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- elliptic operators (9)
- boundary value problems (8)
- index (8)
- K-theory (7)
- manifolds with singularities (6)
- pseudodifferential operators (6)
- relative index (6)
- Atiyah-Patodi-Singer theory (5)
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- index theory (5)
- Boundary value problems (4)
- elliptic operator (4)
- manifold with singularities (4)
- surgery (4)
- 'eta' invariant (3)
- Atiyah-Bott condition (3)
- conical singularities (3)
- conormal symbol (3)
- differential operators (3)
- ellipticity (3)
- eta invariant (3)
- index of elliptic operators in subspaces (3)
- spectral flow (3)
- Atiyah-Bott obstruction (2)
- Carleman matrix (2)
- Chemotaxis (2)
- Fredholm operators (2)
- Hodge theory (2)
- Laplace equation (2)
- Lefschetz fixed point formula (2)
- Mellin transform (2)
- Toeplitz operators (2)
- Zaremba problem (2)
- boundary value problem (2)
- edge singularities (2)
- edge-degenerate operators (2)
- elliptic boundary value problems (2)
- elliptic complexes (2)
- elliptic families (2)
- elliptic family (2)
- elliptic system (2)
- eta-invariant (2)
- holomorphic solution (2)
- homotopy classification (2)
- index formulas (2)
- linking coefficients (2)
- manifolds with conical singularities (2)
- manifolds with edges (2)
- modn-index (2)
- monodromy matrix (2)
- pseudo-differential boundary value problems (2)
- pseudodiferential operators (2)
- quantization (2)
- regularization (2)
- regularizer (2)
- regularizers (2)
- singular partial differential equation (2)
- star-product (2)
- symmetry conditions (2)
- the Cauchy problem (2)
- weighted edge spaces (2)
- (co)boundary operator (1)
- APS problem (1)
- Atiyah-Singer theorem (1)
- Boundary-contact problems (1)
- C0−semigroup (1)
- Calculus of conormal symbols (1)
- Calderón projections (1)
- Casped plates (1)
- Categories of stratified spaces (1)
- Cauchy Riemann operator (1)
- Cauchy problem (1)
- Chern character (1)
- Corner boundary value problems (1)
- Crack theory (1)
- Diophantine Approximation (1)
- Dirac operators (1)
- Edge-degenerate operators (1)
- Elliptic equation with order degeneration (1)
- Elliptic operators in domains with edges (1)
- Euler operator (1)
- Euler's theta functions (1)
- Form (1)
- G-index (1)
- G-trace (1)
- Gevrey classes (1)
- Goursat problem (1)
- Green and Mellin edge operators (1)
- Green operator (1)
- Grushin operator (1)
- Gutzwiller formula (1)
- Hamilton-Jacobi theory (1)
- Hamiltonian group action (1)
- Hardy‘s inequality (1)
- Hyperbolic-parabolic system (1)
- Hypoellipticity (1)
- Integrability (1)
- KS model (1)
- Korn’s weighted inequality (1)
- Lame system (1)
- Lefschetz number (1)
- Logarithmic Sobolev inequality (1)
- Lφ spectrum (1)
- Maslov and Conley–Zehnder index (1)
- Meromorphic operator functions (1)
- Multidimensional nonisentropic hydrodynamic model (1)
- Neumann problem (1)
- Non-linear (1)
- Nonlinear (1)
- Operators on manifolds with conical singularities (1)
- Operators on manifolds with edge (1)
- Operators on manifolds with edge and conical exit to infinity (1)
- Operators on manifolds with second order singularities (1)
- Perron's method (1)
- Pontrjagin duality (1)
- Pseudo-differential operators (1)
- Pseudodifferential operators (1)
- Quasilinear hyperbolic system (1)
- Ramified Cauchy problem (1)
- Riemann-Roch theorem (1)
- Saturation model (1)
- Sobolev problem (1)
- Sobolev spaces with double weights on singular cones (1)
- Surface potentials with asymptotics (1)
- System of nonlocal PDE of first order (1)
- Viscosity solutions (1)
- Volterra symbols (1)
- WKB method (1)
- Weyl algebras bundle (1)
- Weyl symbol (1)
- absorbing set (1)
- analytic continuation (1)
- analytic index (1)
- anisotropic spaces (1)
- asymptotic behavior (1)
- asymptotic stable (1)
- asymptotics of solutions (1)
- bar with variable cross-section (1)
- bending of an orthotropic cusped plate (1)
- boundary values problems (1)
- bundles (1)
- classical and quantum reduction (1)
- cohomology (1)
- comparison principle (1)
- compressible Euler equations (1)
- connections (1)
- conormal asymptotic expansions (1)
- conormal asymptotics (1)
- conormal symbols (1)
- conservation laws (1)
- contact transformations (1)
- continuity in Sobolev spaces with double weights (1)
- corner Sobolev spaces with double weights (1)
- coupled solution (1)
- covering (1)
- cusped bar (1)
- de Rham complex (1)
- de Sitter model ; Fundamental solutions ; Decay estimates (1)
- deformation quantization (1)
- degenerate elliptic equations (1)
- degenerate elliptic systems (1)
- dimension functional (1)
- discrete saymptotic types (1)
- division algebras (1)
- divisors (1)
- domains with singularities (1)
- edge Sobolev spaces (1)
- edge algebra (1)
- edge quantizations (1)
- edge spaces (1)
- edge symbol (1)
- elastic bar (1)
- elliptic functions (1)
- elliptic morphism (1)
- elliptic operators in subspaces (1)
- elliptic operators on non-compact manifolds (1)
- elliptic problem (1)
- elliptic systems (1)
- ellipticity in the edge calculus (1)
- ellipticity of cone operators (1)
- ellipticity of corners operators (1)
- ellipticity with interface conditions (1)
- ellipticity with respect to interior and edge symbols (1)
- energetic space (1)
- exponential function (1)
- exponential stability (1)
- exterior tensor product (1)
- fibre coordinates (1)
- finiteness theorem (1)
- force unification (1)
- fully non-linear degenerate parabolic equations (1)
- fundamental solution (1)
- gauge group (1)
- geodesics (1)
- geometric optics approximation (1)
- global exact boundary controllability (1)
- global solution (1)
- global solutions (1)
- good-inner function (1)
- hyperbolic operators (1)
- ill-posed problem (1)
- illposed problem (1)
- index formula (1)
- index of elliptic operator (1)
- integral formulas (1)
- interfaces with conical singularities (1)
- lifespan (1)
- manifold with edge (1)
- manifolds with cusps (1)
- meromorphic family (1)
- metaplectic operators (1)
- mixed elliptic problems (1)
- mod k index (1)
- moduli space of flat connections (1)
- modulo n index (1)
- moment map (1)
- monotone method (1)
- multiple characteristics (1)
- nondegenerate condition (1)
- nonhomogeneous boundary value problems (1)
- nonlocal problem (1)
- norm estimates with respect to a parameter (1)
- operator algebras on manifolds with singularities (1)
- operators on manifolds with conical and edge singularities (1)
- operators on manifolds with edges (1)
- operators on manifolds with singularities (1)
- order reduction (1)
- parallelizable spheres (1)
- parameter-dependent cone operators (1)
- parameter-dependent ellipticity (1)
- parameter-dependent pseudodifferential operators (1)
- parity condition (1)
- parity conditions (1)
- polydisc (1)
- principal symbolic hierarchies (1)
- problem of classification (1)
- pseudo-diferential operators (1)
- pseudo-differential operators (1)
- pseudo-differentialboundary value problems (1)
- pseudodifferential boundary value problems (1)
- pseudodifferential operator (1)
- pseudodifferential subspace (1)
- pseudodifferential subspaces (1)
- relative cohomology (1)
- relative index formulas (1)
- relative η-invariant (1)
- residue (1)
- semiconductors (1)
- semiprocess (1)
- shock wave (1)
- spectral boundary value problems (1)
- spectral independence (1)
- spectral resolution (1)
- spectral theorem (1)
- star product (1)
- subRiemannian geometry (1)
- symmetry group (1)
- symplectic (canonical) transformations (1)
- symplectic reduction (1)
- system Lame (1)
- systems of partial differential equations (1)
- uniform compact attractor (1)
- vibration (1)
- weighted Sobolev space (1)
- weighted Sobolev spaces with discrete saymptotics (1)
- weighted spaces (1)
- weighted spaces with asymptotics (1)
- η-invariant (1)
- ∂-operator (1)
Institute
- Institut für Mathematik (245)
In this paper, we discuss the global existence of solutions for Chemotaxis models with saturation growth. If the coe±cients of the equations are all positive smooth T-periodic functions, then the problem has a positive T-periodic solution, and meanwhile we discuss here the stability problems for the T-periodic solutions.
In this paper, the problem on formation and construction of a shock wave for three dimensional compressible Euler equations with the small perturbed spherical initial data is studied. If the given smooth initial data satisfies certain nondegenerate condition, then from the results in [20], we know that there exists a unique blowup point at the blowup time such that the first order derivates of smooth solution blow up meanwhile the solution itself is still continuous at the blowup point. From the blowup point, we construct a weak entropy solution which is not uniformly Lipschitz continuous on two sides of shock curve, moreover the strength of the constructed shock is zero at the blowup point and then gradually increases. Additionally, some detailed and precise estimates on the solution are obtained in the neighbourhood of the blowup point.
This note is devoted to the study on the global existence of a shock wave for the supersonic flow past a curved wedge. When the curved wedge is a small perturbation of a straight wedge and the angle of the wedge is less than some critical value, wwe show that a shock attached at the wedge will exist globally.
In this article we construct the fundamental solutions for the wave equation arising in the de Sitter model of the universe. We use the fundamental solutions to represent solutions of the Cauchy problem and to prove the Lp − Lq-decay estimates for the solutions of the equation with and without a source term.
Contents: 1 Introduction 2 Main result 3 Construction of the asymptotic solutions 3.1 Derivation of the equations for the profiles 3.2 Exsistence of the principal profile 3.3 Determination of Usub(2) and the remaining profiles 4 Stability of the samll global solutions. Justification of One Phase Nonlinear Geometric Optics for the Kirchhoff-type equations 4.1 Stability of the global solutions to the Kirchhoff-type symmetric hyperbolic systems 4.2 The nonlinear system of ordinary differential equations with the parameter 4.3 Some energies estimates 4.4 The dependence of the solution W(t, ξ) on the function s(t) 4.5 The oscillatory integrals of the bilinear forms of the solutions 4.6 Estimates for the basic bilinear form Γsub(s)(t) 4.7 Contraction mapping 4.8 Stability of the global solution 4.9 Justification of One Phase Nonlinear Geometric Optics for the Kirchhoff-type equations
It is shown that bounded solutions to semilinear elliptic Fuchsian equations obey complete asymptoic expansions in terms of powers and logarithms in the distance to the boundary. For that purpose, Schuze's notion of asymptotic type for conormal asymptotics close to a conical point is refined. This in turn allows to perform explicit calculations on asymptotic types - modulo the resolution of the spectral problem for determining the singular exponents in the asmptotic expansions.
Edge representations of operators on closed manifolds are known to induce large classes of operators that are elliptic on specific manifolds with edges, cf. [9]. We apply this idea to the case of boundary value problems. We establish a correspondence between standard ellipticity and ellipticity with respect to the principal symbolic hierarchy of the edge algebra of boundary value problems, where an embedded submanifold on the boundary plays the role of an edge. We first consider the case that the weight is equal to the smoothness and calculate the dimensions of kernels and cokernels of the associated principal edge symbols. Then we pass to elliptic edge operators for arbitrary weights and construct the additional edge conditions by applying relative index results for conormal symbols.
Green formulae for elliptic cone differential operators are established. This is achieved by an accurate description of the maximal domain of an elliptic cone differential operator and its formal adjoint; thereby utilizing the concept of a discrete asymptotic type. From this description, the singular coefficients replacing the boundary traces in classical Green formulas are deduced.