Bergmannsche Polynom-Erzeugende erster Art.
Bistable traveling waves for monotone semiflows with applications
This paper is devoted to the study of traveling waves for monotone evolution systems of bistable type. In an abstract setting, we establish the existence of traveling waves for discrete and continuous-time monotone semiflows in homogeneous and periodic habitats. The results are then extended to monotone semiflows with weak compactness. We also apply the theory to four classes of evolution systems.
Blow up and near soliton dynamics for the critical gKdV equation
These notes present the main results of [22, 23, 24] concerning the mass critical (gKdV) equation for initial data in close to the soliton. These works revisit the blow up phenomenon close to the family of solitons in several directions: definition of the stable blow up and classification of all possible behaviors in a suitable functional setting, description of the minimal mass blow up in , construction of various exotic blow up rates in , including grow up in infinite time.
Blow up for the critical gKdV equation. II: Minimal mass dynamics
We consider the mass critical (gKdV) equation for initial data in . We first prove the existence and uniqueness in the energy space of a minimal mass blow up solution and give a sharp description of the corresponding blow up soliton-like bubble. We then show that this solution is the universal attractor of all solutions near the ground state which have a defocusing behavior. This allows us to sharpen the description of near soliton dynamics obtained in [29].
Borel resummation of formal solutions to nonlinear Laplace equations in 2 variables
We consider a nonlinear Laplace equation Δu = f(x,u) in two variables. Following the methods of B. Braaksma [Br] and J. Ecalle used for some nonlinear ordinary differential equations we construct first a formal power series solution and then we prove the convergence of the series in the same class as the function f in x.
Borel summable solutions of the Burgers equation
We give necessary and sufficient conditions for the formal power series solutions to the initial value problem for the Burgers equation to be convergent or Borel summable.
Boundary integral equations and modified Green's functions
Boundary integral equations for mixed boundary value problems in polygonal domains and Galerkin approximation
Boundary integral equations of the logarithmic potential theory for domains with peaks
Integral equations of boundary value problems of the logarithmic potential theory for a plane domain with several peaks at the boundary are studied. We present theorems on the unique solvability and asymptotic representations for solutions near peaks. We also find kernels of the integral operators in a class of functions with a weak power singularity and describe classes of uniqueness.
Boundary layer correctors and generalized polarization tensor for periodic rough thin layers. A review for the conductivity problem
We study the behaviour of the steady-state voltage potential in a material composed of a two-dimensional object surrounded by a rough thin layer and embedded in an ambient medium. The roughness of the layer is supposed to be εα–periodic, ε being the magnitude of the mean thickness of the layer, and α a positive parameter describing the degree of roughness. For ε tending to zero, we determine the appropriate boundary layer correctors which lead to approximate transmission conditions equivalent to...
Boundary layer for Chaffee-Infante type equation
This article is concerned with the nonlinear singular perturbation problem due to small diffusivity in nonlinear evolution equations of Chaffee-Infante type. The boundary layer appearing at the boundary of the domain is fully described by a corrector which is “explicitly" constructed. This corrector allows us to obtain convergence in Sobolev spaces up to the boundary.
Boundary layers and glancing blow-up in nonlinear geometric optics
Boundary singularities of solutions to quasilinear elliptic equations
Asymptotic formulae for solutions to boundary value problems for linear and quasilinear elliptic equations and systems near a boundary point are discussed. The boundary is not necessarily smooth. The main ingredient of the proof is a spectral splitting and reduction of the original problem to a finite-dimensional dynamical system. The linear version of the corresponding abstract asymptotic theory is presented in our new book “Differential equations with operator coefficients”, Springer, 1999.
Boundary value problems and duality between Lp Dirichlet and regularity problems for second order parabolic systems in non-cylindrical domains.
In this paper we consider general second order, symmetric and strongly elliptic parabolic systems with real valued and constant coefficients in the setting of a class of time-varying, non-smooth infinite cylindersΩ = {(x0,x,t) ∈ R x Rn-1 x R: x0 > A(x,t)}.We prove solvability of Dirichlet, Neumann as well as regularity type problems with data in Lp and Lp1,1/2 (the parabolic Sobolev space having tangential (spatial) gradients and half a time derivative in Lp) for p ∈ (2 − ε, 2 + ε) assuming...
Boundary value problems for quasielliptic equations in a half-space.
Boundary value problems for quasi-linear elliptic second order equations in unbounded cone-like domains
We study the behaviour of weak solutions (as well as their gradients) of boundary value problems for quasi-linear elliptic divergence equations in domains extending to infinity along a cone.
Boundary value problems of generalized axially symmetric Helmholtz equations
Boundary-value problems of Hilbert type for linear p-areolar differential equations in the form
Boussinesq/Boussinesq systems for internal waves with a free surface, and the KdV approximation
We study here some asymptotic models for the propagation of internal and surface waves in a two-fluid system. We focus on the so-called long wave regime for one-dimensional waves, and consider the case of a flat bottom. Following the method presented in [J.L. Bona, T. Colin and D. Lannes, Arch. Ration. Mech. Anal. 178 (2005) 373–410] for the one-layer case, we introduce a new family of symmetric hyperbolic models, that are equivalent to the classical Boussinesq/Boussinesq system displayed in [W. Choi...
Boussinesq/Boussinesq systems for internal waves with a free surface, and the KdV approximation
We study here some asymptotic models for the propagation of internal and surface waves in a two-fluid system. We focus on the so-called long wave regime for one-dimensional waves, and consider the case of a flat bottom. Following the method presented in [J.L. Bona, T. Colin and D. Lannes, Arch. Ration. Mech. Anal.178 (2005) 373–410] for the one-layer case, we introduce a new family of symmetric hyperbolic models, that are equivalent to the classical Boussinesq/Boussinesq system displayed in [W. Choi...