By Victor A. Galaktionov
Blow-up for Higher-Order Parabolic, Hyperbolic, Dispersion and Schrödinger Equations exhibits how 4 varieties of higher-order nonlinear evolution partial differential equations (PDEs) have many commonalities via their certain quasilinear degenerate representations. The authors current a unified method of care for those quasilinear PDEs.
The booklet first stories the actual self-similar singularity options (patterns) of the equations. This technique permits 4 various periods of nonlinear PDEs to be handled concurrently to set up their remarkable universal positive factors. The ebook describes many homes of the equations and examines conventional questions of existence/nonexistence, uniqueness/nonuniqueness, international asymptotics, regularizations, shock-wave concept, and numerous blow-up singularities.
Preparing readers for extra complex mathematical PDE research, the ebook demonstrates that quasilinear degenerate higher-order PDEs, even unique and awkward ones, aren't as daunting as they first seem. It additionally illustrates the deep good points shared via various kinds of nonlinear PDEs and encourages readers to increase additional this unifying PDE strategy from different viewpoints.
Read Online or Download Blow-up for Higher-Order Parabolic, Hyperbolic, Dispersion and Schroedinger Equations PDF
Similar geometry books
Designed for advanced undergraduate arithmetic or physics classes, this volume focuses on "practical geometry," emphasizing themes and methods of maximal use in all components of arithmetic. matters comprise algebraic and combinatoric preliminaries, isometries and similarities, an creation to crystallography, fields and vector areas, affine areas, and projective areas.
This e-book offers a scientific account of conformal geometry of n-manifolds, in addition to its Riemannian opposite numbers. A unifying subject is their discrete holonomy teams. particularly, hyperbolic manifolds, in measurement three and better, are addressed. The remedy covers additionally proper topology, algebra (including combinatorial team idea and sorts of crew representations), mathematics concerns, and dynamics.
This e-book collects the papers of the convention held in Berlin, Germany, 27-29 August 2012, on 'Space, Geometry and the mind's eye from Antiquity to the fashionable Age'. The convention was once a joint attempt via the Max Planck Institute for the background of technological know-how (Berlin) and the Centro die Ricerca Matematica Ennio De Giorgi (Pisa).
- A Budget of Trisections
- Current Developments in Differential Geometry and its Related Fields: Proceedings of the 4th International Colloquium on Differential Geometry and its Related Fields
- An Introduction to the Geometry of N Dimensions
- Points and Lines: Characterizing the Classical Geometries (Universitext)
- Algebraic Geometry and Singularities
- Non-Euclidean Geometry (6th Edition)
Additional resources for Blow-up for Higher-Order Parabolic, Hyperbolic, Dispersion and Schroedinger Equations
For utt = (un+1 )xx + un+1 in IR × IR+ (u ≥ 0). (11) Note that this PDE, as a hyperbolic 2 × 2 system, admits solutions with shocks, with quite nontrivial properties. But now we describe its similar blow-up (regional) S-regime separate variables solutions, where the resulting ODE takes the form: 2 uS (x, t) = (T − t)− n f˜(x) =⇒ 2 2 n n + 1 f˜ = (f˜n+1 ) + f˜n+1 . (12) Using an extra scaling, f˜(x) = 2(n+2) n 1 n f (x), (13) yields the same ODE (4) and, hence, the exact localized solution (5).
69) The new functional H(r, v) = 1 2 r2 − 1 β rβ |v|β (70) has an absolute minimum point, where Hr (r, v) ≡ r − rβ−1 |v|β = 0 =⇒ r0 (v) = |v|β 1 2−β . (71) . (72) We then obtain the following functional: 2−β 2 ˜ H(v) = H(r0 (v), v) = − 2−β 2β r0 (v) ≡ − 2β |v|β 2 2−β Obviously, the critical points of the functional (72) on the set (69) coincide with those for ˜ H(v) = |v|β , (73) so we arrive at an even, non-negative, convex, and uniformly diﬀerentiable functional, to which L–S theory applies [252, § 57]; see also [94, p.
Numerical construction of periodic orbits for m = 3 Consider now the second equation in (103), which admits constant equilibria (104) existing for all n > 0. It is easy to check that the equilibria ±ϕ0 are asymptotically stable as s → +∞. Then, the necessary periodic orbit is situated between these stable equilibria, so it is unstable as s → +∞. 7 for n = 15, obtained by shooting from s = 0 with prescribed Cauchy data. 6 Convergence to the stable periodic solution of ODE (114) (the limit value n = +∞).