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A New Proof of Okaji’s Theorem for a Class of Sum of Squares Operators

Paulo D. Cordaro, Nicholas Hanges (2009)

Annales de l’institut Fourier

Let P be a linear partial differential operator with analytic coefficients. We assume that P is of the form “sum of squares”, satisfying Hörmander’s bracket condition. Let q be a characteristic point for P . We assume that q lies on a symplectic Poisson stratum of codimension two. General results of Okaji show that P is analytic hypoelliptic at q . Hence Okaji has established the validity of Treves’ conjecture in the codimension two case. Our goal here is to give a simple, self-contained proof of...

Affine ultraregular generalized functions

Khaled Benmeriem, Chikh Bouzar (2010)

Banach Center Publications

Algebras of ultradifferentiable generalized functions satisfying some regularity assumptions are introduced. We give a microlocal analysis within these algebras related to the affine regularity type and the ultradifferentiability property. As a particular case we obtain new algebras of Gevrey generalized functions.

Application of linear hyperbolic PDE to linear quantum fields in curved spacetimes : especially black holes, time machines and a new semi-local vacuum concept

Bernard Kay (2000)

Journées équations aux dérivées partielles

Several situations of physical importance may be modelled by linear quantum fields propagating in fixed spacetime-dependent classical background fields. For example, the quantum Dirac field in a strong and/or time-dependent external electromagnetic field accounts for the creation of electron-positron pairs out of the vacuum. Also, the theory of linear quantum fields propagating on a given background curved spacetime is the appropriate framework for the derivation of black-hole evaporation (Hawking...

Continuity of the fundamental operations on distributions having a specified wave front set (with a counterexample by Semyon Alesker)

Christian Brouder, Nguyen Viet Dang, Frédéric Hélein (2016)

Studia Mathematica

The pull-back, push-forward and multiplication of smooth functions can be extended to distributions if their wave front sets satisfy some conditions. Thus, it is natural to investigate the topological properties of these operations between spaces Γ ' of distributions having a wave front set included in a given closed cone Γ of the cotangent space. As discovered by S. Alesker, the pull-back is not continuous for the usual topology on Γ ' , and the tensor product is not separately continuous. In this paper,...

Development of small and large compressive pulses in two-phase flow

Nishi Deepa Palo, Jasobanta Jena, Meera Chadha (2024)

Applications of Mathematics

The evolutions of small and large compressive pulses are studied in a two-phase flow of gas and dust particles with a variable azimuthal velocity. The method of relatively undistorted waves is used to study the mechanical pulses of different types in a rotational, axisymmetric dusty gas. The results obtained are compared with that of nonrotating medium. Asymptotic expansion procedure is used to discuss the nonlinear theory of geometrical acoustics. The influence of the solid particles and the rotational...

On geometry of fronts in wave propagations

Susumu Tanabé (1999)

Banach Center Publications

We give a geometric descriptions of (wave) fronts in wave propagation processes. Concrete form of defining function of wave front issued from initial algebraic variety is obtained by the aid of Gauss-Manin systems associated with certain complete intersection singularities. In the case of propagations on the plane, we get restrictions on types of possible cusps that can appear on the wave front.

On microlocal analyticity of solutions of first-order nonlinear PDE

Shif Berhanu (2009)

Annales de l’institut Fourier

We study the microlocal analyticity of solutions u of the nonlinear equation u t = f ( x , t , u , u x ) where f ( x , t , ζ 0 , ζ ) is complex-valued, real analytic in all its arguments and holomorphic in ( ζ 0 , ζ ) . We show that if the function u is a C 2 solution, σ Char L u and 1 i σ ( [ L u , L u ¯ ] ) < 0 or if u is a C 3 solution, σ Char L u , σ ( [ L u , L u ¯ ] ) = 0 , and σ ( [ L u , [ L u , L u ¯ ] ] ) 0 , then σ W F a u . Here W F a u denotes the analytic wave-front set of u and Char L u is the characteristic set of the linearized operator. When m = 1 , we prove a more general result involving the repeated brackets of L u and L u ¯ of any order.

Sheaf theory and regularity. Application to local and microlocal analysis

Jean-André Marti (2010)

Banach Center Publications

A review of some methods in sheaf theory is presented to make precise a general concept of regularity in algebras or spaces of generalized functions. This leads to the local analysis of the sections of sheaves or presheaves under consideration and then to microlocal analysis and microlocal asymptotic analysis.

Wave front set for positive operators and for positive elements in non-commutative convolution algebras

Joachim Toft (2007)

Studia Mathematica

Let WF⁎ be the wave front set with respect to C , quasi analyticity or analyticity, and let K be the kernel of a positive operator from C to ’. We prove that if ξ ≠ 0 and (x,x,ξ,-ξ) ∉ WF⁎(K), then (x,y,ξ,-η) ∉ WF⁎(K) and (y,x,η,-ξ) ∉ WF⁎(K) for any y,η. We apply this property to positive elements with respect to the weighted convolution u B φ ( x ) = u ( x - y ) φ ( y ) B ( x , y ) d y , where B C is appropriate, and prove that if ( u B φ , φ ) 0 for every φ C and (0,ξ) ∉ WF⁎(u), then (x,ξ) ∉ WF⁎(u) for any x.

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