RESULT OF LOCAL EXISTENCE OF SOLUTIONS OF NONLOCAL VISCOELASTIC SYSTEM WITH RESPECT TO THE NONLINEARITY OF SOURCE TERMS

This work deals with the proof of local existence theorem of solutions for coupled nonlocal singular viscoelastic system with respect to the nonlinearity of source terms by using the Faedo–Galerkin method together with energy methods. This work makes a new contribution, since most of the previous works did not address the proof of the theorem of the local existence of solutions. It is also a completed study of Boulaaras et al. [Adv. Differ. Equ. 2020 (2020) 310].

Continuous dependence of the Cauchy problem for the inhomogeneous nonlinear Schrödinger equation in $H^{s} (\mathbb R^{n})$

<p style='text-indent:20px;'>We consider the Cauchy problem for the inhomogeneous nonlinear Schrödinger (INLS) equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ iu_{t} +\Delta u = |x|^{-b} f(u), \;u(0)\in H^{s} (\mathbb R^{n} ), $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ n\in \mathbb N $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ 0&lt;s&lt;\min \{ n, \; 1+n/2\} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ 0&lt;b&lt;\min \{ 2, \;n-s, \;1+\frac{n-2s}{2} \} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ f(u) $\end{document}</tex-math></inline-formula> is a nonlinear function that behaves like <inline-formula><tex-math id="M5">\begin{document}$ \lambda |u|^{\sigma } u $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M6">\begin{document}$ \sigma&gt;0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M7">\begin{document}$ \lambda \in \mathbb C $\end{document}</tex-math></inline-formula>. Recently, the authors in [<xref ref-type="bibr" rid="b1">1</xref>] proved the local existence of solutions in <inline-formula><tex-math id="M8">\begin{document}$ H^{s}(\mathbb R^{n} ) $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M9">\begin{document}$ 0\le s&lt;\min \{ n, \; 1+n/2\} $\end{document}</tex-math></inline-formula>. However even though the solution is constructed by a fixed point technique, continuous dependence in the standard sense in <inline-formula><tex-math id="M10">\begin{document}$ H^{s}(\mathbb R^{n} ) $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M11">\begin{document}$ 0&lt; s&lt;\min \{ n, \; 1+n/2\} $\end{document}</tex-math></inline-formula> doesn't follow from the contraction mapping argument. In this paper, we show that the solution depends continuously on the initial data in the standard sense in <inline-formula><tex-math id="M12">\begin{document}$ H^{s}(\mathbb R^{n} ) $\end{document}</tex-math></inline-formula>, i.e. in the sense that the local solution flow is continuous <inline-formula><tex-math id="M13">\begin{document}$ H^{s}(\mathbb R^{n} )\to H^{s}(\mathbb R^{n} ) $\end{document}</tex-math></inline-formula>, if <inline-formula><tex-math id="M14">\begin{document}$ \sigma $\end{document}</tex-math></inline-formula> satisfies certain assumptions.</p>

Revisiting the characteristic initial value problem for the vacuum Einstein field equations

Using the Newman–Penrose formalism we study the characteristic initial value problem in vacuum General Relativity. We work in a gauge suggested by Stewart, and following the strategy taken in the work of Luk, demonstrate local existence of solutions in a neighbourhood of the set on which data are given. These data are given on intersecting null hypersurfaces. Existence near their intersection is achieved by combining the observation that the field equations are symmetric hyperbolic in this gauge with the results of Rendall. To obtain existence all the way along the null-hypersurfaces themselves, a bootstrap argument involving the Newman–Penrose variables is performed.

Improved existence for the characteristic initial value problem with the conformal Einstein field equations

We adapt Luk’s analysis of the characteristic initial value problem in general relativity to the asymptotic characteristic problem for the conformal Einstein field equations to demonstrate the local existence of solutions in a neighbourhood of the set on which the data are given. In particular, we obtain existence of solutions along a narrow rectangle along null infinity which, in turn, corresponds to an infinite domain in the asymptotic region of the physical spacetime. This result generalises work by Kánnár on the local existence of solutions to the characteristic initial value problem by means of Rendall’s reduction strategy. In analysing the conformal Einstein equations we make use of the Newman–Penrose formalism and a gauge due to J. Stewart.

Existence and uniqueness for a nonlocal internal wave model

In this paper, we establish local existence of solutions of a variant of a system derived by Choi and Camassa [Weakly nonlinear internal waves in a two-fluids system, J. Fluid Mech. 313 (1996) 83–103] to describe the propagation of an internal wave at the interface of two immiscible fluids with constant densities. We also present a numerical solver to approximate the solutions of the Cauchy problem.