Modulus of continuity of the coefficients and loss of derivatives in the strictly hyperbolic Cauchy problem

Abstract

We deal with the Cauchy problem for a strictly hyperbolic second-order operator with nonregular coefficients in the time variable. It is well-known that the problem is well-posed in L 2 in case of Lipschitz continuous coefficients and that the log-Lipschitz continuity is the natural threshold for the well-posedness in Sobolev spaces which, in this case, holds with a loss of derivatives. Here, we prove that any intermediate modulus of continuity between the Lipschitz and the log-Lipschitz one leads to an energy estimate with arbitrary small loss of derivatives. We also provide counterexamples to show that the following classification: modulus of continuity → loss of derivatives * Corresponding author.

Time-Dependent Loss of Derivatives for Hyperbolic Operators with Non Regular Coefficients

“…At that point, the key idea was to perform a dierent approximation of the coecients in dierent zones of the phase space: in particular, they set ε = |ξ| −1 . Finally, they obtained an energy estimate with a xed loss of derivatives: there exists a constant δ > 0 such that, for all s ∈ R, the inequality holds true for all u ∈ C 2 ([0, T ]; H ∞ (R N )), for some constant C s depending only on s. Let us remark that if the coecients a jk are not Lipschitz continuous then a loss of regularity cannot be avoided, as shown by Cicognani and Colombini in [4]. Besides, in this paper the authors prove that, if the regularity of the coecients a jk is measured by a modulus of continuity, any intermediate modulus of continuity between the Lipschitz and the log-Lipschitz ones entails necessarily a loss of regularity, which, however, can be made arbitrarily small.…”

A well-posedness result for hyperbolic operators with Zygmund coefficients

“…More recently (see paper [15]), Tarama analysed instead the problem when coecients satisfy an integral log-Zygmund condition: there exists a constant C > 0 such that, for all j, k and all ε ∈ ]0, T /2[ , one has hyp:int-LZ hyp:int-LZ (4) T −ε ε |a jk (t + ε) + a jk (t − ε) − 2 a jk (t)| dt ≤ C ε log 1 + 1 ε .…”

“…On the other hand, it's obvious that if the a jk 's satisfy (3), then they satisfy also (4): so, a more general class of functions is considered. Both in [5] and [15], the authors proved an energy estimate with a xed loss of derivatives: there exists a constant δ > 0 such that, for all s ∈ R, the inequality sup 0≤t≤T u(t, ·) H s+1−δ + ∂ t u(t, ·) H s−δ ≤ est:c-loss (5)…”

Regularity of hyperbolic magnetic Schrödinger equation with oscillating coefficients

“…The method of instability argument to be used was developed in [4] to show that a Logtype loss really appears for hyperbolic Cauchy problems. Now we further develop this idea to demonstrate that the precise ν-loss of derivatives really appears for the magnetic Schrödinger equation.…”