Sharp regularity theory for second order hyperbolic equations of Neumann type

Summary We consider the mixed problem for a general, time independent, second order hyperbolic equation in the unknown u, with datum g L₂() in the Neumann B.C., with datum f L₂(Q) in the right hand side of the equation and, say, initial conditions u₀=u₁=0. We obtain sharp regularity results for u in Q and ù| in , by a pseudo-differential approach on the half-space.Research partially supported by the National Science Foundation under Grants DMS-83-016668 and DMS-87-96320.     

Time regularity of the solutions to second order hyperbolic equations

We consider the Cauchy problem for a second order weakly hyperbolic equation, with coefficients depending only on the time variable. We prove that if the coefficients of the equation belong to the Gevrey class g^s₀\gamma^{s_{0}} and the Cauchy data belong to g^s₁\gamma^{s_{1}}, then the Cauchy problem has a solution in  g^s₀([0,T^*];g^s₁(\mathbbR))\gamma^{s_{0}}([0,T^{*}];\gamma^{s_{1}}(\mathbb{R})) for some T ^*>0, provided 1≤s ₁≤2−1/s ₀. If the equation is strictly hyperbolic, we may replace the previous condition by 1≤s ₁≤s ₀.     

On second order weakly hyperbolic equations with oscillating coefficients and regularity loss of the solutions

Regularity of the solution for the wave equation with constant propagation speed is conserved with respect to time, but such a property is not true in general if the propagation speed is variable with respect to time. The main purpose of this paper is to describe the order of regularity loss of the solution due to the variable coefficient by the following four properties of the coefficient: “smoothness”, “oscillations”, “degeneration” and “stabilization”. Actually, we prove the Gevrey and C^∞ well‐posedness for the wave equations with degenerate coefficients taking into account the interactions of these four properties. Moreover, we prove optimality of these results by constructing some examples (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Hypoellipticity and higher order Levi conditions

We study the C^∞$C^{\infty }$-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.     

Analytic regularity for a class of semi-linear weakly hyperbolic equations of second order

A class of semi-linear weakly hyperbolic equations of second order in 1 space dimension is considered. Using the properties of the analytic functions we give energy estimates for the solutions and then we prove the propagation of the analytic regularity.     

Circulant preconditioners for second order hyperbolic equations

Linear systems arising from implicit time discretizations and finite difference space discretizations of second-order hyperbolic equations in two dimensions are considered. We propose and analyze the use of circulant preconditioners for the solution of linear systems via preconditioned iterative methods such as the conjugate gradient method. Our motivation is to exploit the fast inversion of circulant systems with the Fast Fourier Transform (FFT). For second-order hyperbolic equations with initial and Dirichlet boundary conditions, we prove that the condition number of the preconditioned system is ofO() orO(m), where is the quotient between the time and space steps andm is the number of interior gridpoints in each direction. The results are extended to parabolic equations. Numerical experiments also indicate that the preconditioned systems exhibit favorable clustering of eigenvalues that leads to a fast convergence rate.     

On a class of second order Fuchsian hyperbolic equations

Summary We study a class of second order Fuchsian hyperbolic operators. The well-posedness of the Cauchy problem in a space of regular distributions is proved, together with results on the propagation of singularities of the solution. Moreover we give a representation formula for the distribution solutions of the homogeneous equation.     

Levi condition for hyperbolic equations with oscillating coefficients

In the present paper we explain new Levi conditions of C^∞ type for second-order hyperbolic Cauchy problems. Our goal is to explain the special influence of oscillations in the coefficients. It turns out that such oscillations have an essential influence coupled with the asymptotic behavior of characteristics around multiple points.     

Decay estimates for second order evolution equations with memory

This paper develops a unified method to derive decay estimates for general second order integro-differential evolution equations with semilinear source terms. Depending on the properties of convolution kernels at infinity, we show that the energy of a mild solution decays exponentially or polynomially as t→+∞. Our approach is based on integral inequalities and multiplier techniques.These decay results can be applied to various partial differential equations. We discuss three examples: a semilinear viscoelastic wave equation, a linear anisotropic elasticity model, and a Petrovsky type system.     

Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients III

In an earlier work of the author it was proved that the Strichartz estimates for second order hyperbolic operators hold in full if the coefficients are of class . Here we strengthen this and show that the same holds if the coefficients have two derivatives in . Then we use this result to improve the local theory for second order nonlinear hyperbolic equations.