Local Well-Posedness of Generalized Maxwell-Chern-Simons-Higgs Equation in ℝ1+1

In this article we study local and global well-posedness of the Lagrangian Averaged Euler equations. We show local well-posedness in Triebel-Lizorkin spaces and further prove a Beale-Kato-Majda type necessary and sufficient condition for global existence involving the stream function. We also establish new sufficient conditions for global existence in terms of mixed Lebesgue norms of the generalized Clebsch variables.

We consider the Cauchy problem for the kinetic derivative nonlinear Schrödinger equation on the torus [Formula: see text] for [Formula: see text], where the constants [Formula: see text] are such that [Formula: see text] and [Formula: see text], and [Formula: see text] denotes the Hilbert transform. This equation has dissipative nature, and the energy method is applicable to prove local well-posedness of the Cauchy problem in Sobolev spaces [Formula: see text] for [Formula: see text]. However, the gauge transform technique, which is useful for dealing with the derivative loss in the nonlinearity when [Formula: see text], cannot be directly adapted due to the presence of the Hilbert transform. In particular, there has been no result on local well-posedness in low regularity spaces or global solvability of the Cauchy problem. In this paper, we shall prove local and global well-posedness of the Cauchy problem for small initial data in [Formula: see text], [Formula: see text]. To this end, we make use of the parabolic-type smoothing effect arising from the resonant part of the nonlocal nonlinear term [Formula: see text], in addition to the usual dispersive-type smoothing effect for nonlinear Schrödinger equations with cubic nonlinearities. As by-products of the proof, we also obtain forward-in-time regularization and backward-in-time ill-posedness results.

The purpose of this paper is to study local and global well-posedness of initial value problem for generalized Korteweg-de Vries (gKdV) equation in ^L^r. We show (large data) local well-posedness, small data global well-posedness, and small data scattering for gKdV equation in the scale critical ^L^r space. A key ingredient is a Stein-Tomas type inequality for the Airy equation, which generalizes usual Strichartz estimates for ^L^r-framework.

Local and global well-posedness for the 2D generalized Zakharov–Kuznetsov equation

In this paper, we consider an incompressible viscous flow without surface tension in a finite-depth domain of three dimensions, with a free top boundary and a fixed bottom boundary. The system is governed by the Navier--Stokes equations in this moving domain. Traditionally, this problem can be analyzed in the Lagrangian coordinates as a perturbation of linear equations in a fixed domain. In the series of papers [Anal. PDE, 6 (2013), pp. 287--369; Anal. PDE, 6 (2013), pp. 1429--1533; Arch. Ration. Mech. Anal., 207 (2013), pp. 459--531], Tice and Guo introduced a new framework using the geometric structure in the Eulerian coordinates to study both local and global well-posedness and decay of this system. Following this path, we extend their result in local well-posedness from the small data case to the general data case. Also, we give a simpler proof of global well-posedness in the small data case with horizontally infinite cross section. Other than the geometric energy estimates, the time-dependent Galerkin method, and the interpolation estimates with Riesz potential and minimal counts, which are introduced in these papers, we utilize three new techniques: (1) using the $\epsilon$-Poisson integral to construct a diffeomorphism between the fixed domain and the moving domain; (2) using a bootstrapping argument to prove the comparison estimates in the steady Navier--Stokes equations for general data of free surface; and (3) redefining the energy and dissipation to simplify the original complicated bootstrapping argument to show the interpolation estimates.

Regularity properties of the cubic nonlinear Schrödinger equation on the half line

Local and global well-posedness for the Cauchy problem associated with the nonlinear Dirac equation $$ i{\partial \psi \over \partial t}+i\alpha \cdot\nabla \psi -m\beta \psi +G(\psi)=0\quad \hbox{in}\ > \hbox{\bf R}^4 $$ are studied in the Sobolev spaces $H^s.$ For regular enough covariant nonlinearities that homogeneous of degree $p\ge 3$, local well-posedness in $H^{s}$ is proved for $s > {3\over 2}- {1\over p-1}$ when p is an odd integer and for ${3\over 2}-{1\over p-1} < s < {p-1\over 2}$ when p is not an odd integer. If $p > 3$, global well-posedness for small initial data in $H^{s(p)},$ $s(p)={3\over 2}-{1\over p-1}$, is also proved. Local and global well-posedness of the Cauchy problem for the nonlinear Klein--Gordon and wave equations are also considered.

In this paper, we address the local well-posedness of the spatially inhomogeneous noncutoff Boltzmann equation when the initial data decays polynomially in the velocity variable. We consider the case of very soft potentials $\gamma + 2s < 0$. Our main result completes the picture for local well-posedness in this decay class by removing the restriction $\gamma + 2s > -3/2$ of previous works. Our approach is entirely based on the Carleman decomposition of the collision operator into a lower order term and an integro-differential operator similar to the fractional Laplacian. Interestingly, this yields a very short proof of local well-posedness when $\gamma \in (-3,0]$ and $s \in (0,1/2)$ in a weighted $C^1$ space that we include as well.

In this paper, we address the question of the hyperbolicity and the local well-posedness of the two-layer shallow water model, with free surface, in two dimensions. We first provide a general criterion that proves the symmetrizability of this model, which implies hyperbolicity and local well-posedness in $\mathcal{H}^s(\mathbb{R}^2)$, with $s>2$. Then, we analyze rigorously the eigenstructure associated with this model and prove a more general criterion for hyperbolicity and local well-posedness, under a weak density-stratification assumption. Finally, we consider a new conservative two-layer shallow water model, prove the hyperbolicity and the local well-posedness, and relate it to the basic two-layer shallow water model.

The local well-posedness and existence of weak solutions for a generalized Camassa–Holm equation

Local well-posedness for the sixth-order Boussinesq equation

Editorial

This paper deals with the Cauchy problem for the interacting system of the Camassa–Holm and Degasperis–Procesi equations $$\begin{aligned} m_t=-3m(2u_x+v_x)-m_x(2u+v), n_t=-2n(2u_x+v_x)-n_x(2u+v), \end{aligned}$$ where $$m=u-u_{xx}$$ and $$n= v-v_{xx}$$ . By the transport equations theory and the classical Friedrichs regularization method, the local well-posedness of solutions for this system in nonhomogeneous Besov spaces $$B^s_{p,r}\times B^s_{p,r}$$ with $$1\le p,r \le +\infty $$ and $$s>\max \left\{ 2+\frac{1}{p},\frac{5}{2}\right\} $$ is obtained, and the local well-posedness in critical Besov space $$B^{5/2}_{2,1}\times B^{5/2}_{2,1}$$ is also established. Moreover, by the approach for approximate solutions and well-posedness estimates, we obtain two sequences of solution for this equation, which are bounded in the Sobolev space $$H^s({\mathbb {R}})\times H^s({\mathbb {R}})$$ with $$s>5/2$$ , and the distance between the two sequences is lower-bounded by a positive constant for any time t, but converges to zero at the initial time. This implies that the solution map is not uniformly continuous.

The Cauchy problem for 1-D nonlinear Schrödinger equations with quadratic nonlinearities are considered in the spaces $H^{s,a}$ defined by $ \| f \|_{H^{s,a}}=\| (1+|\xi|)^{s-a} |\xi|^a \widehat{f} \|_{L^2}, $ and sharp local well-posedness and ill-posedness results are obtained in these spaces for nonlinearities including the term $u\bar{u}$. In particular, when $a=0$ the previous well-posedness result in $H^s$, $s>-1/4$, given by Kenig, Ponce and Vega (1996), is improved to $s\ge -1/4$. This also extends the result in $H^{s,a}$ by Otani (2004). The proof is based on an iteration argument similar to that of Kenig, Ponce and Vega, with a modification of the spaces of the Fourier restriction norm. Our result is also applied to the ``good'' Boussinesq equation and yields local well-posedness in $H^s\times H^{s-2}$ with $s>-1/2$, which is an improvement of the previous result given by Farah (2009).

The local well-posedness and global solution for a modified periodic two-component Camassa–Holm system