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Uniform discrete Sobolev space estimates are proven for a class of finite-difference schemes for singularly-perturbed hyperbolic-parabolic systems. The estimates obtained improve previous results even when the PDEs do not involve singular perturbations. These estimates are used in a companion paper to prove the convergence of solutions as the discretization parameter and/or the singular perturbation parameter tends to zero.

Solutions of certain finite-difference schemes for singularly-perturbed evolutionary PDEs converge as the perturbation parameter and/or the discretization parameters tend to zero. Under suitable hypotheses a sharp convergence rate of order one-half in the time step, uniform in the perturbation parameter, is obtained.

We calculate the critical exponent of nonlinear Schrodinger (NLS) equations with anisotropic negative fourth-order dispersion using an anisotropic Gagliardo–Nirenberg inequality. We also prove global existence, and in some cases uniqueness, for subcritical solutions and for critical solutions with small L2 norm, without making use of Strichartz-type estimates for the linear operator. At exponents equal to or above critical, the blowup profile is anisotropic. Our results imply, in particular, that negative fourth-order temporal dispersion arrests spatio-temporal collapse in Kerr media with anomalous time-dispersion in one transverse dimension but not in two transverse dimensions. We also show that a small negative anisotropic fourth-order dispersion stabilizes the (otherwise unstable) waveguide solutions of the two-dimensional critical NLS.

The O (ϵ) part of solutions obtained via Glimm′s scheme of strictly hyperbolic conservation laws with O (ϵ) periodic BV initial data is shown to tend as ϵ → 0 to the unique entropy solution of the modulation equations of weakly nonlinear geometric optics derived by Majda and Rosales.

Using Glimm's scheme, sufficient conditions are derived for the global existence of a weak solution to a strictly hyperbolic genuinely nonlinear system of partial differential equations in one space dimension when the initial data is a small BV perturbation of a solvable Riemann problem. By using the finite propagation speed of the system, this yields a local existence theorem for arbitrary BV initial data that satisfies the above-mentioned conditions at all large jumps. The case of linearly degenerate fields is also treated, and the results are applied to the p -system and to the 1-D nonisentropic γ-gas-law Euler equations.

An important question in the study of nonlinear equations is the understanding of the asymptotic behavior of families of solutions, or approximate solutions. One of the difficulty is known to be the nonlinear interaction of oscillations. This phenomenon has been studied for special classes of solutions, such as oscillatory functions with phases and amplitudes. This is the domain of geometric optics. In some cases, general families of bounded solutions have also been considered. The present paper enters the later category, but is highly inspired by the results of nonlinear geometric optics. Geometric optics provides a precise description of the asymptotic structure of solutions to both linear (e.g. [Wh]) and nonlinear ([J], [HK], [MR], [HMR], [JMR 1], [JMR 2], [S], . . .) hyperbolic PDEs whose initial data u(0, x) = u0(x) + eu1(x, ~ φ0(x) e ) have rapid oscillations with prescribed phases ~ φ0 and a single prescribed scale 1e . Such solutions are important in both theory and applications ([L], [KK]). In the lowest approximation, oscillations having different spatial scales interact only when m equals zero, since otherwise the amplitudes of higher-frequency oscillations become negligible relative to those of lower frequencies. Taking m = 0 requires L∞ estimates. Systems that are well-posed in L∞ occur mostly in one spatial dimension. For such systems, arbitrary uniformly bounded sequences of initial data may be considered, but comparatively little is known about the asymptotic structure of the resulting solutions. Since the oscillations in such solutions may be extremely complex, a natural problem is to determine their non-oscillatory part, which may be identified with the weak limit of the sequence of solutions. The highly oscillatory nature of the solutions manifests itself in the absence of strong convergence in general. Of course, some information about those oscillations will generally be needed in order to determine the weak limit. The asymptotic structure of uniformly bounded solutions depends strongly on the nature of the nonlinearity: For entropy solutions of a genuinely nonlinear conservation law ([T 1]) or of a pair of such equations in one spatial dimension ([DP]), oscillations of amplitude O(1) do not persist for positive times, so a sequence of solutions converges strongly to the solution whose initial data is the weak limit of the initial data. Although oscillations do persist in linear systems, the weak limit of a sequence of solutions to such

We consider the initial value problem for the Zakharov equations $$\begin{gathered} \left( Z \right)\frac{1}{{\lambda ^2 }}n_{tt} - \Delta (n + \left| {\rm E} \right|^2 ) = 0n(x,0) = n_0 (x) \hfill \\ n_t (x,0) = n_1 (x) \hfill \\ iE_t + \Delta E - nE = 0E(x,0) = E_0 (x) \hfill \\ \end{gathered} $$ (x∈ℝ k ,k=2, 3,t ≧0) which model the propagation of Langmuir waves in plasmas. For suitable initial data solutions are shown to exist for a time interval independent of λ, a parameter proportional to the ion acoustic speed. For such data, solutions of (Z) converge as λ → ∞ to a solution of the cubic nonlinear Schrodinger equation (CSE)iE t +ΔE+|E|2 E=0. We consider both weak and strong solutions. For the case of strong solutions the results are analogous to previous results on the incompressible limit of compressible fluids.