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We show that to every small and decaying solution of the linearized constraint equations about Minkowski spacetime, one can add a quadratically small correction to obtain a solution of the full constraint equations. Near spacelike infinity,…
We consider a class of particular solutions to the (2+1)-dimensional nonlinear partial differential equation (PDE) $u_t +\partial_{x_2}^n u_{x_1} - u_{x_1} u =0$ (here $n$ is any integer) reducing it to the ordinary differential equation…
The set of integrable symmetries of the nonstationary Schr\"{o}dinger equation is shown to admit a natural decomposition into subsets of mutually commuting symmetries. Hierarchies of time evolutions associated with each of these subsets…
We establish the existence of weak solutions $u$ of the semilinear wave equation $\partial_t^2 u-\textrm{div}_x(a(t,x)\nabla_xu)=f_k(u)$ where $a(t,x)$ is equal to $1$ outside a compact set with respect to $x$ and a non-linear term $f_k$…
This work studies the angular component $ \pi_{t} = u_{t} / \| u_{t} \| $ associated to the solution $ u $ of a vector-valued linear hyperviscous SPDE on a $d$-dimensional torus $$\mathrm{d} u^{\alpha} =- \nu^{\alpha} (- \Delta)^{\mathbf{a}…
Well-posedness and a number of qualitative properties for solutions to the Cauchy problem for the following nonlinear diffusion equation with a spatially inhomogeneous source $$ \partial_tu=\Delta u^m+|x|^{\sigma}u^p, $$ posed for…
The Zakharov-Kuznetsov equation in spatial dimension $d\geq 5$ is considered. The Cauchy problem is shown to be globally well-posed for small initial data in critical spaces and it is proved that solutions scatter to free solutions as $t…
The Direct and the Inverse Scattering Problems for the heat operator with a potential being a perturbation of an arbitrary $N$ soliton potential are formulated. We introduce Jost solutions and spectral data and present their properties.…
We study the asymptotic behavior of the Oleinik's solution to the steady Prandtl equation when the outer flow $U(x)=1$. Serrin proved that the Oleinik's solution converges to the famous Blasius solution $\bar u$ in $L^\infty_y$ sense as…
Boundary value problems for integrable nonlinear evolution PDEs formulated on the finite interval can be analyzed by the unified method introduced by one of the authors and used extensively in the literature. The implementation of this…
It is shown that Einstein-Weyl (EW) equations in 2+1 dimensions contain the dispersionless Kadomtsev-Petviashvili (dKP) equation as a special case: If an EW structure admits a constant weighted vector then it is locally given by $h=d y^2-4d…
We consider a perturbed KdV equation: [\dot{u}+u_{xxx} - 6uu_x = \epsilon f(x,u(\cdot)), \quad x\in \mathbb{T}, \quad\int_\mathbb{T} u dx=0.] For any periodic function $u(x)$, let $I(u)=(I_1(u),I_2(u),...)\in\mathbb{R}_+^{\infty}$ be the…
We consider a scalar conservation law with a spatially discontinuous flux at a single point $x=0$, and we study the initial data identification problem for $AB$-entropy solutions associated to an interface connection $(A,B)$. This problem…
Given $2\leq p<\infty$, $s\in (0, 1)$ and $t\in (1, 2s)$, we establish interior $W^{t,p}$ Calderon-Zygmund estimates for solutions of nonlocal equations of the form \[ \int_{\Omega} \int_{\Omega} K\left (x,|x-y|,\frac{x-y}{|x-y|}\right )…
The initial-boundary value problem for the Kundu--Eckhaus equation on the half-line is considered in this paper by using the Fokas method. We will show that the solution $u(x,t)$ can be expressed in terms of the solution of a matrix…
A new class of time-symmetric solutions to the initial value constraints of vacuum General Relativity is introduced. These data are globally regular, asymptotically flat (with possibly several asymptotic ends) and in general have no…
We study the generalization of the dispersionless Kadomtsev - Petviashvili (dKP) equation in n+1 dimensions and with nonlinearity of degree m+1, a model equation describing the propagation of weakly nonlinear, quasi one dimensional waves in…
Motivated by optimal control problems and differential games for functional differential equations of retarded type, the paper deals with a Cauchy problem for a path-dependent Hamilton--Jacobi equation with a right-end boundary condition.…
We are concerned with the Cauchy problem for the KdV equation on the whole line with an initial profile V_0 which is decaying sufficiently fast at +\infty and arbitrarily enough (i.e., no decay or pattern of behavior) at -\infty. We show…
This paper resolves the characteristic initial data problem for the three-dimensional compressible Euler equations - an open problem analogous to Christodoulou's characteristic initial value formulation for the vacuum Einstein field…