Microlocal time decays for hyperbolic eqations with lower order terms.

Ruzhansky, Michael

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Global time estimates for solutions to equations of dissipative type

Michael Ruzhansky, James Smith (2005)

Journées Équations aux dérivées partielles

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Global time estimates of $L^{p} - L^{q}$ norms of solutions to general strictly hyperbolic partial differential equations are considered. The case of special interest in this paper are equations exhibiting the dissipative behaviour. Results are applied to discuss time decay estimates for Fokker-Planck equations and for wave type equations with negative mass.

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Kharibegashvili, S. (1994)

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We compute and justify rigorous geometric optics expansions for linear hyperbolic boundary value problems that do not satisfy the uniform Lopatinskii condition. We exhibit an amplification phenomenon for the reflection of small high frequency oscillations at the boundary. Our analysis has two important consequences for such hyperbolic boundary value problems. Firstly, we make precise the optimal energy estimate in Sobolev spaces showing that losses of derivatives must occur from the...

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The aspects of symmetrization of hyperbolic equations which will be considered in this review have their own history and are related to some classical results from other areas of mathematics ([12]). Here symmetrization means representation of an initial system of equations in the form of a symmetric t-hyperbolic system in the sense of Friedrichs. Some equations of mathematical physics, for example, the equations of acoustics, of gas dynamics, etc. already have this form. In the 70's...

On the well posedness of vanishing viscosity limits

Alberto Bressan (2002)

Journées équations aux dérivées partielles

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This paper provides a survey of recent results concerning the stability and convergence of viscous approximations, for a strictly hyperbolic system of conservation laws in one space dimension. In the case of initial data with small total variation, the vanishing viscosity limit is well defined. It yields the unique entropy weak solution to the corresponding hyperbolic system.

Propagation through trapped sets and semiclassical resolvent estimates

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Motivated by the study of resolvent estimates in the presence of trapping, we prove a semiclassical propagation theorem in a neighborhood of a compact invariant subset of the bicharacteristic flow which is isolated in a suitable sense. Examples include a global trapped set and a single isolated periodic trajectory. This is applied to obtain microlocal resolvent estimates with no loss compared to the nontrapping setting.

Cauchy problem for hyperbolic operators with triple characteristics of variable multiplicity

Enrico Bernardi, Antonio Bove, Vesselin Petkov (2010)

Journées Équations aux dérivées partielles

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We study a class of third order hyperbolic operators $P$ in $G = Ω \cap {0 \leq t \leq T}, Ω \subset ℝ^{n + 1}$ with triple characteristics on $t = 0$ . We consider the case when the fundamental matrix of the principal symbol for $t = 0$ has a couple of non vanishing real eigenvalues and $P$ is strictly hyperbolic for $t > 0 .$ We prove that $P$ is strongly hyperbolic, that is the Cauchy problem for $P + Q$ is well posed in $G$ for any lower order terms $Q$ .

Root arrangements of hyperbolic polynomial-like functions.

Vladimir Petrov Kostov (2006)

Revista Matemática Complutense

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A real polynomial P of degree n in one real variable is hyperbolic if its roots are all real. A real-valued function P is called a hyperbolic polynomial-like function (HPLF) of degree n if it has n real zeros and P⁽ⁿ⁾ vanishes nowhere. Denote by x_k ⁽ⁱ⁾ the roots of P⁽ⁱ⁾, k = 1, ..., n-i, i = 0, ..., n-1. Then in the absence of any equality of the form x_i...