Characteristics Of Variable Multiplicity Research Articles

Hyperbolic systems with non-diagonalisable principal part and variable multiplicities, I: well-posedness

In this paper we analyse the well-posedness of the Cauchy problem for a rather general class of hyperbolic systems with space-time dependent coefficients and with multiple characteristics of variable multiplicity. First, we establish a well-posedness result in anisotropic Sobolev spaces for systems with upper triangular principal part under interesting natural conditions on the orders of lower order terms below the diagonal. Namely, the terms below the diagonal at a distance k to it must be of order -,k. This setting also allows for the Jordan block structure in the system. Second, we give conditions for the Schur type triangularisation of general systems with variable coefficients for reducing them to the form with an upper triangular principal part for which the first result can be applied. We give explicit details for the appearing conditions and constructions for 2times 2 and 3times 3 systems, complemented by several examples.

Exponential Function of Pseudo-Differential Operators in Gevrey Spaces

The article is devoted to the construction of the exponential function of the matrix pseudo-differential operator, which does not satisfy conditions of any known theorem (see, e.g. Sec. 8 Ch. VIII and Sec. 2 Ch. XI of Treves in Introduction to the theory of pseudodifferential and Fourier integral operator, vols. 1 & 2, Plenum Press, New York, 1980). An application of the exponential function to the fundamental solution of the Cauchy problem for the hyperbolic operators with the characteristics of variable multiplicity is given.

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.

On the well-posedness of the mixed problem for hyperbolic operators with characteristics of variable multiplicity

The paper is devoted to the study of the well-posedness of mixed problems for hyperbolic equations with constant coefficients and characteristics of variable multiplicity. The authors distinguish a class of higher-order hyperbolic operators with constant coefficients and characteristics of variable multiplicity for which a generalization of the Sakamoto L2-well-posedness of the mixed problem is obtained.

Viscous boundary value problems for symmetric systems with variable multiplicities

Extending investigations of Métivier and Zumbrun in the hyperbolic case, we treat stability of viscous shock and boundary layers for viscous perturbations of multidimensional hyperbolic systems with characteristics of variable multiplicity, specifically the construction of symmetrizers in the low-frequency regime where variable multiplicity plays a role. At the same time, we extend the boundary-layer theory to “real” or partially parabolic viscosities, Neumann or mixed-type parabolic boundary conditions, and systems with nonconservative form, in addition proving a more fundamental version of the Zumbrun–Serre–Rousset theorem, valid for variable multiplicities, characterizing the limiting hyperbolic system and boundary conditions as a nonsingular limit of a reduced viscous system. The new effects of viscosity are seen to be surprisingly subtle; in particular, viscous coupling of crossing hyperbolic modes may induce a destabilizing effect. We illustrate the theory with applications to magnetohydrodynamics.

Uniform Stability Estimates for Constant-Coefficient Symmetric Hyperbolic Boundary Value Problems

Answering a question left open in Métivier and Zumbrun (2005), we show for general symmetric hyperbolic boundary problems with constant coefficients, including in particular systems with characteristics of variable multiplicity, that the uniform Lopatinski condition implies strong L 2 well-posedness, with no further structural assumptions. The result applies, more generally, to any system that is strongly L 2 well-posed for at least one boundary condition. The proof is completely elementary, avoiding reference to Kreiss symmetrizers or other specific techniques. On the other hand, it is specific to the constant-coefficient case; at least, it does not translate in an obvious way to the variable-coefficient case. The result in the hyperbolic case is derived from a more general principle that can be applied, for example, to parabolic or partially parabolic problems like the Navier–Stokes or viscous MHD equations linearized about a constant state or even a viscous shock.

On real hyperbolic systems with characteristics of variable multiplicity

We consider the linearized hydrodynamic system for an ideal contractible fluid and a linearized system of Maxwell equations with nonlinear current J ( ρ , E ) . We suggest conditions on the symbol of the linear system that generalize the properties of symbols of the linearized systems under consideration. Under these conditions, we prove an existence and uniqueness theorem for linear nonstrictly hyperbolic systems. Linearization of nonlinear hyperbolic systems is used to study waves and their stability under the action of high-frequency perturbations of initial data in physical media. The linearization of a nonlinear real firstorder system for some smooth solution yields a real first-order hyperbolic system L of the form

Parametrices for symmetric systems with multiplicity

We construct microlocal parametrices for generic symmetric systems of partial differential equations having characteristics of variable multiplicity. We apply this to obtain microlocal solutions of Cauchy problems for generic classes of electromagnetic and elasto-dynamical systems.

Hyperbolic boundary value problems for symmetric systems with variable multiplicities

We extend the Kreiss–Majda theory of stability of hyperbolic initial–boundary-value and shock problems to a class of systems, notably including the equations of magnetohydrodynamics (MHD), for which Majda's block structure condition does not hold: namely, simultaneously symmetrizable systems with characteristics of variable multiplicity, satisfying at points of variable multiplicity either a “totally nonglancing” or a “nonglancing and linearly splitting” condition. At the same time, we give a simple characterization of the block structure condition as “geometric regularity” of characteristics, defined as analyticity of associated eigenprojections. The totally nonglancing or nonglancing and linearly splitting conditions are generically satisfied in the simplest case of crossings of two characteristics, and likewise for our main physical examples of MHD or Maxwell equations for a crystal. Together with previous analyses of spectral stability carried out by Gardner–Kruskal and Blokhin–Trakhinin, this yields immediately a number of new results of nonlinear inviscid stability of shock waves in MHD in the cases of parallel or transverse magnetic field, and recovers the sole previous nonlinear result, obtained by Blokhin–Trakhinin by direct “dissipative integral” methods, of stability in the zero-magnetic field limit. We also discuss extensions to the viscous case.

Quasilinear hyperbolic operators with the characteristics of variable multiplicity

Quasilinear hyperbolic operators with the characteristics of variable multiplicity

On the propagation of singularities for pseudo-differential operators with characteristics of variable multiplicity

(1992). On the propagation of singularities for pseudo-differential operators with characteristics of variable multiplicity. Communications in Partial Differential Equations: Vol. 17, No. 9-10, pp. 1709-1736.

Precise propagation of singularities for a hyperbolic system with characteristics of variable multiplicity

In this paper we consider the Cauchy problem for a hyperbolic system with characteristics of variable multiplicity and construct a certain solution whose wave front set propagates precisely along the so-called “broken null bicharacteristic flow”, in other words, along the admissible trajectory if we use the terminology of [6].

Uniqueness of the solution of the Cauchy problem for linear partial differential equations with characteristics of variable multiplicity

Uniqueness of the solution of the Cauchy problem for linear partial differential equations with characteristics of variable multiplicity

Wave fronts of solutions of boundary-value problems for symmetric hyperbolic systems. III. Systems with characteristics of variable multiplicity

Wave fronts of solutions of boundary-value problems for symmetric hyperbolic systems. III. Systems with characteristics of variable multiplicity

Uniqueness of solutions of the Cauchy problem for linear partial differential equations with characteristics of variable multiplicity

The first general results in the study of the uniqueness of the Cauchy problem for partial differential equations with coefficients which are not necessarily analytic were given by Calderon [l] and Hijrmander [6]. Using the theory of singular integral operators Calderon proved uniqueness theorems for partial differential operators with simple characteristics. He [2] later extended the results to hold for operators whose real characteristics are simple, nonreal characteristics are at most double, characteristics never cross each other, and real characteristics never become complex. This last condition is modified by Kumano-go [9] and Nirenberg [12], both allowing the real characteristics to become complex under certain circumstances. In this paper, we shall give another proof of Nirenberg’s extension of Calderon’s theorem. Then we shall prove uniqueness if the multiplicity of the real characteristics (which may turn complex in a certain way) is greater than one, or the multiplicity of the nonreal characteristics is greater than two. We will do this by putting a condition on the lower-order terms. In both cases, we require the multiplicity to be constant.

The parametrix of the Cauchy problem for hyperbolic operators with characteristics of variable multiplicity

The parametrix of the Cauchy problem for hyperbolic operators with characteristics of variable multiplicity