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In this paper, the authors prove global well-posedness of the massless Maxwell-Dirac equation in the Coulomb gauge on \\mathbb{R}^{1+d} (d\\geq 4) for data with small scale-critical Sobolev norm, as well as modified scattering of the solutions. Main components of the authors' proof are A) uncovering null structure of Maxwell-Dirac in the Coulomb gauge, and B) proving solvability of the underlying covariant Dirac equation. A key step for achieving both is to exploit (and justify) a deep analogy between Maxwell-Dirac and Maxwell-Klein-Gordon (for which an analogous result was proved earlier by Krieger-Sterbenz-Tataru, which says that the most difficult part of Maxwell-Dirac takes essentially the same form as Maxwell-Klein-Gordon.

In this paper, we show the existence of the first non trivial family of classical global solutions of the inviscid surface quasi-geostrophic equation.

In this paper, we consider the Cauchy problem for semilinear classical wave equations u tt - Δ u = | u | p S ( n ) μ ( | u | ) with the Strauss exponent p S ( n ) and a modulus of continuity μ = μ ( τ ) , which provides an additional regularity of nonlinearities in u = 0 comparing with the power nonlinearity | u | p S ( n ) . We obtain a sharp condition on μ as a threshold between global (in time) existence of small data radial solutions by deriving polynomial-logarithmic type weighted L t ∞ L r ∞ estimates, and blow-up of solutions in finite time even for small data by applying iteration methods with slicing procedure. These results imply a conjecture for the critical regularity of source nonlinearities for semilinear classical wave equations. We verify this conjecture in the 3d case.

Let M α be the spherical maximal operators of complex order α on R n . In this article we show that when n ≥ 2 , suppose ‖ M α f ‖ L p ( R n ) ≤ C ‖ f ‖ L p ( R n ) holds for some α and p ≥ 2 , then we must have that Re α ≥ max { 1 / p - ( n - 1 ) / 2 , - ( n - 1 ) / p } . In particular, when n = 2 , we prove that ‖ M α f ‖ L p ( R 2 ) ≤ C ‖ f ‖ L p ( R 2 ) if Re α > max { 1 / p - 1 / 2 , - 1 / p } , and consequently the range of α is sharp in the sense that the estimate fails for Re α < max { 1 / p - 1 / 2 , - 1 / p } .

In this paper we study Strichartz estimates for the half wave, the half Klein–Gordon and the Dirac Equations on compact manifolds without boundary, proving in particular for each of these flows local in time estimates both for the wave and Schrödinger admissible couples (in this latter case with an additional loss of regularity). The strategy for the proof is based on a refined version of the WKB approximation.

We study the stabilization and the well-posedness of solutions of the quintic wave equation with locally distributed damping. The novelty of this paper is that we deal with the difficulty that the main equation does not have good nonlinear structure amenable to a direct proof of a priori bounds and a desirable observability inequality. It is well known that observability inequalities play a critical role in characterizing the long time behaviour of solutions of evolution equations, which is the main goal of this study. In order to address this, we approximate weak solutions for regular solutions for which it is possible to obtain a priori bounds and prove the essential observability inequality. The treatment of these approximate solutions is still a challenging task and requires the use of Strichartz estimates and some microlocal analysis tools such as microlocal defect measures.

We study the motion of closed hypersurfaces with friction in central force fields. This kind of closed hypersurfaces can be considered as an electrically charged membrane with a constant charge per area in a radially symmetric potential. The parabolic flow case was first established by Schnürer & Smoczyk (J. Reine. Angew. Math. 500 (2002) 77–95). The hyperbolic flow without central force fields and friction was studied by LeFloch & Smoczyk (J. Math. Pures Appl. 90 (2008) 591–614), and a general case by Notz (Comm. Pure. Appl. Math. 66 (2013) 790–819), and a dissipative model by Shao (Arch. Ration. Mech. Anal. 243 (2022) 501–557). In this paper, we consider a general case of the motion of closed hypersurfaces with friction. It also models the motion of elastic membrane with external friction and internal friction. A quasi-linear degenerate dissipative hyperbolic system including the mean curvature operator is introduced to describe it. Our main result shows that closed hypersurfaces converges smoothly to a uniquely determined smooth immersion (e.g. sphere) if the initial shape of elastic membrane being a small perturbation of this immersion.

The article studies inverse problems of determining unknown coefficients in various semi-linear and quasi-linear wave equations given the knowledge of an associated source-to-solution map. We introduce a method to solve inverse problems for nonlinear equations using interaction of three waves that makes it possible to study the inverse problem in all globally hyperbolic spacetimes of the dimension $n+1\\geqslant 3$ and with partial data. We consider the case when the set $\\Omega _{\\mathrm{in}}$ , where the sources are supported, and the set $\\Omega _{\\mathrm{out}}$ , where the observations are made, are separated. As model problems we study both a quasi-linear equation and a semi-linear wave equation and show in each case that it is possible to uniquely recover the background metric up to the natural obstructions for uniqueness that is governed by finite speed of propagation for the wave equation and a gauge corresponding to change of coordinates. The proof consists of two independent components. In the geometric part of the article we introduce a novel geometrical object, the three-to-one scattering relation. We show that this relation determines uniquely the topological, differential and conformal structures of the Lorentzian manifold in a causal diamond set that is the intersection of the future of the point $p_{in}\\in \\Omega _{\\mathrm{in}}$ and the past of the point $p_{out}\\in \\Omega _{\\mathrm{out}}$ . In the analytic part of the article we study multiple-fold linearisation of the nonlinear wave equation using Gaussian beams. We show that the source-to-solution map, corresponding to sources in $\\Omega _{\\mathrm{in}}$ and observations in $\\Omega _{\\mathrm{out}}$ , determines the three-to-one scattering relation. The methods developed in the article do not require any assumptions on the conjugate or cut points.

We study the Euler-Bernoulli equations with time delay: where represents the time delay. We exhibit the blow-up behavior of solutions with both positive and nonpositive initial energy for the Euler-Bernoulli equations involving time delay.

In the previous paper (Ding et al. in J Differ Equ 385:183–253, 2024), for the 3D quasilinear wave equation - ( 1 + ( ∂ t ϕ ) p ) ∂ t 2 ϕ + Δ ϕ = 0 with short pulse initial data ( ϕ , ∂ t ϕ ) ( 1 , x ) = ( δ 2 - ε 0 ϕ 0 ( r - 1 δ , ω ) , δ 1 - ε 0 ϕ 1 ( r - 1 δ , ω ) ) , where p ∈ N , 0 < ε 0 < 1 , under the outgoing constraint condition ( ∂ t + ∂ r ) k ϕ ( 1 , x ) = O ( δ 2 - ε 0 - k max { 0 , 1 - ( 1 - ε 0 ) p } ) for k = 1 , 2 , the authors establish the global existence of smooth large solution ϕ when p > p c with p c = 1 1 - ε 0 . In the present paper, under the same outgoing constraint condition, when 1 ≤ p ≤ p c , we will show that the smooth solution ϕ may blow up and further the outgoing shock is formed in finite time.