arXiv subjects
Publications and source records attributed to Achenef Tesfahun.
It is shown that the uniform radius of spatial analyticity $σ(t)$ of solutions at time $t$ to the 1d, 2d and 3d cubic nonlinear Schrödinger equations cannot decay faster than $1/|t|$ as $|t| \to \infty$, given initial data that is analytic with fixed radius $σ_0$.
Lower bound on the rate of decrease in time of the uniform radius of spatial analyticity of solutions to the quartic generalized KdV equation is derived, which improves an earlier result by Bona, Grujić and Kalisch.
We study the well-posedness of the Dirac-Klein-Gordon system in one space dimension with initial data that have an analytic extension to a strip around the real axis. It is proved that the radius of analyticity of the solutions at time $t$ cannot decay faster than $1/t^4$ as $t$ tends to infinity.
We prove that the initial value problem for the equation \[ - i\partial_t u + \sqrt{m^2-Δ} \, u= (\frac{e^{-μ_0 |x|}}{|x|} \ast |u|^2)u \ \text{in} \ \mathbb R^{1+3}, \quad m\ge 0, \ μ_0 >0\] is globally well-posed and the solution scatters to free waves asymptotically as $t \to \pm \infty$ if we start with initial data which is small in $H^s(\mathbb R^{3})$ for $s>\frac12$, and if $m>0$. Moreover, if the initial data is radially symmetric we can improve the above result to $m\ge 0$ and $s>0$, which is almost optimal, in the sense that $L^2(\mathbb R^{3})$ is the critical space for the equation. The main ingredients in the proof are certain endpoint Strichartz estimates, $L^2(\mathbb R^{1+3})$ bilinear estimates for free waves and the application of the $U^p$ and $V^p$ function spaces.
We prove that the Yang-Mills equations in the Lorenz gauge (YM-LG) is locally well-posed for data below the energy norm, in particular, we can take data for the gauge potential $A$ and the associated curvature $F$ in $H^s\times H^{s-1}$ and $H^r\times H^{r-1}$ for $s=(\frac67+,-\frac1{14} +)$, respectively. This extends a recent by Selberg and the present author on the local well-posedness of YM-LG for finite energy data (specifically, for $(s, r)=(1-, 0)$). We also prove unconditional uniqueness of the energy class solution, that is, uniqueness in the classical space $C([-T, T]; X_0)$, where $X_0$ is the energy data space. The key ingredient in the proof is the fact that most bilinear terms in YM-LG contain null structure some of which uncovered in the present paper.
The Yang-Mills-Higgs equations (Y-M-H), when written relative to Lorenz gauge, become a system of nonlinear wave equations. The key bilinear terms in the resulting system turn out to be null forms--this is in light of a recent discovery of null structure by Selberg and the present author for Yang-Mills equations in Lorenz gauge. Using the null structure found and bilinear space-time estimates, we prove local well-posedness of Y-M-H in Lorenz gauge for finite energy data.
We demonstrate null structure in the Yang-Mills equations in Lorenz gauge. Such structure was found in Coulomb gauge by Klainerman and Machedon, who used it to prove global well-posedness for finite-energy data. Compared with Coulomb gauge, Lorenz gauge has the advantage---shared with the temporal gauge---that it can be imposed globally in space even for large solutions. Using the null structure and bilinear space-time estimates, we also prove local-in-time well-posedness of the equations in Lorenz gauge, for data with finite energy. The time of existence depends on the initial energy and on the $H^s \times H^{s-1}$-norm of the initial potential, for some $s < 1$.
Recently, Candy and Bournaveas proved local well-posedness of the space-time monopole equation in Lorenz gauge for initial data in $H^s $ with $s>\frac14$. The equation is $L^2$-critical, and hence a $\frac14$ derivative gap is left between their result and the scaling prediction. In this paper, we consider initial data in the Fourier-Lebesgue space $\hat{H_p^s}$ for $1<p\le 2$ which coincides with $H^s$ when $p=2$ but scales like lower regularity Sobolev spaces for $1<p< 2$. In particular, we will see that as $p\rightarrow 1^+$, the critical exponent $s^c_p\rightarrow 1^-$, in which case $\hat{\dot H_{1+}^{1-}}$ is the critical space. We shall prove almost optimal local well-posedness to the space-time monopole equation in Lorenz gauge with initial data in the aforementioned spaces that correspond to $p$ close to 1.
We prove that the Cauchy problem for the Chern-Simons-Higgs equations on the (2+1)-dimensional Minkowski space-time is globally well posed for initial data with finite energy. This improves a result of Chae and Choe, who proved global well-posedness for more regular data. Moreover, we prove local well-posedness even below the energy regularity, using the the null structure of the system in Lorenz gauge and bilinear space-time estimates for wave-Sobolev norms.
Recently, A. Gruenrock and H. Pecher proved global well-posedness of the 2d Dirac-Klein-Gordon equations given initial data for the spinor and scalar fields in $H^s$ and $H^{s+1/2} \times H^{s-1/2}$, respectively, where $s\ge 0$, but uniqueness was only known in a contraction space of Bourgain type, strictly smaller than the natural solution space $C([0,T]; H^s \times H^{s+1/2} \times H^{s-1/2})$. Here we prove uniqueness in the latter space for $s \ge 0$. This improves a recent result of H. Pecher, where the range $s>1/30$ was covered.
It is known that the Maxwell-Klein-Gordon system (M-K-G), when written relative to the Coulomb gauge, is globally well-posed for finite-energy initial data. This result, due to Klainerman and Machedon, relies crucially on the null structure of the main bilinear terms of M-K-G in Coulomb gauge. It appears to have been believed that such a structure is not present in Lorenz gauge, but we prove here that it is, and we use this fact to prove finite-energy global well-posedness in Lorenz gauge. The latter has the advantage, compared to Coulomb gauge, of being Lorentz invariant, hence M-K-G in Lorenz gauge is a system of nonlinear wave equations, whereas in Coulomb gauge the system has a less symmetric form, as it contains also a nonlinear elliptic equation.
We prove that the Cauchy problem for the Dirac-Klein-Gordon system of equations in 1D is globally well-posed in a range of Sobolev spaces of negative index for the Dirac spinor and positive index for the scalar field. The main ingredient in the proof is the theory of almost conservation law and I-method introduced by Colliander, Keel, Staffilani, Takaoka and Tao. Our proof also relies on the null structure in the system, and bilinear spacetime estimates of Klainerman-Machedon type.
We prove that the Cauchy problem for the Dirac-Klein-Gordon system of equations in 1+3 dimensions is locally well-posed in a range of Sobolev spaces for the Dirac spinor and the meson field. The result contains and extends the earlier known results for the same problem. Our proof relies on the null structure in the system, and bilinear spacetime estimates of Klainerman-Machedon type.
We extend recent results of S. Machihara and H. Pecher on low regularity well-posedness of the Dirac-Klein-Gordon (DKG) system in one dimension. Our proof, like that of Pecher, relies on the null structure of DKG, recently completed by D'Ancona, Foschi and Selberg, but we show that in 1d the argument can be simplified by modifying the choice of projections for the Dirac operator. We also show that the result is best possible up to endpoint cases, if one iterates in Bourgain-Klainerman-Machedon spaces.