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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]

Consider n ducts having a common origin and filled with a fluid. Along each duct, the full Euler system describes the evolution of the fluid. At the junction, suitable physical conditions couple the n Euler systems. In this paper we prove the well posedness of the Cauchy problem for the model so obtained, provided the total varaiatio of the initial data is sufficiently small. [ABSTRACT FROM AUTHOR]

We consider smooth solutions of radial exterior Cauchy–Neumann problems with small initial data for radial quasilinear wave equations in three space dimensions, when the size of the initial data tends to 0. We obtain rather precise information on the lifespan, analogous with well known Cauchy problem results of Hörmander and John. [ABSTRACT FROM AUTHOR]

We investigate the Cauchy problem for homogeneous equations of order m in the (t,x)-plane, with coefficients depending only on x. Assuming that the characteristic roots satisfy the condition \[ \lambda_i^2(x) + \lambda_j^2(x) \le M\,\bigl(\lambda_i(x)-\lambda_j(x)\bigr)^2, \quad 1\le i < j \le m, \] we succeed in constructing a smooth symmetrizer which behaves like a diagonal matrix: this allows us to prove the well-posedness in $\mathcal{C}^\infty$. [ABSTRACT FROM AUTHOR]

Benney introduced a general strategy for deriving systems of nonlinear partial differential equations associated with long- and short-wave solutions. The semi-linear Benney system was studied recently by Tsutsumi and Hatano. Here, we tackle the nonlinear version of it and using compensated compactness techniques, we prove the global existence of weak solutions to the Cauchy problem, in the case that the equation for the amplitude of the long wave is a quasilinear conservation law with flux f(v) = av2 - bv3 where a, b are constants with b > 0. [ABSTRACT FROM AUTHOR]

For a class of first-order weakly hyperbolic pseudo-differential systems with finite time degeneracy, well-posedness of the Cauchy problem is proved in an adapted scale of Sobolev spaces. These Sobolev spaces are constructed in correspondence to the hyperbolic operator under consideration, making use of ideas from the theory of elliptic boundary value problems on manifolds with singularities. In addition, an upper bound for the loss of regularity that occurs when passing from the Cauchy data to the solutions is established. In many examples, this upper bound turns out to be sharp. [ABSTRACT FROM AUTHOR]