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Abstract: Abstract This paper is motivated by the non-linear stability problem for the expanding region of Kerr de Sitter cosmologies in the context of Einstein’s equations with positive cosmological constant. We show that under dynamically realistic assumptions the conformal Weyl curvature of the spacetime decays towards future null infinity. More precisely we establish decay estimates for Weyl fields which are (i) uniform (with respect to a global time function) (ii) optimal (with respect to the rate) and (iii) consistent with a global existence proof (in terms of regularity). The proof relies on a geometric positivity property of compatible currents which is a manifestation of the global redshift effect capturing the expansion of the spacetime. PubDate: 2022-05-04

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Abstract: Abstract In this paper we study the deterministic and stochastic homogenisation of free-discontinuity functionals under linear growth and coercivity conditions. The main novelty of our deterministic result is that we work under very general assumptions on the integrands which, in particular, are not required to be periodic in the space variable. Combining this result with the pointwise Subadditive Ergodic Theorem by Akcoglu and Krengel, we prove a stochastic homogenisation result, in the case of stationary random integrands. In particular, we characterise the limit integrands in terms of asymptotic cell formulas, as in the classical case of periodic homogenisation. PubDate: 2022-04-07

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Abstract: Abstract We construct mixing solutions to the incompressible porous media equation starting from Muskat type data in the partially unstable regime. In particular, we consider bubble and turned type interfaces with Sobolev regularity. As a by-product, we prove the continuation of the evolution of IPM after the Rayleigh–Taylor and smoothness breakdown exhibited in (Castro et al. in Arch Ration Mech Anal 208(3):805–909, 2013, Castro et al. in Ann Math. (2) 175(2):909–948, 2012). At each time slice the space is split into three evolving domains: two non-mixing zones and a mixing zone which is localized in a neighborhood of the unstable region. In this way, we show the compatibility between the classical Muskat problem and the convex integration method. PubDate: 2022-04-07

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Abstract: Abstract In this paper, we prove that the translating solitons of the mean curvature flow in \(\mathbb {R}^4\) which arise as blow-up limit of embedded, mean convex mean curvature flow must have SO(2) symmetry. PubDate: 2022-03-25

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Abstract: Abstract We analyse an operator arising in the description of singular solutions to the two-dimensional Keller-Segel problem. It corresponds to the linearised operator in parabolic self-similar variables, close to a concentrated stationary state. This is a two-scale problem, with a vanishing thin transition zone near the origin. Via rigorous matched asymptotic expansions, we describe the eigenvalues and eigenfunctions precisely. We also show a stability result with respect to suitable perturbations, as well as a coercivity estimate for the non-radial part. These results are used as key arguments in a new rigorous proof of the existence and refined description of singular solutions for the Keller–Segel problem by the authors [8]. The present paper extends the result by Dejak, Lushnikov, Yu, Ovchinnikov and Sigal [11]. Two major difficulties arise in the analysis: this is a singular limit problem, and a degeneracy causes corrections not being polynomial but logarithmic with respect to the main parameter. PubDate: 2022-03-19

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Abstract: Abstract We consider the two-species Vlasov-Maxwell-Boltzmann (VMB) system with the scaling under which the moments of the fluctuations to the global Maxwellians formally converge to two-fluid incompressible Navier-Stokes-Fourier-Maxwell (NSFM) system with Ohm’s law. We prove the uniform estimates with respect to Knudsen number \(\varepsilon \) for the fluctuations by employing two types of micro-macro decompositions, and furthermore a hidden damping effect from the microscopic Ohm’s law. As consequences, the existence of the global-in-time classical solutions of VMB with all \(\varepsilon \in (0,1]\) is established. Moreover, the convergence of the fluctuations of the solutions of VMB to the classical solutions of NSFM with Ohm’s law is rigorously justified. This limit was justified in the recent breakthrough of Arsénio and Saint-Raymond (From the Vlasov-Maxwell-Boltzmann system to incompressible viscous electro-magneto-hydrodynamics. Vol. 1. EMS Monographs in Mathematics, European Mathematical Society (EMS), Zürich, 2019) from renormalized solutions of VMB to dissipative solutions of incompressible viscous electro-magneto-hydrodynamics under the suitable scalings. In this sense, our result provides a classical solution analogue of the corresponding limit in Arsénio and Saint-Raymond (From the Vlasov-Maxwell-Boltzmann system to incompressible viscous electro-magneto-hydrodynamics. Vol. 1. EMS Monographs in Mathematics, European Mathematical Society (EMS), Zürich, 2019) . PubDate: 2022-02-13 DOI: 10.1007/s40818-022-00117-6

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Abstract: Abstract We provide a self-contained proof of a trapped surface formation theorem, which simplifies the previous results by Christodoulou and by An–Luk. Our argument is based on a systematic approach for the scale-critical estimates in An–Luk and it connects Christodoulou’s short-pulse method and Klainerman–Rodnianski’s signature counting argument to the peeling properties previously studied in the small-data regime such as Klainerman–Nicolo. In particular this allows us to avoid elliptic estimates and geometric renormalizations, and gives us new technical improvements and simplifications. PubDate: 2022-01-16 DOI: 10.1007/s40818-021-00114-1

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Abstract: Abstract This work is concerned with the mathematical analysis of the bulk rheology of random suspensions of rigid particles settling under gravity in viscous fluids. Each particle generates a fluid flow that in turn acts on other particles and hinders their settling. In an equilibrium perspective, for a given ensemble of particle positions, we analyze both the associated mean settling speed and the velocity fluctuations of individual particles. In the 1970s, Batchelor gave a proper definition of the mean settling speed, a 60-year-old open problem in physics, based on the appropriate renormalization of long-range particle contributions. In the 1980s, a celebrated formal calculation by Caflisch and Luke suggested that velocity fluctuations in dimension \(d=3\) should diverge with the size of the sedimentation tank, contradicting both intuition and experimental observations. The role of long-range self-organization of suspended particles in form of hyperuniformity was later put forward to explain additional screening of this divergence in steady-state observations. In the present contribution, we develop the first rigorous theory that allows to justify all these formal calculations of the physics literature. PubDate: 2022-01-11 DOI: 10.1007/s40818-021-00115-0

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Abstract: Abstract We present a systematic approach to regularity theory of the multi-dimensional Euler alignment systems with topological diffusion introduced in [35]. While these systems exhibit flocking behavior emerging from purely local communication, bearing direct relevance to empirical field studies, global and even local well-posedness has proved to be a major challenge in multi-dimensional settings due to the presence of topological effects. In this paper we reveal two important classes of global smooth solutions—parallel shear flocks with incompressible velocity and stationary density profile, and nearly aligned flocks with close to constant velocity field but arbitrary density distribution. Existence of such classes is established via an efficient continuation criterion requiring control only on the Lipschitz norm of state quantities, which makes it accessible to the applications of fractional parabolic theory. The criterion presents a major improvement over the existing result of [28], and is proved with the use of quartic paraproduct estimates. PubDate: 2021-12-22 DOI: 10.1007/s40818-021-00116-z

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Abstract: Abstract In this paper, we study the linear stability properties of perturbations around the homogeneous Couette flow for a 2D isentropic compressible fluid in the domain \(\mathbb {T}\times \mathbb {R}\) . In the inviscid case there is a generic Lyapunov type instability for the density and the irrotational component of the velocity field. More precisely, we prove that their \(L^2\) norm grows as \(t^{1/2}\) and this confirms previous observations in the physics literature. On the contrary, the solenoidal component of the velocity field experiences inviscid damping, namely it decays to zero even in the absence of viscosity. For a viscous compressible fluid, we show that the perturbations may have a transient growth of order \(\nu ^{-1/6}\) (with \(\nu ^{-1}\) being proportional to the Reynolds number) on a time-scale \(\nu ^{-1/3}\) , after which it decays exponentially fast. This phenomenon is also called enhanced dissipation and our result appears to be the first to detect this mechanism for a compressible flow, where an exponential decay for the density is not a priori trivial given the absence of dissipation in the continuity equation. PubDate: 2021-10-19 DOI: 10.1007/s40818-021-00112-3

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Abstract: Abstract We consider the stability of transonic contact discontinuity for the two-dimensional steady compressible Euler flows in a finitely long nozzle. This is the first work on the mixed-type problem of transonic flows across a contact discontinuity as a free boundary in nozzles. We start with the Euler-Lagrangian transformation to straighten the contact discontinuity in the new coordinates. However, the upper nozzle wall in the subsonic region depending on the mass flux becomes a free boundary after the transformation. Then we develop new ideas and techniques to solve the free-boundary problem in three steps: (1) we fix the free boundary and generate a new iteration scheme to solve the corresponding fixed boundary value problem of the hyperbolic-elliptic mixed type by building some powerful estimates for both the first-order hyperbolic equation and a second-order nonlinear elliptic equation in a Lipschitz domain; (2) we update the new free boundary by constructing a mapping that has a fixed point; (3) we establish via the inverse Lagrangian coordinate transformation that the original free interface problem admits a unique piecewise smooth transonic solution near the background state, which consists of a smooth subsonic flow and a smooth supersonic flow with a contact discontinuity. PubDate: 2021-09-23 DOI: 10.1007/s40818-021-00113-2

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Abstract: Abstract A rigorous derivation of the incompressible Euler equations with the no-penetration boundary condition from the Boltzmann equation with the diffuse reflection boundary condition has been a challenging open problem. We settle this open question in the affirmative when the initial data of fluid are well-prepared in a real analytic space, in 3D half space. As a key of this advance, we capture the Navier-Stokes equations of $$\begin{aligned} \textit{viscosity} \sim \frac{\textit{Knudsen number}}{\textit{Mach number}} \end{aligned}$$ satisfying the no-slip boundary condition, as an intermediary approximation of the Euler equations through a new Hilbert-type expansion of the Boltzmann equation with the diffuse reflection boundary condition. Aiming to justify the approximation we establish a novel quantitative \(L^p\) - \(L^\infty \) estimate of the Boltzmann perturbation around a local Maxwellian of such viscous approximation, along with the commutator estimates and the integrability gain of the hydrodynamic part in various spaces; we also establish direct estimates of the Navier-Stokes equations in higher regularity with the aid of the initial-boundary and boundary layer weights using a recent Green’s function approach. The incompressible Euler limit follows as a byproduct of our framework. PubDate: 2021-08-27 DOI: 10.1007/s40818-021-00108-z

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Abstract: Abstract We obtain a general sufficient condition on the geometry of possibly singular planar domains that guarantees global uniqueness for any weak solution to the Euler equations on them whose vorticity is bounded and initially constant near the boundary. While similar existing results require domains that are \(C^{1,1}\) except at finitely many convex corners, our condition involves much less domain smoothness, being only slightly more restrictive than the exclusion of corners with angles greater than \(\pi \) . In particular, it is satisfied by all convex domains. The main ingredient in our approach is showing that constancy of the vorticity near the boundary is preserved for all time because Euler particle trajectories on these domains, even for general bounded solutions, cannot reach the boundary in finite time. We then use this to show that no vorticity can be created by the boundary of such possibly singular domains for general bounded solutions. We also show that our condition is essentially sharp in this sense by constructing domains that come arbitrarily close to satisfying it, and on which particle trajectories can reach the boundary in finite time. In addition, when the condition is satisfied, we find sharp bounds on the asymptotic rate of the fastest possible approach of particle trajectories to the boundary. PubDate: 2021-08-25 DOI: 10.1007/s40818-021-00107-0

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Abstract: Abstract We prove global well-posedness of the fifth-order Korteweg-de Vries equation on the real line for initial data in \(H^{-1}(\mathbb {R})\) . Global well-posedness in \(L^2({\mathbb {R}})\) was shown previously in [8] using the method of commuting flows. Since this method is insensitive to the ambient geometry, it cannot go beyond the sharp \( L^2\) threshold for the torus demonstrated in [3]. To prove our result, we introduce a new strategy that integrates dispersive effects into the method of commuting flows. PubDate: 2021-08-25 DOI: 10.1007/s40818-021-00111-4

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Abstract: Abstract In this paper we investigate maximal \(L^q\) -regularity for time-dependent viscous Hamilton–Jacobi equations with unbounded right-hand side and superlinear growth in the gradient. Our approach is based on the interplay between new integral and Hölder estimates, interpolation inequalities, and parabolic regularity for linear equations. These estimates are obtained via a duality method à la Evans. This sheds new light on the parabolic counterpart of a conjecture by P.-L. Lions on maximal regularity for Hamilton–Jacobi equations, recently addressed in the stationary framework by the authors. Finally, applications to the existence problem of classical solutions to Mean Field Games systems with unbounded local couplings are provided. PubDate: 2021-08-22 DOI: 10.1007/s40818-021-00109-y

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Abstract: Abstract We revisit the proof of Landau damping near stable homogenous equilibria of Vlasov–Poisson systems with screened interactions in the whole space \(\mathbb {R}^d\) (for \(d\ge 3\) ) that was first established by Bedrossian, Masmoudi and Mouhot in [5]. Our proof follows a Lagrangian approach and relies on precise pointwise in time dispersive estimates in the physical space for the linearized problem that should be of independent interest. This allows to cut down the smoothness of the initial data required in [5] (roughly, we only need Lipschitz regularity). Moreover, the time decay estimates we prove are essentially sharp, being the same as those for free transport, up to a logarithmic correction. PubDate: 2021-08-20 DOI: 10.1007/s40818-021-00110-5

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Abstract: Abstract We extend the variational approach to regularity for optimal transport maps initiated by Goldman and the first author to the case of general cost functions. Our main result is an \(\epsilon \) -regularity result for optimal transport maps between Hölder continuous densities slightly more quantitative than the result by De Philippis–Figalli. One of the new contributions is the use of almost-minimality: if the cost is quantitatively close to the Euclidean cost function, a minimizer for the optimal transport problem with general cost is an almost-minimizer for the one with quadratic cost. This further highlights the connection between our variational approach and De Giorgi’s strategy for \(\epsilon \) -regularity of minimal surfaces. PubDate: 2021-08-18 DOI: 10.1007/s40818-021-00106-1

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Abstract: Abstract In this paper we give a new and simplified proof of the theorem on selection of standing waves for small energy solutions of the nonlinear Schrödinger equations (NLS) that we gave in [6]. We consider a NLS with a Schrödinger operator with several eigenvalues, with corresponding families of small standing waves, and we show that any small energy solution converges to the orbit of a time periodic solution plus a scattering term. The novel idea is to consider the “refined profile”, a quasi–periodic function in time which almost solves the NLS and encodes the discrete modes of a solution. The refined profile, obtained by elementary means, gives us directly an optimal coordinate system, avoiding the normal form arguments in [6], giving us also a better understanding of the Fermi Golden Rule. PubDate: 2021-07-20 DOI: 10.1007/s40818-021-00105-2

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Abstract: Abstract We consider the spatially inhomogeneous non-cutoff Boltzmann equation with moderately soft potentials and any singularity parameter \(s\in (0,1)\) , i.e. with \(\gamma +2s\in (0,2)\) on the whole space \({\mathbb {R}}^3\) . We prove that if the initial data \(f_{{{\,\mathrm{in}\,}}}\) are close to the vacuum solution \(f_{\text {vac}}=0\) in an appropriate weighted norm then the solution f remains regular globally in time and approaches a solution to a linear transport equation. Our proof uses \(L^2\) estimates and we prove a multitude of new estimates involving the Boltzmann kernel without angular cut-off. Moreover, we rely on various previous works including those of Gressman–Strain, Henderson–Snelson–Tarfulea and Silvestre. From the point of view of the long time behavior we treat the Boltzmann collisional operator perturbatively. Thus an important challenge of this problem is to exploit the dispersive properties of the transport operator to prove integrable time decay of the collisional operator. This requires the most care and to successfully overcome this difficulty we draw inspiration from Luk’s work [Stability of vacuum for the Landau equation with moderately soft potentials, Annals of PDE (2019) 5:11] and that of Smulevici [Small data solutions of the Vlasov-Poisson system and the vector field method, Ann. PDE, 2(2):Art. 11, 55, 2016]. In particular, to get at least integrable time decay we need to consolidate the decay coming from the space-time weights and the decay coming from commuting vector fields. PubDate: 2021-06-05 DOI: 10.1007/s40818-021-00103-4

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Abstract: Abstract We study the stability of smooth and peaked solitary waves to the modified Camassa-Holm equation. This quasilinear equation with cubic nonlinearity is completely integrable and arises as a model for the unidirectional propagation of shallow water waves. Based on the phase portrait analysis, we demonstrate the existence of unique localized smooth solcontra1itary-wave solution with certain range of the linear dispersive parameter. We then show orbital stability of the smooth solitary-wave solution under small disturbances by means of variational methods, considering a minimization problem with an appropriate constraint. Using the variational approach with suitable conservation laws, we also establish the orbital stability of peakons in the Sobolev space \( H^1 \cap W^{1, 4} \) without the assumption on the positive momentum density initially. Finally we demonstrate spectral stability of such smooth solitary waves using refined spectral analysis of the linear operator corresponding to the second-order variational derivative of the local Hamiltonian. PubDate: 2021-06-05 DOI: 10.1007/s40818-021-00104-3