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The interaction of nonlinear analysis and modern applied mathematics. (English) Zbl 1153.76300

Proc. Int. Congr. Math., Kyoto/Japan 1990, Vol. I, 175-191 (1991).
The common theme of this review paper is the discussion of weak convergence phenomena for solutions of nonlinear partial differential equations. The weak convergence usually follows from \(L^p\)-bounds for the solutions, and it occurs along a minimizing sequence for problems from the Calculus of Variations, as a blow-up time is approached, as amplitudes and periods of perturbations go to zero, as a viscosity goes to zero, or as irregular data are approximated by smooth ones. After briefly introducing some of the mathematical tools for the study of the gap between weak and strong convergence in \(L^p\)-spaces (Young measures for oscillations, defect measures for concentrations), the author first gives an overview of applications such as the decription of defects in materials (phase transitions in solid crystals and defects in liquid crystals), the behavior of focussing solutions of nonlinear Schrödinger equations with critical nonlinear terms past blow-up, formal and rigorous theories in nonlinear geometric optics, and general oscillation and concentration phenomena for three-dimensional incompressible fluid flow in the high Reynolds number limit. The second part of the paper is specifically concerned with two-dimensional inviscid incompressible flow, in particular with problems whose vorticity is in the limit concentrated on a one-dimensional vortex sheet. Here more detailed results are given, including results on the nature of the weak limit: Only concentration is possible, occurring on a set of small Hausdorff dimension; concentration-cancellation is shown to occur under additional assumptions and is conjectured to occur in general, leading to a weak limit that is still a solution. Heuristics and proof ideas are given, and open problems are formulated. There are 61 references, including recent preprints.
For the entire collection see [Zbl 0741.00019].

MSC:

76-02 Research exposition (monographs, survey articles) pertaining to fluid mechanics
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35Q30 Navier-Stokes equations
76B47 Vortex flows for incompressible inviscid fluids
35Q35 PDEs in connection with fluid mechanics
76Q05 Hydro- and aero-acoustics
76F99 Turbulence
76M25 Other numerical methods (fluid mechanics) (MSC2010)
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