Analysis of PDEs (math.AP)

Abstract: This paper deals with the Vlasov-Stokes' system in three dimensions with periodic boundary conditions in the spatial variable. We prove the existence of a unique strong solution to this two-phase model under the assumption that initial velocity moments of certain order are bounded. We use a fixed point argument to arrive at a global-in-time solution.

Abstract: In this paper, we prove the global existence of analytical solutions to the compressible Oldroyd-B model without retardation near a non-vacuum equilibrium in ${\mathbb R}^n$ $(n=2,3)$. Zero retardation results in zero dissipation in the velocity equation, which is the main difficulty that prevents us to obtain the long time well-posedness of solutions. Through dedicated analysis, we find that the linearized equations of this model have damping effects, which ensures the global-in-time existence of small data solutions. However, the nonlinear quadratic terms have one more order derivative than the linear part and no good structure is discovered to overcome this derivative loss problem. So we can only build the result in the analytical energy space rather than Sobolev space with finite order derivatives.

Abstract: We prove the validity of a regularizing property on the boundary of the double layer potential associated to the fundamental solution of a {\em nonhomogeneous} second order elliptic differential operator with constant coefficients in Schauder spaces of exponent greater or equal to two that sharpens classical results of N.M.~G\"{u}nter, S.~Mikhlin, V.D.~Kupradze, T.G.~Gegelia, M.O.~Basheleishvili and T.V.~Bur\-chuladze, U.~Heinemann and extends the work of A.~Kirsch who has considered the case of the Helmholtz operator.

Abstract: In this paper, we study the existence and stability of random transition waves for time heterogeneous Fisher-KPP Equations with nonlocal diffusion. More specifically, we consider general time heterogeneities both for the nonlocal diffusion kernel and the reaction term. We use the comparison principle of the scalar equation and the method of upper and lower solutions to investigate the existence of random transition wave solution when the wave speed is large enough. In addition, we show the stability of random transition fronts for non-autonomous Fisher-KPP equations with nonlocal diffusion.

Abstract: We study the existence and non-existence of positive solutions for the following class of nonlinear elliptic problems in the hyperbolic space $$ -\Delta_{\mathbb{B}^N} u-\lambda u=a(x)u^{p-1} \, + \, \varepsilon u^{2^*-1} \,\;\;\text{in}\;\mathbb{B}^{N}, \quad u \in H^{1}{(\mathbb{B}^{N})}, $$ where $\mathbb{B}^N$ denotes the hyperbolic space, $2<p<2^*:=\frac{2N}{N-2}$, if $N \geqslant 3; 2<p<+\infty$, if $N = 2,\;\lambda < \frac{(N-1)^2}{4}$, and $0< a\in L^\infty(\mathbb{B}^N).$ We first prove the existence of a positive radially symmetric ground-state solution for $a(x) \equiv 1.$ Next, we prove that for $a(x) \geq 1$, there exists a ground-state solution for $\varepsilon$ small. For proof, we employ ``conformal change of metric" which allows us to transform the original equation into a singular equation in a ball in $\mathbb R^N$. Then by carefully analysing the energy level using blow-up arguments, we prove the existence of a ground-state solution. Finally, the case $a(x) \leq 1$ is considered where we first show that there is no ground-state solution, and prove the existence of a \it bound-state solution \rm (high energy solution) for $\varepsilon$ small. We employ variational arguments in the spirit of Bahri-Li to prove the existence of high energy-bound-state solutions in the hyperbolic space.

Abstract: In this paper, we consider the focusing fractional nonlinear Schr\"{o}dinger equation (FNLS) on the waveguide manifolds $\mathbb{R}^d\times\mathbb{T}^m$ both in the isotropic and anisotropic case. Under different conditions, we establish the existence and periodic dependence of the ground states of the focusing FNLS. In the intercritical regime, we also establish the large data scattering for the anisotropic focusing FNLS by appealing to the framework of semivirial vanishing geometry.

Abstract: In this work we prove a sharp quantitative form of Liouville's theorem, which asserts that, for all $n\geq 3$, the weakly conformal maps of $\mathbb S^{n-1}$ with degree $\pm 1$ are M\"obius transformations. In the case $n=3$ this estimate was first obtained by Bernand-Mantel, Muratov and Simon (Arch. Ration. Mech. Anal. 239(1):219-299, 2021), with different proofs given later on by Topping, and by Hirsch and the third author. The higher-dimensional case $n\geq 4$ requires new arguments because it is genuinely nonlinear: the linearized version of the estimate involves quantities which cannot control the distance to M\"obius transformations in the conformally invariant Sobolev norm. Our main tool to circumvent this difficulty is an inequality introduced by Figalli and Zhang in their proof of a sharp stability estimate for the Sobolev inequality.

Abstract: We prove the absence of a Lavrentiev gap for vectorial integral functionals of the form $$ F: g+W_0^{1,1}(\Omega)^m\to\mathbb{R}\cup\{+\infty\},\qquad F(u)=\int_\Omega W(x,\mathrm{D} u)\,\mathrm{d}x, $$ where the boundary datum $g:\Omega\subset \mathbb{R}^d\to\mathbb{R}^m$ is sufficiently regular, $\xi\mapsto W(x,\xi)$ is convex and lower semicontinuous, satisfies $p$-growth from below and suitable growth conditions from above. More precisely, if $p\leq d-1$, we assume $q$-growth from above with $q\leq \frac{(d-1)p}{d-1-p}$, while for $p>d-1$ we require essentially no growth conditions from above and allow for unbounded integrands. Concerning the $x$-dependence, we impose a well-known local stability estimate that is redundant in the autonomous setting, but in the general non-autonomous case can further restrict the growth assumptions.

Abstract: For the obstacle problem with a nonlinear operator, we characterize the space of global solutions with compact contact sets. This is achieved by constructing a bijection onto a class of quadratic polynomials describing the asymptotic behavior of solutions.

Abstract: We prove a general clustering result for the fractional Sobolev space W^{s,p}. Then we show how corresponding results in W^{1,p} and BV, respectively, can be deduced as special cases.