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We study the Cauchy problem for the simplest first-order Hamilton–Jacobi equation in one space dimension, with a bounded and Lipschitz continuous Hamiltonian which only depends on the spatial derivative. Uniqueness of discontinuous viscosity solutions is proven, if the initial data function has a finite number of jump discontinuities. Main ingredients of the proof are the barrier effect of spatial discontinuities of a solution (which is linked to the boundedness of the Hamiltonian), and a comparison theorem for semicontinuous viscosity subsolution and supersolution. These are defined in the spirit of the paper [H. Ishii, Perron's method for Hamilton–Jacobi equations, Duke Math. J.55 (1987) 368–384], yet using essential limits to introduce semicontinuous envelopes. The definition is shown to be compatible with Perron's method for existence and is crucial in the uniqueness proof. We also describe some properties of the time evolution of spatial jump discontinuities of the solution, and obtain several results about singular Neumann problems which arise in connection with the above referred barrier effect. [ABSTRACT FROM AUTHOR]

The goal of this paper is three-fold. First, we prove that the Cauchy problem for a generalized KP-I equation u t + | D x | α ∂ x u + ∂ x − 1 ∂ y 2 u + 1 2 ∂ x (u 2) = 0 , α ≥ 4 is locally well-posed in the anisotropic Sobolev spaces H s 1 , s 2 ( R 2) with s 1 > − α − 1 4 and s 2 ≥ 0. Second, we prove that the Cauchy problem is globally well-posed in H s 1 , 0 ( R 2) with s 1 > − (α − 1) (3 α − 4) 4 (5 α + 3) if 4 ≤ α ≤ 5. Finally, we show that the Cauchy problem is globally well-posed in H s 1 , 0 ( R 2) with s 1 > − α (3 α − 4) 4 (5 α + 4) if α > 5. Our result improves the result of Saut and Tzvetkov [The Cauchy problem for the fifth order KP equations, J. Math. Pures Appl.79 (2000) 307–338] and Li and Xiao [Well-posedness of the fifth order Kadomtsev–Petviashvili-I equation in anisotropic Sobolev spaces with nonnegative indices, J. Math. Pures Appl.90 (2008) 338–352]. [ABSTRACT FROM AUTHOR]

We consider an Oldroyd-B model which is derived in Ref. 4 [J. W. Barrett, Y. Lu and E. Süli, Existence of large-data finite-energy global weak solutions to a compressible Oldroyd-B model, Commun. Math. Sci.15 (2017) 1265–1323] via micro–macro-analysis of the compressible Navier–Stokes–Fokker–Planck system. The global well posedness of strong solutions as well as the associated time-decay estimates in Sobolev spaces for the Cauchy problem are established near an equilibrium state. The terms related to η , in the equation for the extra stress tensor and in the momentum equation, lead to new technical difficulties, such as deducing L t 2 L x 2 -norm dissipative estimates for the polymer number density and its spatial derivatives. One of the main objectives of this paper is to develop a way to capture these dissipative estimates via a low–medium–high-frequency decomposition. [ABSTRACT FROM AUTHOR]