Analysis of PDEs (math.AP)

Abstract: In this paper we prove the existence of a weak solution to a doubly nonlinear parabolic fractional $p$-Laplacian equation, which has general doubly non-linearlity including not only the Sobolev subcritical/critical/supercritical cases but also the slow/fast diffusion ones. Our proof reveals the weak convergence method for the doubly nonlinear fractional $p$-Laplace operator.

Abstract: In this note we investigate interior regularity criteria for suitable weak solutions to the 3D Naiver-Stokes equations, and obtain the solutions are regular in the interior if the $L^p_tL_x^q(Q_1)$ norm of the velocity is sufficiently small, where $1\leq \frac{2}{p}+\frac{3}{q}<2$ and $2\leq p\leq \infty$. It improves the recent result of $p,q>2 $ by Kwon \cite{Kwon} (J. Differential Equations 357 (2023), 1--31.), and also generalizes Chae-Wolf's $L_t^\infty L_x^{\frac32+}$ criterion \cite{CW2017} (Arch. Ration. Mech. Anal. 225 (2017), no. 1, 549--572.).

Abstract: This paper is devoted to the proof of the existence of the principal eigenvalue and related eigenfunctions for fully nonlinear uniformly elliptic equations posed in a punctured ball, in presence of a singular potential. More precisely, we analyze existence, uniqueness and regularity of solutions $( \bar\lambda_\gamma, u_\gamma)$ of the equation $$F( D^2 u_\gamma)+ \bar \lambda_\gamma \frac{u_\gamma}{r^\gamma} = 0\ {\rm in} \ B(0,1)\setminus \{0\}, \ u_\gamma = 0 \ {\rm on} \ \partial B(0,1)$$ where $u_\gamma>0$ in $B(0,1)\setminus \{0\}$, and $\gamma >0$. We prove existence of radial solutions which are continuous on $\overline{ B(0,1)}$ in the case $\gamma <2$, existence of unbounded solutions in the case $\gamma = 2$ and a non existence result for $\gamma >2$. We also give the explicit value of $\bar \lambda_2$ in the case of Pucci's operators, which generalizes the Hardy--Sobolev constant for the Laplacian.

Abstract: We show convergence of minimizers of weighted inertia-energy functionals to solutions of initial value problems for a class of nonlinear wave equations. The result is given for the nonhomogeneous case under a natural growth assumption on the forcing term.

Abstract: We present extensions of rigidity estimates and of Korn's inequality to the setting of (mixed) variable exponents growth. The proof techniques, based on a classical covering argument, rely on the log-H\"older continuity of the exponent to get uniform regularity estimates on each cell of the cover, and on an extension result \`a la Nitsche in Sobolev spaces with variable exponents. As an application, by means of $\Gamma$-convergence we perform a passage from nonlinear to linearized elasticity under variable subquadratic energy growth far from the energy well.

Abstract: We use our earlier Brownian-time framework to formulate and establish global uniqueness and local-in-time existence of the Burgers incarnation of the Kuramoto-Sivashinsky PDE on $\mathbb{R}_+\times\mathbb{R}^d$, in the class of time-continuous $L^{2p}$-valued solutions, $p\ge1$, for every $d<6p$. We assume neither space compactness, nor spatial coordinates dependence, nor smallness of initial data. The surprising discovery of the $6p$-th dimension bound, even for local solutions, is revealed by our approach and the Brownian-time kernel -- the Brownian average of an angled $d$-dimensional Schr\"odinger propagator -- at its heart. We use this kernel to give a systematic approach, for all dimensions simultaneously, including a novel formulation -- even in the well-known $d=1$ case -- of the KS equation. This yields the estimates leading to this article's conclusions. We achieve the stated results by fusing some of our earlier Brownian-time stochastic processes constructions and ideas -- encoded in the aforementioned kernel -- with analytic ones, including complex and harmonic analysis; by employing suitable $N$-ball approximations together with fixed point theory; and by an adaptation of the stochastic analytic stopping-time technique to our deterministic setting. Using a separate strategy, that is also built on our Brownian-time paradigm, we treat the global wellposedness of the multidimensional KS equation in a followup upcoming article. This work also serves as a template for another forthcoming article in which we prove similar results for the time-fractional Burgers equation in multidimensional space.

Abstract: This work studies the Cauchy problem for the energy-critical inhomogeneous Hartree equation with inverse square potential $$i\partial_t u-\mathcal K_\lambda u=\pm |x|^{-\tau}|u|^{p-2}(I_\alpha *|\cdot|^{-\tau}|u|^p)u, \quad \mathcal K_\lambda=-\Delta+\frac\lambda{|x|^2}$$ in the energy space $H_\lambda^1:=\{f\in L^2,\quad\sqrt{\mathcal{K}_\lambda}f\in L^2\}$. In this paper, we develop a well-posedness theory and investigate the blow-up of solutions in $H_\lambda^1$. Furthermore we present a dichotomy between energy bounded and non-global existence of solutions under the ground state threshold. To this end, we use Caffarelli-Kohn-Nirenberg weighted interpolation inequalities and some equivalent norms considering $\mathcal K_\lambda$, which make it possible to control the non-linearity involving the singularity $|x|^{-\tau}$ as well as the inverse square potential. The novelty here is the investigation of the energy critical regime which remains still open and the challenge is to deal with three technical problems: a non-local source term, an inhomogeneous singular term $|\cdot|^{-\tau}$, and the presence of an inverse square potential.

Abstract: In this paper, the relationships between Lie symmetry groups and fundamental solutions for a class of conformable time fractional partial differential equations (PDEs) with variable coefficients are investigated. Specifically, the group-invariant solutions to the considered equations are constructed applying symmetry group method and the corresponding fundamental solutions for these systems are established with the help of the above obtained group-invariant solutions and inverting Laplace transformation. In addition, the connections between fundamental solutions for two conformable time fractional systems are given by equivalence transformation. Furthermore, the conservation laws of these fractional systems are provided using new Noether theorem and obtained Lie algebras.

Abstract: We show the global existence of a weak solution for the Navier-Stokes equations for compressible fluids with slip boundary conditions of friction type.

Abstract: We study the $d$-dimensional ($d\geq2$) incompressible Oldroyd-B model with only stress tensor diffusion and without velocity dissipation as well as the damping mechanism on the stress tensor. Firstly, based upon some new observations on the model, we develope the pure energy argument (independent of spectral analysis) in general $L^p$ framework, and present a small initial data global existence and uniqueness of solutions to the model. Our results yield that the coupling and interaction of the velocity and the non-Newtonian stress actually enhances the regularity of the system. Later, by adding some additional $L^2$ type conditions on the low frequencies of the initial data $(u_0,\tau_0)$, %but without any more smallness restrictions, we obtain the optimal time-decay rates of the global solution $(u,\tau)$. Our result solves the problem proposed in Wang, Wu, Xu and Zhong \cite{Wang-Wu-Xu-Zhong} ({\it J. Funct. Anal.}, 282 (2022), 109332.).

Abstract: Motivated by the study of relativistic atoms, we consider the Hardy operator $(-\Delta)^{\alpha/2}-\kappa|x|^{-\alpha}$ acting on functions of the form $u(|x|) |x|^{\ell} Y_{\ell,m}(x/|x|)$ in $L^2(\mathbb{R}^d)$, when $\kappa\geq0$ and $\alpha\in(0,2]\cap(0,d+2\ell)$. We give a ground state representation of the corresponding form on the half-line (Theorem 1.5). For the proof we use subordinated Bessel heat kernels.

Abstract: We consider a $p$-fractional Choquard-type equation \[ (-\Delta)_p^s u+a|u|^{p-2}u=b(K\ast F(u))F'(u)+\varepsilon_g |u|^{p_g-2}u \quad\text{in $\mathbb{R}^N$}, \] where $0<s<1<p<N/s$, $p<p_g\leq p_s^*$, $(a,b,\varepsilon_g)\in (0,\infty)^3$, $K(x)= |x|^{-(N-\alpha)}$, $\alpha\in (0,N)$, and $F$ is a doubly critical nonlinearity in the sense of the Hardy-Littlewood-Sobolev inequality. It is noteworthy that the local nonlinearity may also have critical growth. Combining Brezis-Nirenberg's method with some new ideas, we obtain the ground state solutions via the mountain pass lemma and a generalized Lions-type theorem.

Abstract: Considered herein is a particular nonlinear dispersive stochastic system consisting of Dirac and Klein-Gordon equations. They are coupled by nonlinear terms due to the Yukawa interaction. We consider a case of homogeneous multiplicative noise that seems to be very natural from the perspective of the least action formalism. We are able to show existence and uniqueness of a corresponding Cauchy problem in Bourgain spaces. Moreover, the regarded model implies charge conservation, known for the deterministic analogue of the system, and this is used to prove a global existence result for suitable initial data.

Abstract: In this paper we will derive an integral equation which transform a three-dimensional acoustic transmission problem with \textit{variable} coefficients, non-zero absorption, and mixed boundary conditions to a non-local equation on the skeleton of the domain $\Omega\subset\mathbb{R}^{3}$, where ``skeleton'' stands for the union of the interfaces and boundaries of a Lipschitz partition of $\Omega$. To that end, we introduce and analyze abstract layer potentials as solutions of auxiliary coercive full space variational problems and derive jump conditions across domain interfaces. This allows us to formulate the non-local skeleton equation as a \textit{direct method} for the unknown Cauchy data of the original partial differential equation. We establish coercivity and continuity of the variational form of the skeleton equation without based on an auxiliary full space variational problem. Explicit knowledge of Green's functions is not required and our estimates are explicit in the complex wave number.

Abstract: The purpose of this work is to study the $3D$ energy-critical inhomogeneous generalized Hartree equation in $$ i\partial_tu+\Delta u+|x|^{-b}(I_\alpha\ast|\cdot|^{-b}|u|^{p})|u|^{p-2}u=0,\;\ x\in\mathbb{R}^3, $$ where $p=3+\alpha-2b$. We show global well-posedness and scattering below the ground state threshold with general initial data in $\dot{H}^1$. To this end, we exploit the decay of the nonlinearity, which together with the Kenig--Merle roadmap, allows us to treat the non-radial case as the radial case. In particular, we also show scattering for the classical generalized Hartree equation ($b=0$) assuming initial radial data. Moreover, in the defocusing case, we show scattering with non-radial data. At the end of the introduction, we discuss some open problems related to this equation.