There is an interesting open question: for $n$-D ($n\ge 1$) semilinear Euler-Poisson-Darboux equation $\partial_t^2u-Δu+\fracμ{t}\partial_tu=|u|^p$, where $t\ge 1$, $p>1$ and $μ>0$, the global small data weak solution $u$ will exist when $p>p_{crit}(n,μ)=\max\{p_s(n+μ), p_f(n)\}$ with the Strauss exponent $p_{s}(n+μ)=\frac{n+μ+1+\sqrt{(n+μ)^2+10(n+μ)-7}}{2(n+μ-1)}$ and the Fujita exponent $p_f(n)=1+\frac{2}{n}$. The blowup of weak solution $u$ has been shown when $1<p\le p_{crit}(n,μ)$ meanwhile this open question has been solved for $n=1,2$. In the present paper, we focus on this open question for $n=3$ and establish the global existence of small data solution $u$ for $μ\geq\frac{14}{5}$ (equivalent to $p_{crit}(3,μ)=p_f(3)=\frac53$) and $p>\max\{\frac53, 1+\frac{2}μ\}$.
Degenerate hyperbolic equations are dealing with many important issues for applied nature. While a variety of degenerate equations and boundary conditions, successfully matched to these differential equation, most in the characteristic coordinates reduced to Euler-Darboux one. Some boundary value problems, in particular Cauchy problem, for the specified equation demanded the introduction of special classes in which formulae are simple and can be used to meet the new challenges, including Delta-problems in squares that contain singularity line for equation coefficients with data on adjacent or …
The inverse problem for the Euler-Poisson-Darboux equation deals with reconstruction of the Cauchy data for this equation from incomplete information about its solution. In the present article, this problem is studied in connection with the injectivity of the shifted $k$-plane transform, which assigns to functions in $L^p(\mathbb {R}^n)$ their mean values over all k-planes at a fixed distance from the given $k$-planes. Several generalizations, including the Radon transform over strips of fixed width in $\mathbb {R}^2$ and a similar transform over tubes of fixed diameter in $\mathbb {R}^3$, are…
In Einstein's theory of relativity, the interaction of two collinearly polarized plane gravitational waves can be described by a Goursat problem for the Euler--Darboux equation in a triangular domain. In this paper, using a representation of the solution in terms of Abel integrals, we give a full asymptotic expansion of the solution near the diagonal of the triangle. The expansion is related to the formation of a curvature singularity of the spacetime. In particular, our framework allows for boundary data with derivatives which are singular at the corners. This level of generality is crucial f…
In this paper, we study desingularization of steady solutions of 3D incompressible Euler equation with helical symmetry in a general helical domain. We construct a family of steady Euler flows with helical symmetry, such that the associated vorticities tend asymptotically to a helical vortex filament. The solutions are obtained by solving a semilinear elliptic problem in divergence form with a parameter. By using the stream-function method, we show the existence and asymptotic behavior of ground state solutions concentrating near a single point as the parameter $ \varepsilon\to 0 $. Qualitativ…
In this article, we construct traveling-rotating helical vortices with small cross-section to the 3D incompressible Euler equations in an infinite pipe, which tend asymptotically to singular helical vortex filament evolved by the binormal curvature flow. The construction is based on studying a general semilinear elliptic problem in divergence form \begin{equation*} \begin{cases} -\varepsilon^2\text{div}(K(x)\nabla u)= (u-q|\ln\varepsilon|)^{p}_+,\ \ &x\in Ω,\\ u=0,\ \ &x\in\partial Ω, \end{cases} \end{equation*} for small values of $ \varepsilon. $ Helical vortex solutions concentrating near s…
In this paper, we investigate steady Euler flows in a two-dimensional bounded domain. By an adaption of the vorticity method, we prove that for any nonconstant harmonic function $q$, which corresponds to a nontrivial irrotational flow, there exists a family of steady Euler flows with small circulation in which the vorticity is continuous and supported in a small neighborhood of the set of maximum points of $q$ near the boundary, and the corresponding stream function satisfies a semilinear elliptic equation with a given profile function. Moreover, if $q$ has $k$ isolated maximum points $\{\bar{…
In this paper, we consider steady Euler flows in two-dimensional bounded annuli, as well as in exterior circular domains, in punctured disks and in the punctured plane. We always assume rigid wall boundary conditions. We prove that, if the flow does not have any stagnation point, and if it satisfies further conditions at infinity in the case of an exterior domain or at the center in the case of a punctured disk or the punctured plane, then the flow is circular, namely the streamlines are concentric circles. In other words, the flow then inherits the radial symmetry of the domain. The proofs ar…
We are concerned with the existence of periodic travelling-wave solutions for the generalized surface quasi-geostrophic (gSQG) equation(including incompressible Euler equation), known as von Kármán vortex street. These solutions are of $C^1$ type, and are obtained by studying a semilinear problem on an infinite strip whose width equals to the period. By a variational characterization of solutions, we also show the relationship between vortex size, travelling speed and street structure. In particular, the vortices with positive and negative intensity have equal or unequal scaling size in our co…
We analyze the Cauchy problem for symmetric hyperbolic equations with a time singularity of Fuchsian type and establish a global existence theory along with decay estimates for evolutions towards the singular time under a small initial data assumption. We then apply this theory to semilinear wave equations near spatial infinity on Minkowski and Schwarzschild spacetimes, and to the relativistic Euler equations with Gowdy symmetry on Kasner spacetimes.
