Decay of solutions for a semilinear system of elastic waves in an exterior domain with damping near infinity

We show that the solutions of a semilinear system of elastic waves in an exterior domain with a localized damping near infinity decay in an algebraic rate to zero. We impose an additional condition on the Lamé coefficients. It seems that this restriction cannot be overcome by using the two-finite-speed propagation of the elastic model, since we do not assume compact support on the initial data and because the dissipation does not have compact support. The decay rates obtained for the total energy of the linear problem and the L²$L^2$-norm of the solution improve previous results. For the semilinear problem the decay rates in this paper seem to be the first contribution, mainly in the context of initial data without compact support and localized dissipation.     

Optimal decay rates for the system of elastic waves in R n with structural damping

We consider the Cauchy problem in R ⁿ for the system of elastic waves with structural damping. We derive (almost) optimal decay rates in time for the L ²-norm and the total energy which improves previous results for this system. To derive the estimates for elastic waves, we employ an improvement in a method in the Fourier space, which was developed in our previous works. Our estimates came from those for a generalized energy of α-order in the Fourier space.     

Detection of point-like scatterers using one type of scattered elastic waves

In this paper, we are concerned with the detection of point-like obstacles using elastic waves. We show that one type of waves, either the P or the S scattered waves, is enough for localizing the points. We also show how the use of S incident waves gives better resolution than the P waves. These affirmations are demonstrated by several numerical examples using a MUSIC type algorithm.     

Orbital stability for periodic standing waves of the Klein–Gordon–Zakharov system and the beam equation

The existence and stability of spatially periodic waves ${\left(e^{i{\omega}t}\varphi_\omega, \psi_\omega\right)}$ in the Klein–Gordon–Zakharov (KGZ) system are studied. We show a local existence result for low regularity initial data. Then, we construct a one-parameter family of periodic dnoidal waves for (KGZ) system when the period is bigger than ${\sqrt{2}\pi}$ . We show that these waves are stable whenever an appropriate function satisfies the standard Grillakis–Shatah–Strauss (Grillakis et al. J Funct Anal 74(1):160–197, 1987; Grillakis et al. J Funct Anal 94(2):308–348, 1990) type condition. We compute the intervals for the parameter ω explicitly in terms of L and by taking the limit L → ∞ we recover the previously known stability results for the solitary waves in the whole line case. For the beam equation, we show the existence of spatially periodic standing waves and show that orbital stability holds if an appropriate functional satisfies Grillakis–Shatah–Strauss type condition.     

The evolution of traveling waves in a simple isothermal chemical system modeling quadratic autocatalysis with strong decay

In this paper, we study a reaction–diffusion system for an isothermal chemical reaction scheme governed by a quadratic autocatalytic step A+B→2B$A+B\to 2B$ and a decay step B→C$B\to C$, where A, B, and C are the reactant, the autocatalyst, and the inner product, respectively. Previous numerical studies and experimental evidences demonstrate that if the autocatalyst is introduced locally into this autocatalytic reaction system where the reactant A initially distributes uniformly in the whole space, then a pair of waves will be generated and will propagate outwards from the initial reaction zone. One crucial feature of this phenomenon is that for the strong decay case, the formation of waves is independent of the amount of the autocatalyst B introduced into the system. It is this phenomenon of KPP-type which we would like to address in this paper. To study the propagation of reactant and autocatalyst analytically, we first use the tail behavior of waves to construct a pair of generalized super-/sub-solutions for the approximate system of the autocatalytic reaction system. Note that the autocatalytic reaction system does not enjoy comparison principle. Together with a family of truncated problems, we can establish the existence of a family of traveling waves with the minimal speed. Second, we use this pair of generalized super-/sub-solutions to show that the propagation of waves is fully determined by the rate of decay of the initial data at infinity in the sense of Aronson–Weinberger formulation, which in turn confirms the aforementioned numerical and experimental results.     

Existence locale et unicité d'écoulements de fluides viscoélastiques dans des domaines non bornés

In this Note, we present a result of local existence and uniqueness, for any initial data, of the solutions to the equations of viscoelastic fluids of Jeffreys type (differential constitutive law). The system of equations is supposed to be verified in an unbounded domain Ω ⊂ ℝ^N (N = 2 or 3)), uniformly regular. The difficulty comes essentially from the loss of compactness in the case of unbounded domains. To overcome this difficulty we use a local compactness method, which allows us to construct a sequence of solutions on subdomains ;inn whi_n which union covers Ω After that, we pass to the limit to define a solution over the whole domain. Finally we show the uniqueness of this solution in its class of regularity, by using an energy estimate.     

Stability of non-monotone traveling waves for a discrete diffusion equation with monostable convolution type nonlinearity

This paper is concerned with the stability of non-monotone traveling waves for a discrete diffusion equation with monostable convolution type nonlinearity. By using the anti-weighted energy method and nonlin-ear Halanay’s inequality, we prove that all noncritical traveling waves (waves with speeds c > c_*, c_* is minimal speed) are time-exponentially stable, when the initial perturbations around the waves are small. As a corollary of our stability result, we immediately obtain the uniqueness of the traveling waves.     

Uniform Decay properties of a model in structural acoustics

We investigate decay properties for a system of coupled partial differential equations which model the interaction between acoustic waves in a cavity and the walls of the cavity. In this system a wave equation is coupled to a structurally damped plate or beam equation. The underlying semigroup for this system is not uniformly stable, but when the system is appropriately restricted we obtain some uniform stability. We present two results of this type. For the first result, we assume that the initial wave data is zero, and the initial plate or beam data is in the natural energy space; then the corresponding solution to system decays uniformly to zero. For the second result, we assume that the initial condition is in the natural energy space and the control function is L²(0,∞) (in time) into the control space; then the beam displacement and velocity are both L²(0,∞) into a space with two spatial derivatives.     

Higher order shadow waves and delta shock blow up in the Chaplygin gas

The introductory part of this paper contains an overview of known results about elementary and delta shock solutions to Riemann problem for well known Chaplygin gas model (nowadays used in cosmological theories for dark energy) in terms of entropic shadow waves. Shadow waves are introduced in [17] and they are represented by shocks depending on a small parameter ε with unbounded amplitudes having a distributional limit involving the Dirac delta function. In a search for admissible solutions to all possible cases of mutual interactions of waves arising from double Riemann initial data we found same cases that cannot be resolved with already known types of elementary or shadow wave solutions. These cases are resolved by introducing a sequence of higher order shadow waves depending on integer powers of ε. It is shown that such waves have a distributional limit but only until some finite time T.     

Asymptotics in nonlinear evolution system with dissipation and ellipticity on quadrant

In this paper, we consider an initial boundary value problem for some nonlinear evolution system with dissipation and ellipticity. We establish the global existence and furthermore obtain the Lp (p?2) decay rates of solutions corresponding to diffusion waves. The analysis is based on the energy method and pointwise estimates.     

Analyticity and Smoothing Effect for the Coupled System of Equations of Korteweg-de Vries Type with a Single Point Singularity

Using Bourgain spaces and the generator of dilation P=3t ? _t +x ? _x , which almost commutes with the linear Korteweg-de Vries operator, we show that a solution of the initial value problem associated for the coupled system of equations of Korteweg-de Vries type which appears as a model to describe the strong interaction of weakly nonlinear long waves, has an analyticity in time and a smoothing effect up to real analyticity if the initial data only have a single point singularity at x=0.     

Large time WKB approximation for multi-dimensional semiclassical Schrödinger-Poisson system

We consider the semiclassical Schrödinger-Poisson system with a special initial data of WKB type such that the solution of the limiting hydrodynamical equation becomes time-global in dimensions at least three. We give an example of such initial data in the focusing case via the analysis of the compressible Euler-Poisson equations. This example is a large data with radial symmetry, and is beyond the reach of the previous results because the phase part decays too slowly. Extending previous results in this direction, we justify the WKB approximation of the solution with this data for an arbitrarily large interval of R₊.     

On the ill-posedness of a weakly dispersive one-dimensional Boussinesq system

We study the Cauchy problem for a one-dimensional dispersive system of Boussinesq type which models weakly nonlinear long wave surface waves. We prove that such a system is ill-posed in H ^s (?) for s < 0 in the sense that the solution does not depend continuously on the initial data. We also provide criteria for the formation of singularities.     

The Cahn–Hilliard–Hele–Shaw system with singular potential

The Cahn–Hilliard–Hele–Shaw system is a fundamental diffuse-interface model for an incompressible binary fluid confined in a Hele–Shaw cell. It consists of a convective Cahn–Hilliard equation in which the velocity u is subject to a Korteweg force through Darcy's equation. In this paper, we aim to investigate the system with a physically relevant potential (i.e., of logarithmic type). This choice ensures that the (relative) concentration difference φ takes values within the admissible range. To the best of our knowledge, essentially all the available contributions in the literature are concerned with a regular approximation of the singular potential. Here we first prove the existence of a global weak solution with finite energy that satisfies an energy dissipative property. Then, in dimension two, we further obtain the uniqueness and regularity of global weak solutions. In particular, we show that any two-dimensional weak solution satisfies the so-called strict separation property, namely, if φ is not a pure state at some initial time, then it stays instantaneously away from the pure states. When the spatial dimension is three, we prove the existence of a unique global strong solution, provided that the initial datum is regular enough and sufficiently close to any local minimizer of the free energy. This also yields the local Lyapunov stability of the local minimizer itself. Finally, we prove that under suitable assumptions any global solution converges to a single equilibrium as time goes to infinity.     

A Carleman estimate for the linear magnetoelastic waves system and an inverse source problem in a bounded conductive medium

ABSTRACT

In this paper, we consider an inverse problem for the simultaneous diffusion process of elastic and electromagnetic waves in an isotropic heterogeneous elastic body which is identified with an open bounded domain. From the mathematical point of view, the system under consideration can be viewed as the coupling between the hyperbolic system of elastic waves and a parabolic system for the magnetic field. We study an inverse problem of determining the external source terms by observations data in a neighborhood of the boundary and we prove the Hölder stability. For the proof, we show a Carleman estimate for the displacement and the magnetic field of the magnetoelastic system.     

The incompressible limit in $$L^p$$ type critical spaces

This paper aims at justifying the low Mach number convergence to the incompressible Navier–Stokes equations for viscous compressible flows in the ill-prepared data case. The fluid domain is either the whole space, or the torus. A number of works have been dedicated to this classical issue, all of them being, to our knowledge, related to \(L^2\) spaces and to energy type arguments. In the present paper, we investigate the low Mach number convergence in the \(L^p\) type critical regularity framework. More precisely, in the barotropic case, the divergence-free part of the initial velocity field just has to be bounded in the critical Besov space \(\dot{B}^{d/p-1}_{p,r}\cap \dot{B}^{-1}_{\infty ,1}\) for some suitable \((p,r)\in [2,4]\times [1,+\infty ].\) We still require \(L^2\) type bounds on the low frequencies of the potential part of the velocity and on the density, though, an assumption which seems to be unavoidable in the ill-prepared data framework, because of acoustic waves. In the last part of the paper, our results are extended to the full Navier–Stokes system for heat conducting fluids.     

On the role of quadratic oscillations in nonlinear Schrödinger equations

We consider a nonlinear semi-classical Schrödinger equation for which it is known that quadratic oscillations lead to focusing at one point, described by a nonlinear scattering operator. If the initial data is an energy bounded sequence, we prove that the nonlinear term has an effect at leading order only if the initial data have quadratic oscillations; the proof relies on a linearizability condition (which can be expressed in terms of Wigner measures). When the initial data is a sum of such quadratic oscillations, we prove that the associate solution is the superposition of the nonlinear evolution of each of them, up to a small remainder term. In an appendix, we transpose those results to the case of the nonlinear Schrödinger equation with harmonic potential.