Theorems on the Fredholm alternative and well-posedness of the Darboux problem ∂2u(t, x) ∂t ∂x = `(u)(t, x) + q(t, x), u(t, x0) = φ(t) for t ∈ [a, b], u(t0, x) = ψ(x) for x ∈ [c, d] are established, where ` : C(D;R) → L(D;R) is a linear bounded operator, q ∈ L(D;R), t0 ∈ [a, b], x0 ∈ [c, d], φ : [a, b]→ R, ψ : [c, d]→ R are absolutely continuous functions, and D = [a, b] × [c, d]. New sufficien...

We analyze a diffuse interface type approximation, known as the diffuse domain approach, of a linear coupled bulk-surface elliptic PDE system. The well-posedness of the diffuse domain approximation is shown using weighted Sobolev spaces and we prove that the solution to the diffuse domain approximation converges weakly to the solution of the coupled bulk-surface elliptic system as the approxima...

We construct solutions of a free boundary value problem for a hyperbolic equation with Dirichlet boundary data. This problem arises from a model of deformation of granular media.

We prove that the Schrödinger map initial-value problem { ∂ts = s×∆xs on R × [−1, 1]; s(0) = s0 is locally well-posed for small data s0 ∈ H σ0 Q (R ; S), σ0 > (d+ 1)/2, Q ∈ S.

We consider a spherical grain which may be growing by accretion or dissolving in a dilute solution of the same substance where one also has reaction and diiusion. The resulting free boundary problem is shown to be well-posed and additional regularity is obtained.

This paper deals with the global well-posedness of the 3D axisymmetric Euler equations for initial data lying in some critical Besov spaces.

We prove an endpoint multilinear estimate for the X spaces associated to the periodic Airy equation. As a consequence we obtain sharp local well-posedness results for periodic generalized KdV equations, as well as some global well-posedness results below the energy norm.

In this paper, we deal with the inversion of a physical model of a trumpet, i.e. how should the player control the model in order to obtain a given sound ? After having shown that the inversion is a ill-posed problem, we add a physically based constraint which leads to a physically pertinent solution .

We prove that the Benjamin–Ono initial-value problem is locally well-posed for small data in the Banach spaces H̃σ(R), σ ≥ 0, of complex-valued Sobolev functions with special low-frequency structure.

We present a class of spatially interconnected systems with boundary conditions that have close links with their spatially invariant extensions. In particular, well-posedness, stability, and performance of the extension imply the same characteristics for the actual, finite extent system. In turn, existing synthesis methods for control of spatially invariant systems can be extended to this class...