Search

A uniform existence theorem is proven for quasilinear symmetric hyperbolic systems containing two small parameters tending to zero at different rates for more general initial data than required in ...

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.

The equatorial shallow-water equations at low Froude number form a symmetric hyperbolic system with large variable-coefficient terms. Although such systems are not covered by the classical Klainerman-Majda theory of singular limits, the first two authors recently proved that solutions exist uniformly and converge to the solutions of the long-wave equations as the height and Froude number tend to 0. Their proof exploits the special structure of the equations by expanding solutions in series of parabolic cylinder functions. A simpler proof of a slight generalization is presented here in the spirit of the classical theory. © 2008 Wiley Periodicals, Inc.

In this article we consider the evolution of vortex sheets in the plane both as a weak solution of the two dimensional incompressible Euler equations and as a (weak) solution of the Birkhoff-Rott equations. We begin by discussing the classical Birkhoff-Rott equations with respect to arbitrary parametrizations of the sheet. We introduce a notion of weak solution to the Birkhoff-Rott system, and we prove consistency of this notion with the classical formulation of the equations. Our main purpose in this paper is to present a sharp criterion for the equivalence of the weak Euler and weak Birkhoff-Rott descriptions of vortex sheet dynamics.

We show that chemical kinetics relations can be used to describe processes that involve binding and dissociation reactions that take place on surfaces. From a mathematical perspective, the problem we study is a singular limit of a reaction‐diffusion system in which one of the variables concentrates on a lower‐dimensional set in the limit, while the other continues to diffuse in a fixed domain.

A sufficient condition is given for smooth solutions of a scalar conservation law with a saturating viscous term to develop shocks.

The mathematical theory of the passage from compressible to incompressible fluid flow is reviewed.

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.

We study the Euler equations for slightly compressible fluids, that is, after rescaling, the limits of the Euler equations of fluid dynamics as the Mach number tends to zero. In this paper, we consider the general non-isentropic equations and general data. We first prove the existence of classical solutions for a time independent of the small parameter. Then, on the whole space ℝ d , we prove that the solution converges to the solution of the incompressible Euler equations.

The weak limit of a sequence of approximate solutions of the 2-D Euler equations will be a solution if the approximate vorticities concentrate only along a curve x(t) that is Holder continuous with exponent ½. A new proof is given of the theorem of DiPerna and Majda that weak limits of steady approximate solutions are solutions provided that the singularities of the inhomogeneous forcing term are sufficiently mild. An example shows that the weaker condition imposed here on the forcing term is sharp. A simplified formula for the kernel in Delort's weak vorticity formulation of the two-dimensional Euler equations makes the properties of that kernel readily apparent, thereby simplying Delort's proof of the existence of one-signed vortex sheets.

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.

Rosenau has recently proposed a regularized version of the Chapman-Enskog expansion of hydrodynamics. This regularized expansion resembles the usual Navier-Stokes viscosity terms at law wave-numbers, but unlike the latter, it has the advantage of being a bounded macroscopic approximation to the linearized collision operator. The behavior of Rosenau regularization of the Chapman-Enskog expansion (RCE) is studied in the context of scalar conservation laws. It is shown that thie RCE model retains the essential properties of the usual viscosity approximation, e.g., existence of traveling waves, monotonicity, upper-Lipschitz continuity..., and at the same time, it sharpens the standard viscous shock layers. It is proved that the regularized RCE approximation converges to the underlying inviscid entropy solution as its mean-free-path epsilon approaches 0, and the convergence rate is estimated.

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.

The equations governing incompressible fluid flow differ from those for compressible flow in the evolution equation for the density, or the pressure is replaced by the constraint that the flow be divergence-free. The equations of incompressible flow can be derived by taking the limit of an appropriately rescaled version of the equations for compressible flow, as the scaling parameter tends to zero. In physical terms, the scaling parameter is the ratio of the fluid particle speed to the sound speed, and is called “Mach number.” The limit of the rescaled compressible equations as the Mach number tends to zero is known as “incompressible limit.” When the equations of compressible fluid flow tend to goes zero, one expects that solutions of the compressible fluid equations tend to solutions of the incompressible equations in that limit. This expectation has been justified in a wide variety of circumstances. Various extensions of these results and analogous results have been obtained for limits of other systems. This chapter also discusses the rescaling process and the formal limit of the resulting 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 demonstrate that certain strictly anharmonic one-dimensional FPU lattices with a suitable quartic site potential appended support almost-compact discrete breathers over a macroscopic localized domain that is essentially fixed independently of the sparseness of the lattice. Beyond that domain the discrete breather tails decay at a double-exponential rate in the lattice-cell index, becoming truly compact in the continuum limit. Furthermore, the discrete breather is stable for amplitudes below a sharp threshold that depends on the sparseness of the lattice. For the two-dimensional version of the problem, the continuum limit of a planar hexagonal lattice with a purely quartic interaction potential begets an isotropic multidimensional nonlinear wave equation. When a quartic site potential of the appropriate sign is appended, the continuum equation has a compactly supported radial breather solution.

(1986). Symmetric hyperbolic systems with a large parameter. Communications in Partial Differential Equations: Vol. 11, No. 15, pp. 1627-1651.

Explicit solutions are found for the viscous version of the model vorticity equation recently proposed by P. Constantin, P. D. Lax, and A. Majda: where H(w) is the Hilbert transform of w, and v is a positive constant. Various properties of these solutions, including the fact that they blow up after a finite time, are discussed.

On demontre des resultats sur les conditions posees par Diperna sur les solutions a valeur de mesure pour des lois de conservation

The incompressible limit in nonlinear elasticity is shown to fall under the theory of singular limits of quasilinear symmetric hyperbolic systems developed by Klainerman and Majda. Specifically, initial-value problems for a family of hyperelastic materials with stored energy functions $$W\left( {\frac{{\partial x}}{{\partial X}}} \right) = W_\infty \left( {\frac{{\partial x}}{{\partial X}}} \right) + \lambda ^2 w\left( {\det \frac{{\partial x}}{{\partial X}}} \right)$$ are considered, whereX andx are reference and deformed coordinates respectively. Under the assumption that the elasticity tensor $$A_{kl}^{ij} \equiv \frac{{\partial ^2 W_\infty }}{{\partial \left( {\frac{{\partial x^i }}{{\partial X^k }}} \right)\partial \left( {\frac{{\partial x^j }}{{\partial X^l }}} \right)}}$$ is positive definite near the identity matrix and thatw″(1)>0, the following results are proven for appropriate initial data

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.

The asymptotic behavior as λ → ∞ is analysed for solutions u(λ) of quasilinear symmetric hyperbolic systems of the form A0ut + Ajuxi + λCjuxj = Fu(0, x, λ) = u0(x, λ), where the Ak and F depend on t, x, u, and uλ, but the Cj are constant. The results are applied to the equations of slightly compressible fluid dynamics to obtain a generalized form of the acoustics equations that describe the first-order correction to incompressible flow. The special case of a barotropic fluid was analyzed previously by Klainerman and Majda.

A generic averaging theorem is proven for systems of ODEs with two-time scales that cannot be globally transformed into the usual action-angle variable normal form for such systems. This theorem is shown to apply to certain Fourier-space truncations of the non-isentropic slightly compressible Euler equations of fluid mechanics. For the full Euler equations, we derive formally the generic limit equations and analyze some of their properties. In the one-dimensional case, we prove a generic converic convergence result for the full Euler equations, analogous to the result for ODEs. By making use of special properties of the one-dimensional equations, we prove convergence to the solution of a more complicated set of averaged equations when the genericity assumptions fail.

A short-time existence theorem is proven for the initial-boundary-value problem for a class of quasilinear symmetric hyperbolic systems containing a constant-coefficient spatial operator multiplied by a large parameter λ. Solutions u are shown to remain bounded independently of λ for a time independent of λ provided that u (0) and u l (0) are bounded independently of λ and certain structural conditions are satisfied. This result is applied to the quasigeostrophic approximation of the shallow water equations and the constant-pressure approximation in combustion, and solutions for these cases are shown to converge to solutions of limiting equations as λ → ∞.