Celebratio Mathematica — Morawetz

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C. S. Morawetz: “An alternative proof of DiPerna’s theorem,” Comm. Pure Appl. Math. 44 : 8–9 (1991), pp. 1081–1090. To Natascha in love and affection. MR 1127051 Zbl 0763.35056 article

The method of compensated compactness is the most elegant and general way to prove the existence of a solution of the initial boundary value problem for a genuinely nonlinear, strictly hyperbolic system of equations for two unknowns and two independent variables. The technique is as follows: Add viscous terms and prove the existence of the solution to the new system. Establish estimates independent of the coefficient of viscosity that are sufficient to yield a weak limit to the viscous solutions as the viscosity goes to zero. Represent the weak limit and the weak limit of functions of the solutions in terms of the Young measure. Construct a large family of entropy pairs which satisfy certain conservation inequalities. Apply the Tartar–Murat relation (see [2], [3]) for such entropy pairs and conclude that the Young measure is a Dirac measure. From this it is easy to see that the weak limit does satisfy the differential equations and initial data weakly. Finding the right family and concluding that the Young measure is Dirac was the work of Di Perna (see [1]) and it is this theorem that is represented in the title. Although no new theorems about nonlinear hyperbolic pairs of equations resulted, the method provided much insight and a new challenge to find the necessary estimates to apply it to other problems. It was extended with some necessary assumptions to the mixed type equations of transonic flow by Morawetz; see [5]. A more elegant way of proving the theorem using entropy pairs of compact support has been provided by D. Serre; see [4]. Here we simplify Serre’s approach and extend it to equations such as those for flow that is not subsonic.

C. S. Morawetz, D. C. Stevens, and H. Weitzner: “A numerical experiment on a second-order partial differential equation of mixed type,” Comm. Pure Appl. Math. 44 : 8–9 (1991), pp. 1091–1106. In appreciation of Natascha Brunswick’s many contributions to the Courant Institute and particularly to this journal. MR 1127052 Zbl 0836.35102 article

In an earlier article in this journal one of us examined a problem of linear electromagnetic wave propagation in a plasma based on a particular plasma model (see [1]) and commented on the mathematically rather non-standard system of linear partial differential equations for the problem. Since then our attempts to solve the system numerically have not led to entirely satisfactory results. In this note we report on numerical experiments on a much simpler model partial differential equation which can be studied in much greater detail and which clarifies the numerical and mathematical difficulties inherent in the original problem.

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