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We study the limiting behavior of systems of hyperbolic conservation laws with stiff relaxation terms. Reduced systems, inviscid and viscous local conservation laws, and weakly nonlinear limits are derived through asymptotic expansions. An entropy condition is introduced for N x N systems that ensures the hyperbolicity of the reduced inviscid system. The resulting characteristic speeds are shown to be interlaced with those of the original system. Moreover, the first correction to the reduced system is shown to be dissipative. A partial converse is proved for 2 x 2 systems. This structure is then applied to study the convergence to the reduced dynamics for the 2 X 2 case. 0 1994 John Wiley & Sons, Inc.

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To the memory of R. J. DiPerna. AbstractUsing relative entropy estimates about an absolute Maxwellian, it is shown that any properly scaled sequence of DiPerna-Lions renormalized solutions of some classical Boltzmann equations has fluctuations that converge to an infinitesimal Maxwellian with fluid variables that satisfy the incompressibility and Boussinesq relations. Moreover, if the initial fluctuations entropically converge to an infinitesimal Maxwellian then the limiting fluid variables satisfy a version of the Leray energy inequality. If the sequence satisfies a local momentum conservation assumption, the momentum densities globaly converge to a solution of the Stokes equation. A similar discrete time version of this result holds for the Navier-Stokes limit with an additional mild weak compactness assumption. The continuous time Navier-Stokes limit is also discussed.

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In Part I* we have shown, see Theorem 2.10, that as the coefficient of u,,, tends to zero, the solution of the initial value problem for the KdV equation tends to a limit ii in the distribution sense. We have expressed by formula (3.59), where I)? is the partial derivative with respect to x of the function $* defined in Theorem 3.9 as the solution of the variational problem formulated in (2.16), (2.17). I)* is uniquely characterized by the variational condition (3.34); its partial derivatives satisfy (3.51) and (3.52), where I is the set I' defined in The Solution Until Break-timeThe variational problem (2.16)-(2.18) contains x and t as parameters. We first investigate the case t = 0. We make the assumption, justified a posteriori, that for t < t b the set I consists of a single interval

We consider the so-called lake and great lake equations, which are shallow water equations that describe the long-time motion of an inviscid, incompressible fluid contained in a shallow basin with a slowly spatially varying bottom, a free upper surface, and vertical side walls, under the influence of gravity and in the limit of small characteristic velocities and very small surface amplitude. If these equations are posed on a space-periodic domain and the initial data are real analytic, the solution remains real analytic for all times. The proof is based on a characterization of Gevrey classes in terms of decay of Fourier coefficients. In particular, our result recovers known results for the Euler equations in two and three spatial dimensions. We believe the proof is new. Academic Press

In Part I* we have shown, see Theorem 2.10, that as the coefficient of uxxx tends to zero, the solution of the initial value problem for the KdV equation tends to a limit u in the distribution sense. We have expressed u by formula (3.59), where ψx is the partial derivative with respect to x of the function ψ* defined in Theorem 3.9 as the solution of the variational problem formulated in (2.16), (2.17). ψ* is uniquely characterized by the variational condition (3.34); its partial derivatives satisfy (3.51) and (3.52), where I is the set Io defined in (3.36). In Section 4 we show that for t