The failure of uniform dependence on the data is an interesting property classical solution for a hyperbolic system. In this paper, we consider map Cauchy problem to 2D viscous shallow water equations, which hyperbolic–parabolic We give new approach studying issue nonuniform initial these equations. prove that not uniformly continuous in Sobolev spaces H s × $H^s\times H^{s}$ > 2 $s>2$ .

We present a new proof of Banaschewski's theorem stating that the completion lift of a uniform surjection is a surjection. The new procedure allows to extend the fact (and, similarly, the related theorem on closed uniform sublocales of complete uniform frames) to quasi-uniformities ("not necessarily symmetric uniformities"). Further, we show how a (regular) Cauchy point on a closed uniform subl...

Whether or not the data-to-solution map of Cauchy problem for Camassa–Holm equation and Novikov in critical Besov space $$B_{2,1}^{3/2}({\mathbb {R}})$$ is uniformly continuous remains open. In paper, we aim at solving open question left previous works (Li et al. J Differ Equ 269:8686–8700, 2020a; Math Fluid Mech 22:50, 2020b) giving a negative answer to this problem.

Let Ω be a bounded C2 domain in R and φ : ∂Ω → R be a continuous map. The Dirichlet problem for the minimal surface system asks whether there exists a Lipschitz map f : Ω → R with f |∂Ω = φ and with the graph of f a minimal submanifold in Rn+m. For m = 1, the Dirichlet problem was solved more than thirty years ago by Jenkins-Serrin [13] for any mean convex domains and the solutions are all smoo...

We complete the known results on the Cauchy problem in Sobolev spaces for the KdV-Burgers equation by proving that this equation is well-posed in H−1(R) with a solution-map that is analytic from H−1(R) to C([0, T ];H−1(R)) whereas it is ill-posed in Hs(R), as soon as s < −1, in the sense that the flow-map u0 7→ u(t) cannot be continuous from H s(R) to even D′(R) at any fixed t > 0 small enough....

We study uniqueness of the recovery a time-dependent magnetic vector-valued potential and an electric scalar-valued on Riemannian manifold from knowledge Dirichlet to Neumann map hyperbolic equation. The Cauchy data is observed time-like parts space-time boundary proved up natural gauge for problem. proof based Gaussian beams inversion light ray transform Lorentzian manifolds under assumptions ...

Let F (z) = z − H(z) with o(H(z)) ≥ 2 be a formal map from Cn to Cn and G(z) the formal inverse of F (z). In this paper, we give two recurrent formulas for the formal inverse G(z). The first formula not only provides an efficient method for the calculation of G(z), but also reduces the inversion problem to a Cauchy problem of a partial differential equation. The second one is differential free ...

We prove that the Cauchy problem for the three-dimensional Navier–Stokes equations is ill-posed in Ḃ −1,∞ ∞ in the sense that a “norm inflation” happens in finite time. More precisely, we show that initial data in the Schwartz class S that are arbitrarily small in Ḃ−1,∞ ∞ can produce solutions arbitrarily large in Ḃ−1,∞ ∞ after an arbitrarily short time. Such a result implies that the solution ...

Consider the Cauchy problem for a strictly hyperbolic, N×N quasilinear system in one space dimension ut + A(u)ux = 0, u(0, x) = ū(x), (1) where u 7→ A(u) is a smooth matrix-valued map, and the initial data u is assumed to have small total variation. We investigate the rate of convergence of approximate solutions of (1) constructed by the Glimm scheme, under the assumption that, letting λk(u), r...

Considering the Cauchy problem for the modified finite-depthfluid equation ∂tu− Gδ(∂ 2 xu)∓ u 2ux = 0, u(0) = u0, where Gδf = −iF [coth(2πδξ)− 1 2πδξ ]Ff , δ&1, and u is a real-valued function, we show that it is uniformly globally well-posed if u0 ∈ Hs (s ≥ 1/2) with ‖u0‖L2 sufficiently small for all δ&1. Our result is sharp in the sense that the solution map fails to be C in Hs(s < 1/2). More...