite time blow-up for the focusing energy critical non-linear wave equation in three space dimensions.
Carlos Kenig

A Simple Energy Pump for the Surface Quasi-geostrophic Equation

We consider the question of growth of high order Sobolev norms of solutions of the conservative surface quasi-geostrophic equation. We show that if s>0 is large then for every given A there exists initial data with a norm that is small in H s such that the H s norm of corresponding solution at some time exceeds A. The idea of the construction is quasilinear. We use a small perturbation of a stable shear flow. The shear flow can be shown to create small scales in the perturbation part of the flow. The control is lost once the nonlinear effects become too large.
Alexander Kiselev, Fedor Nazarov

On the Formation of Trapped Surfaces

In a recent important breakthrough D. Christodoulou (The Formation of Black Holes in General Relativity. Monographs in Mathematics. Eur. Math. Soc., Zurich, 2009) has solved a long standing problem of General Relativity of evolutionary formation of trapped surfaces in the Einstein-vacuum space-times. He has identified an open set of regular initial conditions on a finite outgoing null hypersurface leading to a formation a trapped surface in the corresponding vacuum space-time to the future of the initial outgoing hypersurface and another incoming null hypersurface with the prescribed Minkowskian data. He also gave a version of the same result for data given on part of past null infinity. His proof is based on an inspired choice of the initial condition, an ansatz which he calls short pulse, and a complex argument of propagation of estimates, consistent with the ansatz, based, largely, on the methods used in the global stability of the Minkowski space (Christodoulou and Klainerman in The Global Nonlinear Stability of the Minkowski Space. Princeton Mathematical Series, vol. 41, 1993). Once such estimates are established in a sufficiently large region of the space-time the actual proof of the formation of a trapped surface is quite straightforward.
Christodoulou’s result has been significantly simplified and extended in my joint works with I. Rodnianski (Klainerman and Rodnianski in On the formation of trapped surfaces, Acta Math. 2011, in press) and (Klainerman and Rodnianski in Discrete Contin. Dyn. Syst. 28(3):1007–1031, 2010). In this note I will give a short survey of these results.
Sergiu Klainerman

Surface Relaxation Below the Roughening Temperature: Some Recent Progress and Open Questions

We discuss two recent projects concerning the evolution of a crystal surface below the roughening temperature. One addresses the evolution of a monotone one-dimensional step train (joint work with Hala Al Hajj Shehadeh and Jonathan Weare). The other addresses the finite-time flattening predicted by a fourth-order PDE model (joint work with Yoshikazu Giga). For each project we begin with a discussion of the mathematical model; then we summarize the recent results, the main ideas behind them, and some related open problems.
Robert V. Kohn

Climate Science, Waves and PDEs for the Tropics

A reader’s guide to recent applied mathematics development in multi-scale modeling in the tropics is provided here including the mathematical theory of precipitation fronts as well as singular limits with variable coefficients in the fast variables.
Andrew J. Majda

On the Propagation of Oceanic Waves Driven by a Strong Macroscopic Flow

In this work we study oceanic waves in a shallow water flow subject to strong wind forcing and rotation, and linearized around an inhomogeneous (nonzonal) stationary profile. This extends the study (Cheverry et al. in Semiclassical and spectral analysis of oceanic waves, Duke Math. J., accepted), where the profile was assumed to be zonal only and where explicit calculations were made possible due to the 1D setting.
Here the diagonalization of the system, which allows to identify Rossby and Poincaré waves, is proved by an abstract semi-classical approach. The dispersion of Poincaré waves is also obtained by a more abstract and more robust method using Mourre estimates. Only some partial results however are obtained concerning the Rossby propagation, as the two dimensional setting complicates very much the study of the dynamical system.
Isabelle Gallagher, Thierry Paul, Laure Saint-Raymond

Hierarchical Construction of Bounded Solutions of div U=F in Critical Regularity Spaces

We implement the hierarchical decomposition introduced in (Tadmor in Hierarchical construction of bounded solutions in critical regularity spaces, arXiv:​1003.​1525v2), to construct uniformly bounded solutions of the problem div U=F, where the two-dimensional data is in the critical regularity space, \(F\in L^{2}_{\#}(\mathbb{T}^{2})\). Criticality in this context, manifests itself by the lack of linear mapping, \(F\in L^{2}_{\#}(\mathbb{T}^{2})\mapsto U\in L^{\infty}(\mathbb{T}^{2})\), (Bourgain and Brezis in J. Am. Math. Soc. 16(2):393–426, 2003). Thus, the intriguing aspect here is that although the problem is linear, the construction of its uniformly bounded solutions is not.
Eitan Tadmor, Changhui Tan

Nonlinear Diffusion with Fractional Laplacian Operators

We describe two models of flow in porous media including nonlocal (long-range) diffusion effects. The first model is based on Darcy’s law and the pressure is related to the density by an inverse fractional Laplacian operator. We prove existence of solutions that propagate with finite speed. The model has the very interesting property that mass preserving self-similar solutions can be found by solving an elliptic obstacle problem with fractional Laplacian for the pair pressure-density. We use entropy methods to show that these special solutions describe the asymptotic behavior of a wide class of solutions.
The second model is more in the spirit of fractional Laplacian flows, but nonlinear. Contrary to usual Porous Medium flows (PME in the sequel), it has infinite speed of propagation. Similarly to them, an L 1-contraction semigroup is constructed and it depends continuously on the exponent of fractional derivation and the exponent of the nonlinearity.
Juan Luis Vázquez

(Ir)reversibility and Entropy

In the 1860’s emerges a revolutionary idea: many properties of the world around us can be explained by combining the atomistic hypothesis with the statistical theory. Some of the great scientific conquests from this time are the Boltzmann equation, which triggers one of the first qualitative studies of a complicated nonlinear partial differential equation; the notion of statistical entropy, which would later be fundamental in other areas of physics and mathematics, including information theory; and the notion of macroscopic irreversibility emerging from microscopically reversible laws. Thus the basic rules of statistical physics were set until Boltzmann’s irreversibility paradigm was shaken by Landau’s discovery of the Landau damping effect, about 80 years later, which opened the idea that equilibration is compatible with preservation of information, and led to a number of problems concerning the statistical theory of matter.
Cédric Villani
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