Levi Condition Research Articles

On the Wave Equation with Space Dependent Coefficients: Singularities and Lower Order Terms

This paper complements the study of the wave equation with discontinuous coefficients initiated in (Discacciati et al. in J. Differ. Equ.319 (2022) 131–185) in the case of time-dependent coefficients. Here we assume that the equation coefficients are depending on space only and we formulate Levi conditions on the lower order terms to guarantee the existence of a very weak solution as defined in (Garetto and Ruzhansky in Arch. Ration. Mech. Anal.217 (2015) 113–154). As a toy model we study the wave equation in conservative form with discontinuous velocity and we provide a qualitative analysis of the corresponding very weak solution via numerical methods.

Local solvability for a quasilinear wave equation with the far field degeneracy: 1D case

We study the Cauchy problem for the quasilinear wave equation $ \partial^2 _t u = u^{2a} \partial^2_x u + F(u) u_x $ with $a \geq 0$ and show a result for the local in time existence under new conditions. In the previous results, it is assumed that $u(0,x) \geq c_0>0$ for some constant $c_0$ to prove the existence and the uniqueness. This assumption ensures that the equation does not degenerate. In this paper, we allow the equation to degenerate at spacial infinity. Namely we consider the local well-posedness under the assumption that $u(0,x)>0$ and $u(0,x) \rightarrow 0$ as $|x| \rightarrow \infty$. Furthermore, to prove the local well-posedness, we find that the so-called Levi condition appears. Our proof is based on the method of characteristic and the contraction mapping principle via weighted $L^\infty$ estimates.

Inhomogeneous wave equation with t-dependent singular coefficients

This paper is devoted to the study of the inhomogeneous wave equation with singular (less than continuous) time dependent coefficients. Particular attention is given to the role of the lower order terms and suitable Levi conditions are formulated in order to obtain a very weak solution as introduced in [15]. Very weak solutions for this kind of equations are also investigated from a numerical point of view in two toy models: the wave equation with a Heaviside function and a delta distribution, respectively, as coefficient in its principal part.

Explicit solutions of a characteristic Goursat problem without the generalized Levi's condition

ABSTRACT In this paper we investigate solutions to the characteristic Goursat problem with analytic data for a class of linear operators of second order without the Levi's generalized condition. We give an explicit representation of the solutions of the problem involving Kummer functions and show that they are generally singular on the surface K tangent to the characteristic surface S.

A generalized Levi condition for weakly hyperbolic Cauchy problems with coefficients low regular in time and smooth in space

We consider the Cauchy problem for weakly hyperbolic m-th order partial differential equations with coefficients low-regular in time and smooth in space. It is well-known that in general one has to impose Levi conditions to get $$C^\infty $$ or Gevrey well-posedness even if the coefficients are smooth. We use moduli of continuity to describe the regularity of the coefficients with respect to time, weight sequences for the characterization of their regularity with respect to space and weight functions to define the solution spaces. Furthermore, we propose a generalized Levi condition that models the influence of multiple characteristics more freely. We establish sufficient conditions for the well-posedness of the Cauchy problem, that link the Levi condition as well as the modulus of continuity and the weight sequence of the coefficients to the weight function of the solution space. Additionally, we obtain that the influences of the Levi condition and the low regularity of coefficients on the weight function of the solution space are independent of each other.

Global Cauchy problems on [formula omitted] for weakly hyperbolic systems with coefficients admitting superlinear growth for |x| → ∞

We show well-posedness in weighted Sobolev spaces on Rn of the global Cauchy problem for classes of nonstrictly hyperbolic systems of (pseudo)differential equations with double characteristic of variable multiplicity admitting superlinear growth for x∈Rn, |x|→∞. We need a global version of the Levi condition introduced by the first author in 1979 for the local case (cf. [8]) and logarithmic type Wintner assumption on the superlinear growth of the symbols, introduced by the authors in [10]. Finally, we exhibit classes of Cauchy problems for divergence free differential equations in space dimension two which are globally well-posed in L2(R2) and C∞(R2) without any restrictions on the growth. Moreover, if the coefficients are entire functions we get global entire solutions of the corresponding Cauchy–Kovalevski problem.

The Cauchy Problem for Schrödinger Equations with Time-Dependent Hamiltonian

We consider the Cauchy problem for a Schrodinger equation with an Hamiltonian depending also on the time variable and that may vanish at t = 0. We find optimal Levi conditions for well-posedness in Sobolev and Gevrey spaces.

Hypoellipticity and higher order Levi conditions

We study the C∞-hypoellipticity for a class of double characteristic operators with symplectic characteristic manifold, in the case the classical condition of minimal loss of derivatives is violated.

LOCAL AND MICROLOCAL CAUCHY PROBLEM FOR NON-EFFECTIVELY HYPERBOLIC OPERATORS

We study differential operators of order 2 and establish new energy estimates which ensure that the micro supports of solutions to the Cauchy problem propagate with finite speed. We then study the Cauchy problem for non-effectively hyperbolic operators with no null bicharacteristic tangent to the doubly characteristic set and with zero positive trace. By checking the energy estimates, we ensure the propagation with finite speed of the micro supports of solutions, and we prove that the Cauchy problem for such non-effectively hyperbolic operators is C∞ well-posed if and only if the Levi condition holds.

A note on weakly hyperbolic equations with analytic principal part

In this note we show how to include low order terms in the C∞ well-posedness results for weakly hyperbolic equations with analytic time-dependent coefficients. This is achieved by doing a different reduction to a system from the previously used one. We find the Levi conditions such that the C∞ well-posedness continues to hold.

Conditions of hyperbolicity of systems with constant multiplicity

We define the Levi conditions for an N × N linear first order system of differential operators. We prove that these conditions are sufficient for the well-posedness of the Cauchy problem and are necessary up to the multiplicity five.

THIN PLATE SPLINES FOR TRANSFINITE INTERPOLATION AT CONCENTRIC CIRCLES

We propose a new method for constructing a polyspline on annuli, i.e. a C 2 surface on ℝ2 \ {0}, which is piecewise biharmonic on annuli centered at 0 and interpolates smooth data at all interface circles. A unique surface is obtained by imposing Beppo Levi conditions on the innermost and outermost annuli, and one additional restriction at 0: either prescribing an extra data value, or asking that the surface is non-singular. We show that the resulting Beppo Levi polysplines on annuli are in fact thin plate splines, i.e. they minimize Duchon's bending energy.

Weakly hyperbolic equations with non-analytic coefficients and lower order terms

In this paper we consider weakly hyperbolic equations of higher orders in arbitrary dimensions with time-dependent coefficients and lower order terms. We prove the Gevrey well-posedness of the Cauchy problem under $C^k$-regularity of coefficients of the principal part and natural Levi conditions on lower order terms which may be only continuous. In the case of analytic coefficients in the principal part we establish the $C^\infty$ well-posedness. The proofs are based on using the quasi-symmetriser for the corresponding system. The main novelty compared to the existing literature is the possibility to include lower order terms to the equation as well as considering any space dimensions. We also give results on the ultradistributional and distributional well-posedness of the problem, and we look at new effects for equations with discontinuous lower order terms.

Conditions of hyperbolicity of linear differentiable systems with constant multiplicity

Let h be a system with characteristics of constant multiplicity. We prove that if there exists an operator A′ such that h∘A′ has diagonal principal part and admits a good decomposition, then h must satisfy the Levi conditions.

On energy estimates for second order hyperbolic equations with Levi conditions for higher order regularity

We consider energy estimates for second order homogeneous hyperbolic equations with time dependent coefficients. The property of energy conservation, which holds in the case of constant coefficients, does not hold in general for variable coefficients; in fact, the energy can be unbounded as t → ∞ in this case. The conditions to the coefficients for the generalized energy conservation (GEC), which is an equivalence of the energy uniformly with respect to time, has been studied precisely for wave type equations, that is, only the propagation speed is variable. However, it is not true that the same conditions to the coefficients conclude (GEC) for general homogeneous hyperbolic equations. The main purpose of this paper is to give additional conditions to the coefficients which provide (GEC); they will be called as Ck-type Levi conditions due to the essentially same meaning of usual Levi condition for the well-posedness of weakly hyperbolic equations.

The Cauchy problem for finitely degenerate hyperbolic equations with polynomial coefficients

We consider hyperbolic Cauchy problems with characteristics of variable multiplicity and coefficients of polynomial growth in the space variables; we focus on second order equations and admit finite order intersections between the characteristics. We obtain well posedness results in $\mathcal{S}(\mathbb{R}^{n})$, $\mathcal{S}'(\mathbb{R}^{n})$ by imposing suitable Levi conditions on the lower order terms. By an energy estimate in weighted Sobolev spaces we show that regularity and behavior at infinity of the solution are different from the ones of the data.

Arzelà's Theorem and strong uniform convergence on bornologies

In 1883 Arzelà (1983/1984) [2] gave a necessary and sufficient condition via quasi-uniform convergence for the pointwise limit of a sequence of real-valued continuous functions on a compact interval to be continuous. Arzelà's work paved the way for several outstanding papers. A milestone was the P.S. Alexandroff convergence introduced in 1948 to tackle the question for a sequence of continuous functions from a topological space (not necessarily compact) to a metric space. In 2009, in the realm of metric spaces, Beer and Levi (2009) [10] found another necessary and sufficient condition through the novel notion of strong uniform convergence on finite sets. We offer a direct proof of the equivalence of Arzelà, Alexandroff and Beer–Levi conditions. The proof reveals the internal gear of these important convergences and sheds more light on the problem. We also study the main properties of the topology of strong uniform convergence of functions on bornologies, initiated in Beer and Levi (2009) [10].

A Fokker–Planck–Kolmogorov equation approach for the monthly affluence forecast of Betania hydropower reservoir

This paper presents a finite difference, time-layer-weighted, bidirectional algorithm that solves the Fokker–Planck–Kolmogorov (FPK) equation in order to forecast the probability density curve (PDC) of the monthly affluences to the Betania hydropower reservoir in the upper part of the Magdalena River in Colombia. First, we introduce a deterministic kernel to describe the basic dynamics of the rainfall–runoff process and show its optimisation using the S/σΔ performance criterion as a goal function. Second, we introduce noisy parameters into this model, configuring a stochastic differential equation that leads to the corresponding FPK equation. We discuss the set-up of suitable initial and boundary conditions for the FPK equation and the introduction of an appropriate Courant–Friederich–Levi condition for the proposed numerical scheme that uses time-dependent drift and diffusion coefficients. A method is proposed to identify noise intensities. The suitability of the proposed numerical scheme is tested against an analytical solution and the general performance of the stochastic model is analysed using a combination of the Kolmogorov, Pearson and Smirnov statistical criteria.

A sufficient condition for well-posedness for systems with time-dependent coefficients

We consider linear, smooth, hyperbolic systems with time-dependent coefficients and size N. We give a condition sufficient for the well-posedness of the Cauchy Problem in some Gevrey classes. We present some Levi conditions to improve the Gevrey index of well-posedness for the scalar equation of order N, using the transformation in [DAS] and following the technique introduced in [CT]. By using this result and adding some assumptions on the form of the first-order term, we can improve the well-posedness for systems. A similar condition has been studied in [DAT] for systems with size 3.

The global Cauchy problem for a vibrating beam equation

We consider the global Cauchy problem for an evolution equation which models an Euler–Bernoulli vibrating beam with time dependent elastic modulus under a force linear function of the displacement u, of the slope ∂ x u , of ∂ x 2 u and ∂ x 3 u . These two last derivatives are proportional to the bending moment and to the shear respectively. We show results of well-posedness in Sobolev spaces assuming that the coefficient of the shear term has a decay rate | x | − σ , σ ⩾ 1 , for the position x → ± ∞ and that all the coefficients of ∂ x k u , 1 ⩽ k ⩽ 3 , satisfy suitable Levi conditions since we allow the elastic modulus to vanish at some time t = t 0 .