In this paper we study some key effects of a discontinuous forcing term in a fourth order wave equation on a bounded domain, modeling the adhesion of an elastic beam with a substrate through an elastic-breakable interaction.
By using a spectral decomposition method we show that the main effects induced by the nonlinearity at the transition from attached to detached states can be traced in a loss of regularity of the solution and in a migration of the total energy through the scales.
We study a Euler-Bernoulli beam equation with localized discontinuous structural damping. As our main result, we prove that the associated $C_0$-semigroup $(S(t))_{t\geq0}$ is of Gevrey class $δ>24$ for $t>0$, hence immediately differentiable. Moreover, we show that $(S(t))_{t\geq0}$ is exponentially stable.
We consider a bi-dimensional viscous incompressible fluid in interaction with a beam located at its boundary. We show the existence of strong solutions for this fluid-structure interaction system, extending a previous result where we supposed that the initial deformation of the beam was small. The main point of the proof consists in the study of the linearized system and in particular in proving that the corresponding semigroup is of Gevrey class.
We prove observability estimates for oscillatory Cauchy data modulo a small kernel for $n$-dimensional wave equations with space and time dependent $C^2$ and $C^{1,1}$ coefficients using Gaussian beams. We assume the domains and observability regions are in $\mathbb{R}^n$, and the GCC applies. This work generalizes previous observability estimates to higher dimensions and time dependent coefficients. The construction for the Gaussian beamlets solving $C^{1,1}$ wave equations represents an improvement and simplification over Waters (2011).
We study the well-posedness and regularity theory for the Radiative Transfer equation in the peaked regime posed in the half-space. An average lemma for the transport equation in the half-space is stablished and used to generate interior regularity for solutions of the model. The averaging also shows a fractional regularization gain up to the boundary for the spatial derivatives.
We study the non-autonomous variational problem: \begin{equation*} \inf_{(ϕ,θ)} \bigg\{\int_0^1 \bigg(\frac{k}{2}ϕ'^2 + \frac{(ϕ-θ)^2}{2}-V(x,θ)\bigg)\text{d}x\bigg\} \end{equation*} where $k>0$, $V$ is a bounded continuous function, $(ϕ,θ)\in H^1([0,1])\times L^2([0,1])$ and $ϕ(0)=0$ in the sense of traces. The peculiarity of the problem is its setting in the product of spaces of different regularity order. Problems with this form arise in elastostatics, when studying the equilibria of a nonlinear Timoshenko beam under distributed load, and in classical dynamics of coupled particles in time-d…
In this paper, we are concerned with the precise relationship between the Hausdorff dimension of possible singular point set $\mathcal{S}$ of suitable weak solutions and the parameter $α$ in the nonlinear term in the following parabolic equation $$h_t+h_{xxxx}+\partial_{xx}|h_x|^α=f.$$ It is shown that when $5/3\leqα<7/3$, the $\frac{3α-5}{α-1}$-dimensional parabolic Hausdorff measure of $\mathcal{S}$ is zero, which generalizes the recent corresponding work of Ozánski and Robinson in [31,SIAM J. Math. Anal. 51: 228--255, 2019] for $α=2$ and $f=0$. The same result is valid for a 3D modified Nav…
If the cost function is not too far from the Euclidean cost, then the optimal map transporting Gaussians restricted to a ball will be regular. \ Similarly, given any cost function which is smooth in a neighborhood of two points on a manifold, there are small neighborhoods near each such that a Gaussian restricted to one is transported smoothly to a Gaussian on the other
In this paper, we study the regularity of optimal mappings in Monge's mass transfer problem. Using the approximation to Monge's cost function given by the Euclidean distance c(x,y)=dist(x,y) through the costs c_\eps(x,y)=(\eps^2+dist(x,y)^2)^{1/2}, we consider the optimal mappings T_\eps for these costs, and we prove that the eigenvalues of the Jacobian matrix DT_\eps, which are all positive, are locally uniformly bounded. By an example we prove that T_\eps is in general not uniformly Lipschitz continuous as \eps-0, even if the mass distributions are positive and smooth, and the domains are c-…
Spacetime decay of mild solutions and conditional quantitative transfer of regularity of the incompressible Navier--Stokes Equations from $\mathbb{R}^n$ to bounded domains
We investigate Strichartz estimates for the nonlinear beam equation with initial data $f\in\dot{H}^s, g\in\dot{H}^{s-2}$ and $f\in H^s, g\in H^{s-2}$. We extend results of H. Lindblad and C. D.Sogge [10] and T. Cazenave and F. B. Weissler [4] to nonlinear beam equations to determine the minimal regularity that is needed to prove well-posedness and scattering results with low regularity data. Finally, we also use small dispersion analysis of M. Christ, J. Colliander and T. Tao [2] to prove the nonlinear beam equation is ill-posed in defocusing case $ω=-1$ when $ 0<s<s_c=\frac{n}{2}-\frac{4}{κ-1…
We construct a frame of complex Gaussians for the space of $L^2(\mathbb{R}^n)$ functions. When propagated along bicharacteristics for the wave equation, the frame can be used to build a parametrix with suitable error terms. When the coefficients of the wave equation have more regularity, propagated frame functions become Gaussian beams.
This is a generalization of our prior work on the compact fixed point theory for the elliptic Rosseland-type equations. We obtain the maximum principle without the technical Steklov techniques. Inspired by the Rosseland equation in the conduction-radiation coupled heat transfer, we use the locally arbitrary growth conditions instead of the common global restricted growth conditions. Its physical meaning is: the absolute temperature should be positive and bounded. There exists a fixed point for the linearized map (compact and continuous in $L^2$) in a closed convex set. We also consider the non…
We shall study special regularity properties of solutions to some nonlinear dispersive models. The goal is to show how regularity on the initial data is transferred to the solutions. This will depend on the spaces where regularity is measured.
This work is devoted to radiative transfer equations with long-range interactions. Such equations arise in the modeling of high frequency wave propagation in random media with long-range dependence. In the regime we consider, the singular collision operator modeling the interaction between the wave and the medium is conservative, and as a consequence wavenumbers take values on the unit sphere. Our goals are to investigate the regularizing effects of grazing collisions, the diffusion limit, and the peaked forward limit. As in the case where wavenumbers take values in R^{d+1}, we show that the t…
This paper considers solutions $u_α$ of the three-dimensional Navier--Stokes equations on the periodic domains $Q_α:=(-α,α)^3$ as the domain size $α\to\infty$, and compares them to solutions of the same equations on the whole space. For compactly-supported initial data $u_α^0\in H^1(Q_α)$, an appropriate extension of $u_α$ converges to a solution $u$ of the equations on ${\mathbb R}^3$, strongly in $L^r(0,T;H^1({\mathbb R}^3))$, $r\in[1,\infty)$. The same also holds when $u_α^0$ is the velocity corresponding to a fixed, compactly-supported vorticity. A consequence is that if an initial compact…
Illuminating the surface of a convex body with parallel beams of light in a given direction generates a shadow region. We prove sharp regularity results for the boundary of this shadow in every direction of illumination. Moreover, techniques are developed for investigating the regularity of the region generated by orthogonally projecting a convex set onto another. As an application we study the geometry and Hausdorff dimension of the singular set corresponding to a Monge-Ampere equation.
Solutions of stationary Fokker-Planck equations in the narrow beam regime are commonly approximated by either ballistic linear transport or by a Fermi pencil-beam equation. We present a rigorous approximation analysis of these three models in a half-space geometry. Error estimates are obtained in a 1-Wasserstein sense, which is an adapted metric to quantify beam spreading. The required well-posedness and regularity results for the stationary Fokker-Planck equation with singular internal and boundary sources are also presented in detail.
We consider a one-parameter family of beam equations with Hamiltonian non-linearity in one space dimension under periodic boundary conditions. In a unified functional framework we study the long time evolution of initial data in two categories of differentiability: (i) a subspace of Sobolev regularity, (ii) a subspace of infinitely many differentiable functions which is strictly contained in the Sobolev space but which strictly contains the Gevrey one. In both cases we prove exponential type times of stability. The result holds for almost all mass parameters and it is obtained by combining nor…
We analyse an algorithm of transition between Cauchy problems for second-order wave equations and first-order symmetric hyperbolic systems in case the coefficients as well as the data are non-smooth, even allowing for regularity below the standard conditions guaranteeing well-posedness. The typical operations involved in rewriting equations into systems are then neither defined classically nor consistently extendible to the distribution theoretic setting. However, employing the nonlinear theory of generalized functions in the sense of Colombeau we arrive at clear statements about the transfer …
This paper concerns the reconstruction of properties of narrow laser beams propagating in turbulent atmospheres. We consider the setting of off-axis measurements, based on light detection away from the main path of the beam. We first model light propagation in the beam itself by macroscopic approximations of radiative transfer equations that take the form of Fermi pencil beam or fractional Fermi pencil beam equations. Such models are effective in the small mean-free-path, large transport-mean-free path regime. The reconstruction of their constitutive parameters is also greatly simplified compa…
We consider the regularity of stationary solutions to the linearized Boltzmann equations in bounded $C^1$ convex domains in $\mathbb{R}^3$ for gases with cutoff hard potential and cutoff Maxwellian gases. We prove that the stationary solutions solutions are Hölder continuous with order $\frac1{2}^-$ away from the boundary provided the incoming data have the same regularity. The key idea is to partially transfer the regularity in velocity obtained by collision to space through transport and collision.
Consider the time-periodic viscous incompressible fluid flow past a body with non-zero velocity at infinity. This article gives sufficient conditions such that weak solutions to this problem are smooth. Since time-periodic solutions do not have finite kinetic energy in general, the well-known regularity results for weak solutions to the corresponding initial-value problem cannot be transferred directly. The established regularity criterion demands a certain integrability of the purely periodic part of the velocity field or its gradient, but it does not concern the time mean of these quantities…
This is a simplification of our prior work on the existence theory for the Rosseland-type equations. Inspired by the Rosseland equation in the conduction-radiation coupled heat transfer, we use the locally arbitrary growth conditions instead of the common global restricted growth conditions. In the Lebesgue square integrable space, the solution to the linear elliptic equation depends continuously on the coefficients matrix. This is a simple version of the maximal regularity. There exists a fixed point for the linearized map (compact and continuous) in a closed convex set. We also consider the …
In this paper we consider the nonlinear beam equations accounting for rotational inertial forces. Under suitable hypotheses we prove the existence, regularity and finite dimensionality of a compact global attractor and an exponential attractor. The main purpose is to trace the behavior of solutions of the nonlinear beam equations when the effect of the rotational inertia fades away gradually. A natural question is whether there are qualitative differences would appear or not. To answer the question, we deal with the rotational inertia with a parameter alpha and consider the difference of behav…
We consider three-dimensional models for rate-independent processes describing materials undergoing phase transformations with heat transfer. The problem is formulated within the framework of generalized standard solids by the coupling of the momentum equilibrium equation and the flow rule with the heat transfer equation. Under appropriate regularity assumptions on the initial data, we prove the existence a global solution for this thermodynamically consistent system, by using a fixed-point argument combined with global energy estimates.
We study existence and uniqueness of distributional solutions to the differential equation of the Euler-Bernoulli rod with discontinuous coefficients and right-hand side. Upon checking the validity of a solution the occurring products of singular coefficients with the distributional solution have no obvious meaning. When interpreted on the most general level of the so-called hierarchy of distributional products, it turns out that existence of a solution forces a minimum regularity to hold. Curiously, the choice of the distributional product concept is thus incompatible with the possibility of …
