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We establish local-in-time Strichartz estimates for solutions of the model case Dirichlet wave equation inside cylindrical convex domains $\Omega \subset \mathbb {R}^ 3$ with smooth boundary $\partial \Omega \neq \emptyset $. The key ingredients to prove Strichartz estimates are dispersive estimates, energy estimates, interpolation and $TT^*$ arguments. Strichartz estimates for waves inside an arbitrary domain $\Omega $ have been proved by Blair, Smith and Sogge [‘Strichartz estimates for the wave equation on manifolds with boundary’, Ann. Inst. H. Poincaré Anal. Non Linéaire26 (2009), 1817–1829]. We provide a detailed proof of the usual Strichartz estimates from dispersive estimates inside cylindrical convex domains for a certain range of the wave admissibility.
In this work, we study a high order derivative in time problem. First, we show that there exists a sequence of elements of the spectrum which tends to infinity and therefore, it is ill posed. Then, we prove the uniqueness of solutions for this problem by adapting the logarithmic arguments to this situation. Finally, the results are applied to the backward in time problem for the generalized linear Burgers’ fluid, a couple of heat conduction problems and a viscoelastic model.
We study in $\mathbb {R}^{3+1}$ a system of nonlinearly coupled Klein-Gordon equations under the null condition, with (possibly vanishing) mass varying in the interval $[0, 1]$. Our goal is three-fold, which extends the results in the earlier work of [5, 3]: 1) we want to establish the global well-posedness result to the system that is uniform in terms of the mass parameter (i.e., the smallness of the initial data is independent of the mass parameter); 2) we want to obtain a unified pointwise decay result for the solution to the system, in the sense that the solution decays more like a wave component (independent of the mass parameter) in a certain range of time, while the solution decays as a Klein-Gordon component with a factor depending on the mass parameter in the other part of the time range; 3) the solution to the Klein-Gordon system converges to the solution to the corresponding wave system in a certain sense when the mass parameter goes to 0. In order to achieve these goals, we will rely on both the flat and hyperboloidal foliation of the spacetime and prove a mass-independent $L^2$–type energy estimate for the Klein-Gordon equations with possibly vanishing mass. In addition, the case of the Klein-Gordon equations with certain restricted large data is discussed.
We prove that solutions to the quintic semilinear wave equation with variable coefficients in ${{\mathbb {R}}}^{1+3}$ scatter to a solution to the corresponding linear wave equation. The coefficients are small and decay as $|x|\to \infty$, but are allowed to be time dependent. The proof uses local energy decay estimates to establish the decay of the $L^{6}$ norm of the solution as $t\to \infty$.
The article studies inverse problems of determining unknown coefficients in various semi-linear and quasi-linear wave equations given the knowledge of an associated source-to-solution map. We introduce a method to solve inverse problems for nonlinear equations using interaction of three waves that makes it possible to study the inverse problem in all globally hyperbolic spacetimes of the dimension
$n+1\geqslant 3$
and with partial data. We consider the case when the set
$\Omega _{\mathrm{in}}$
, where the sources are supported, and the set
$\Omega _{\mathrm{out}}$
, where the observations are made, are separated. As model problems we study both a quasi-linear equation and a semi-linear wave equation and show in each case that it is possible to uniquely recover the background metric up to the natural obstructions for uniqueness that is governed by finite speed of propagation for the wave equation and a gauge corresponding to change of coordinates. The proof consists of two independent components. In the geometric part of the article we introduce a novel geometrical object, the three-to-one scattering relation. We show that this relation determines uniquely the topological, differential and conformal structures of the Lorentzian manifold in a causal diamond set that is the intersection of the future of the point
$p_{in}\in \Omega _{\mathrm{in}}$
and the past of the point
$p_{out}\in \Omega _{\mathrm{out}}$
. In the analytic part of the article we study multiple-fold linearisation of the nonlinear wave equation using Gaussian beams. We show that the source-to-solution map, corresponding to sources in
$\Omega _{\mathrm{in}}$
and observations in
$\Omega _{\mathrm{out}}$
, determines the three-to-one scattering relation. The methods developed in the article do not require any assumptions on the conjugate or cut points.
The cell transmission model (CTM) is a macroscopic model that describes the dynamics of traffic flow over time and space. The effectiveness and accuracy of the CTM are discussed in this paper. First, the CTM formula is recognized as a finite-volume discretization of the kinematic traffic model with a trapezoidal flux function. To validate the constructed scheme, the simulation of shock waves and rarefaction waves as two important elements of traffic dynamics was performed. Adaptation of the CTM for intersecting and splitting cells is discussed. Its implementation on the road segment with traffic influx produces results that are consistent with the analytical solution of the kinematic model. Furthermore, a simulation on a simple road network shows the back and forth propagation of shock waves and rarefaction waves. Our numerical result agrees well with the existing result of Godunov’s finite-volume scheme. In addition, from this accurately proven scheme, we can extract information for the average travel time on a certain route, which is the most important information a traveller needs. It appears from simulations of different scenarios that, depending on the circumstances, a longer route may have a shorter travel time. Finally, there is a discussion on the possible application for traffic management in Indonesia during the Eid al-Fitr exodus.
Motivated by the manufacture of carbon fibre components, this paper considers the smooth draping of loosely woven fabric over rigid obstacles, both smooth and nonsmooth. The draped fabric is modelled as the continuum limit of a Chebyshev net of two families of short rigid rods that are freely pivoted at their joints. This approach results in a system of nonlinear hyperbolic partial differential equations whose characteristics are the fibres in the fabric. The analysis of this system gives useful information about the drapability of obstacles of many shapes and also poses interesting theoretical questions concerning well-posedness, smoothness and computability of the solutions.
The study of spherically symmetric motion is important for the theory of explosion waves. In this paper, we consider a ‘spherical piston’ problem for the relativistic Euler equations, which describes the wave motion produced by a sphere expanding into an infinite surrounding medium. We use the reflected characteristics method to construct a global piecewise smooth solution with a single shock of this spherical piston problem, provided that the speed of the sphere is a small perturbation of a constant speed.
In this note, we study the hyperbolic stochastic damped sine-Gordon equation (SdSG), with a parameter β2 > 0, and its associated Gibbs dynamics on the two-dimensional torus. After introducing a suitable renormalization, we first construct the Gibbs measure in the range 0 < β2 < 4π via the variational approach due to Barashkov-Gubinelli (2018). We then prove almost sure global well-posedness and invariance of the Gibbs measure under the hyperbolic SdSG dynamics in the range 0 < β2 < 2π. Our construction of the Gibbs measure also yields almost sure global well-posedness and invariance of the Gibbs measure for the parabolic sine-Gordon model in the range 0 < β2 < 4π.
Von Neumann’s original proof of the ergodic theorem is revisited. A uniform convergence rate is established under the assumption that one can control the density of the spectrum of the underlying self-adjoint operator when restricted to suitable subspaces. Explicit rates are obtained when the bound is polynomial, with applications to the linear Schrödinger and wave equations. In particular, decay estimates for time averages of solutions are shown.
The aim of the paper is to introduce and investigate a dynamical system which consists of a variational–hemivariational inequality of hyperbolic type combined with a non-linear evolution equation. Such a dynamical system arises in studies of complicated contact problems in mechanics. Existence, uniqueness and regularity of a global solution to the system are established. The approach is based on a new semi-discrete approximation with an application of a surjectivity result for a pseudomonotone perturbation of a maximal monotone operator. A new dynamic viscoelastic frictional contact model with adhesion is studied as an application, in which the contact boundary condition is described by a generalised normal damped response condition with unilateral constraint and a multivalued frictional contact law.
We consider the nonlinear wave equation (NLW) on the $d$-dimensional torus $\mathbb{T}^{d}$ with a smooth nonlinearity of order at least 2 at the origin. We prove that, for almost any mass, small and smooth solutions of high Sobolev indices are stable up to arbitrary long times with respect to the size of the initial data. To prove this result, we use a normal form transformation decomposing the dynamics into low and high frequencies with weak interactions. While the low part of the dynamics can be put under classical Birkhoff normal form, the high modes evolve according to a time-dependent linear Hamiltonian system. We then control the global dynamics by using polynomial growth estimates for high modes and the preservation of Sobolev norms for the low modes. Our general strategy applies to any semilinear Hamiltonian Partial Differential Equations (PDEs) whose linear frequencies satisfy a very general nonresonance condition. The (NLW) equation on $\mathbb{T}^{d}$ is a good example since the standard Birkhoff normal form applies only when $d=1$ while our strategy applies in any dimension.
We prove that any simple planar travelling wave solution to the membrane equation in spatial dimension $d\geqslant 3$ with bounded spatial extent is globally nonlinearly stable under sufficiently small compactly supported perturbations, where the smallness depends on the size of the support of the perturbation as well as on the initial travelling wave profile. The main novelty of the argument is the lack of higher order peeling in our vector-field-based method. In particular, the higher order energies (in fact, all energies at order $2$ or higher) are allowed to grow polynomially (but in a controlled way) in time. This is in contrast with classical global stability arguments, where only the ‘top’ order energies used in the bootstrap argument exhibit growth, and reflects the fact that the background travelling wave solution has ‘infinite energy’ and the coefficients of the perturbation equation are not asymptotically Lorentz invariant. Nonetheless, we can prove that the perturbation converges to zero in $C^{2}$ by carefully analysing the nonlinear interactions and exposing a certain ‘vestigial’ null structure in the equations.
We present a model for a class of non-local conservation laws arising in traffic flow modelling at road junctions. Instead of a single velocity function for the whole road, we consider two different road segments, which may differ for their speed law and number of lanes (hence their maximal vehicle density). We use an upwind type numerical scheme to construct a sequence of approximate solutions, and we provide uniform L∞ and total variation estimates. In particular, the solutions of the proposed model stay positive and below the maximum density of each road segment. Using a Lax–Wendroff type argument and the doubling of variables technique, we prove the well-posedness of the proposed model. Finally, some numerical simulations are provided and compared with the corresponding (discontinuous) local model.
The study of radially symmetric motion is important for the theory of explosion waves. We construct rigorously self-similar entropy solutions to Riemann initial-boundary value problems for the radially symmetric relativistic Euler equations. We use the assumption of self-similarity to reduce the relativistic Euler equations to a system of nonlinear ordinary differential equations, from which we obtain detailed structures of solutions besides their existence. For the ultra-relativistic Euler equations, we also obtain the uniqueness of the self-similar entropy solution to the Riemann initial-boundary value problems.
The numerical entropy production (NEP) for shallow water equations (SWE) is discussed and implemented as a smoothness indicator. We consider SWE in three different dimensions, namely, one-dimensional, one-and-a-half-dimensional, and two-dimensional SWE. An existing numerical entropy scheme is reviewed and an alternative scheme is provided. We prove the properties of these two numerical entropy schemes relating to the entropy steady state and consistency with the entropy equality on smooth regions. Simulation results show that both schemes produce NEP with the same behaviour for detecting discontinuities of solutions and perform similarly as smoothness indicators. An implementation of the NEP for an adaptive numerical method is also demonstrated.
The present paper concerns the system ut + [ϕ(u)]x = 0, vt + [ψ(u)v]x = 0 having distributions as initial conditions. Under certain conditions, and supposing ϕ, ψ: ℝ → ℝ functions, we explicitly solve this Cauchy problem within a convenient space of distributions u,v. For this purpose, a consistent extension of the classical solution concept defined in the setting of a distributional product (not constructed by approximation processes) is used. Shock waves, δ-shock waves, and also waves defined by distributions that are not measures are presented explicitly as examples. This study is carried out without assuming classical results about conservation laws. For reader's convenience, a brief survey of the distributional product is also included.
This paper is concerned with the periodic (in time) solutions to an one-dimensional semilinear wave equation with x-dependent coefficients. Such a model arises from the forced vibrations of a nonhomogeneous string and propagation of seismic waves in nonisotropic media. By combining variational methods with saddle point reduction technique, we obtain the existence of at least three periodic solutions whenever the period is a rational multiple of the length of the spatial interval. Our method is based on a delicate analysis for the asymptotic character of the spectrum of the wave operator with x-dependent coefficients, and the spectral properties play an essential role in the proof.
We investigate the Cauchy problem of the viscous liquid-gas two-phase flow model in ℝ3. Under the assumption that the initial data is close to the constant equilibrium state in the framework of Sobolev space H2(ℝ3), the Cauchy problem is shown to be globally well-posed by an energy method. If additionally, for 1 ⩽ p < 6/5, Lp-norm of the initial perturbation is bounded, the optimal convergence rates of the solutions in Lq-norm with 2 ⩽ q ⩽ 6 and optimal convergence rates of their spatial derivatives in L2-norm are also obtained by combining spectral analysis with energy methods.
We demonstrate the global existence of weak solutions to a class of semilinear strongly damped wave equations possessing nonlinear hyperbolic dynamic boundary conditions. The associated linear operator is $(-\unicode[STIX]{x1D6E5}_{W})^{\unicode[STIX]{x1D703}}\unicode[STIX]{x2202}_{t}u$, where $\unicode[STIX]{x1D703}\in [\frac{1}{2},1)$ and $\unicode[STIX]{x1D6E5}_{W}$ is the Wentzell–Laplacian. A balance condition is assumed to hold between the nonlinearity defined on the interior of the domain and the nonlinearity on the boundary. This allows for arbitrary (supercritical) polynomial growth of each potential, as well as mixed dissipative/antidissipative behaviour.