Axisymmetric solutions of the two-dimensional Euler equations with a two-constant equation of state

A family of self-similar and global bounded weak solutions are constructed rigorously for all positive time to the two-dimensional isentropic Euler equations with the equation of state $p=A_1{\rho }^{{\gamma }_1}+A_2{\rho }^{{\gamma }_2}$ for initial pure radial velocity. The main difficulty is that the equations cannot be directly reduced to an autonomous system of ordinary differential equations. To deal with it, we first establish the global existence and the detailed structures of solutions to a new system under the axisymmetric and self-similar assumptions.     

Lump solutions to nonlinear partial differential equations via Hirota bilinear forms

Lump solutions are analytical rational function solutions localized in all directions in space. We analyze a class of lump solutions, generated from quadratic functions, to nonlinear partial differential equations. The basis of success is the Hirota bilinear formulation and the primary object is the class of positive multivariate quadratic functions. A complete determination of quadratic functions positive in space and time is given, and positive quadratic functions are characterized as sums of squares of linear functions. Necessary and sufficient conditions for positive quadratic functions to solve Hirota bilinear equations are presented, and such polynomial solutions yield lump solutions to nonlinear partial differential equations under the dependent variable transformations $u=2{(\ln ?f)}_x$ and $u=2{(\ln ?f)}_{xx}$, where x is one spatial variable. Applications are made for a few generalized KP and BKP equations.     

A regularity theory for quasi-linear Stochastic PDEs in weighted Sobolev spaces

We study the second-order quasi-linear stochastic partial differential equations (SPDEs) defined on $C^1$-domains. The coefficients are random functions depending on $t,x$ and the unknown solutions. We prove the uniqueness and existence of solutions in appropriate Sobolev spaces, and in addition, we obtain $L_p$ and Hölder estimates of both the solution and its gradient.     

Dispersive estimates for rational symbols and local well-posedness of the nonzero energy NV equation. II

We continue our study on the Cauchy problem for the two-dimensional Novikov–Veselov (NV) equation, integrable via the inverse scattering transform for the two dimensional Schrödinger operator at a fixed energy parameter. This work is concerned with the more involved case of a positive energy parameter. For the solution of the linearized equation we derive smoothing and Strichartz estimates by combining new estimates for two different frequency regimes, extending our previous results for the negative energy case [18]. The low frequency regime, which our previous result was not able to treat, is studied in detail. At non-low frequencies we also derive improved smoothing estimates with gain of almost one derivative. Then we combine the linear estimates with a Fourier decomposition method and $X^{s,b}$ spaces to obtain local well-posedness of NV at positive energy in $H^s$, $s>\frac{1}{2}$. Our result implies, in particular, that at least for $s>\frac{1}{2}$, NV does not change its behavior from semilinear to quasilinear as energy changes sign, in contrast to the closely related Kadomtsev–Petviashvili equations. As a complement to our LWP results, we also provide some new explicit solutions of NV at zero energy, generalizations of the lumps solutions, which exhibit new and nonstandard long time behavior. In particular, these solutions blow up in infinite time in $L^2$.     

Universality in the bulk holds close to given points

Let $\mu$ be a measure with compact support. Assume that $\xi$ is a Lebesgue point of $\mu$ and that ${\mu }^{\prime }$ is positive and continuous at $\xi$. Let $\left\{A_n\right\}$ be a sequence of positive numbers with limit $\infty$. We show that one can choose ${\xi }_n\in [\xi ?\frac{A_n}{n},\xi +\frac{A_n}{n}]$ such that $\underset{n\to \infty }{lim}\frac{K_n({\xi }_n,{\xi }_n+\frac{a}{{\stackrel{?}{K}}_n({\xi }_n,{\xi }_n)})}{K_n({\xi }_n,{\xi }_n)}=\frac{sin\pi a}{\pi a}\text{,}$ uniformly for $a$ in compact subsets of the plane. Here $K_n$ is the $n$th reproducing kernel for $\mu$, and ${\stackrel{?}{K}}_n$ is its normalized cousin. Thus universality in the bulk holds on a sequence close to $\xi$, without having to assume that $\mu$ is a regular measure. Similar results are established for sequences of measures.     

Global solvability of the Navier–Stokes equations with a free surface in the maximal Lp-Lq regularity class

We consider the motion of incompressible viscous fluids bounded above by a free surface and below by a solid surface in the N-dimensional Euclidean space for $N\ge 2$. The aim of this paper is to show the global solvability of the Navier–Stokes equations with a free surface, describing the above-mentioned motion, in the maximal $L_p\text{-}{L}_q$ regularity class. Our approach is based on the maximal $L_p\text{-}{L}_q$ regularity with exponential stability for the linearized equations, and also it is proved that solutions to the original nonlinear problem are exponentially stable.     

Quasi-linear PDEs and forward–backward stochastic differential equations: Weak solutions

In this paper, we study the existence, uniqueness and the probabilistic representation of the weak solutions of quasi-linear parabolic and elliptic partial differential equations (PDEs) in the Sobolev space $H_{\rho }^1({\mathbb{R}}^d)$. For this, we study first the solutions of forward–backward stochastic differential equations (FBSDEs) with smooth coefficients, regularity of solutions and their connection with classical solutions of quasi-linear parabolic PDEs. Then using the approximation procedure, we establish their convergence in the Sobolev space to the solutions of the FBSDES in the space $L_{\rho }^2({\mathbb{R}}^d;{\mathbb{R}}^d)?L_{\rho }^2({\mathbb{R}}^d;{\mathbb{R}}^k)?L_{\rho }^2({\mathbb{R}}^d;{\mathbb{R}}^{k\times d})$. This gives a connection with the weak solutions of quasi-linear parabolic PDEs. Finally, we study the unique weak solutions of quasi-linear elliptic PDEs using the solutions of the FBSDEs on infinite horizon.