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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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Abstract: We study the solvability of the second boundary value problem for a class of highly singular fourth order equations of Monge–Ampère type. They arise in the approximation of convex functionals subject to a convexity constraint using Abreu type equations. Both the Legendre transform and partial Legendre transform are used in our analysis. In two dimensions, we establish global solutions to the second boundary value problem for highly singular Abreu equations where the right hand sides are of q-Laplacian type for all \(q>1\) . We show that minimizers of variational problems with a convexity constraint in two dimensions that arise from the Rochet–Choné model in the monopolist’s problem in economics with q-power cost can be approximated in the uniform norm by solutions of the Abreu equation for a full range of q. PubDate: 2021-05-27

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Abstract: We prove that there are stationary solutions to the 2D incompressible free boundary Euler equations with two fluids, possibly with a small gravity constant, that feature a splash singularity. More precisely, in the solutions we construct the interface is a \(\mathcal {C}^{2,\alpha }\) smooth curve that intersects itself at one point, and the vorticity density on the interface is of class \(\mathcal {C}^\alpha \) . The proof consists in perturbing Crapper’s family of formal stationary solutions with one fluid, so the crux is to introduce a small but positive second-fluid density. To do so, we use a novel set of weighted estimates for self-intersecting interfaces that squeeze an incompressible fluid. These estimates will also be applied to interface evolution problems in a forthcoming paper. PubDate: 2021-04-10

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Abstract: We study stability properties of kinks for the (1+1)-dimensional nonlinear scalar field theory models $$\begin{aligned} \partial _t^2\phi -\partial _x^2\phi + W'(\phi ) = 0, \quad (t,x)\in \mathbb {R}\times \mathbb {R}. \end{aligned}$$ The orbital stability of kinks under general assumptions on the potential W is a consequence of energy arguments. Our main result is the derivation of a simple and explicit sufficient condition on the potential W for the asymptotic stability of a given kink. This condition applies to any static or moving kink, in particular no symmetry assumption is required. Last, motivated by the Physics literature, we present applications of the criterion to the \(P(\phi )_2\) theories and the double sine-Gordon theory. PubDate: 2021-04-08

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Abstract: In this paper, the first in a series, we study the deformed Hermitian–Yang–Mills (dHYM) equation from the variational point of view as an infinite dimensional GIT problem. The dHYM equation is mirror to the special Lagrangian equation, and our infinite dimensional GIT problem is mirror to Thomas’ GIT picture for special Lagrangians. This gives rise to infinite dimensional manifold \({\mathcal {H}}\) closely related to Solomon’s space of positive Lagrangians. In the hypercritical phase case we prove the existence of smooth approximate geodesics, and weak geodesics with \(C^{1,\alpha }\) regularity. This is accomplished by proving sharp with respect to scale estimates for the Lagrangian phase operator on collapsing manifolds with boundary. As an application of our techniques we give a simplified proof of Chen’s theorem on the existence of \(C^{1,\alpha }\) geodesics in the space of Kähler metrics. In two follow up papers, these results will be used to examine algebraic obstructions to the existence of solutions to dHYM [26] and special Lagrangians in Landau–Ginzburg models [27]. PubDate: 2021-04-08

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Abstract: This paper is devoted to the study of asymptotic behaviors of solutions to the one-dimensional defocusing semilinear wave equations. We prove that finite energy solution tends to zero in the pointwise sense, hence improving the averaged decay of Lindblad and Tao [4]. Moreover, for sufficiently localized data belonging to some weighted energy space, the solution decays in time with an inverse polynomial rate. This confirms a conjecture raised in the mentioned work. The results are based on new weighted vector fields as multipliers applied to regions bounded by light rays. The key observation for the first result is an integrated local energy decay for the potential energy, while the second result relies on a type of weighted Gagliardo-Nirenberg inequality. PubDate: 2021-04-05

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Abstract: In this paper we deal with the Cauchy problem for the incompressible Euler equations in the three-dimensional periodic setting. We prove non-uniqueness for an \(L^2\) -dense set of Hölder continuous initial data in the class of Hölder continuous admissible weak solutions for all exponents below the Onsager-critical 1/3. Along the way, and more importantly, we identify a natural condition on “blow-up” of the associated subsolution, which acts as the signature of the non-uniqueness mechanism. This improves previous results on non-uniqueness obtained in (Daneri in Comm. Math. Phys. 329(2):745–786, 2014; Daneri and Székelyhidi in Arch. Rat. Mech. Anal. 224: 471–514, 2017) and generalizes (Buckmaster et al. in Comm. Pure Appl. Math. 72(2):229–274, 2018). PubDate: 2021-04-04

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Abstract: We prove that the Cauchy problem for the Muskat equation is well-posed locally in time for any initial data in the critical space of Lipschitz functions with three-half derivative in \(L^2\) . Moreover, we prove that the solution exists globally in time under a smallness assumption. PubDate: 2021-04-03

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Abstract: We consider the SQG equation with dissipation given by a fractional Laplacian of order \(\alpha <\frac{1}{2}\) . We introduce a notion of suitable weak solution, which exists for every \(L^2\) initial datum, and we prove that for such solution the singular set is contained in a compact set in spacetime of Hausdorff dimension at most \(\frac{1}{2\alpha } \left( \frac{1+\alpha }{\alpha } (1-2\alpha ) + 2\right) \) . PubDate: 2021-03-26

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Abstract: Gajic–Warnick [8] have recently proposed a definition of scattering resonances based on Gevrey-2 regularity at infinity and introduced a new class of potentials for which resonances can be defined. We show that standard methods based on complex scaling apply to a larger class of potentials and provide a definition of resonances in wider angles. PubDate: 2021-03-22

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Abstract: We study patch solutions of a family of transport equations given by a parameter \(\alpha \) , \(0< \alpha <2\) , with the cases \(\alpha =0\) and \(\alpha =1\) corresponding to the Euler and the surface quasi-geostrophic equations respectively. In this paper, using several new cancellations, we provide the following new results. First, we prove local well-posedness for \(H^{2}\) patches in the half-space setting for \(0<\alpha < 1/3\) , allowing self-intersection with the fixed boundary. Furthermore, we are able to extend the range of \(\alpha \) for which finite time singularities have been shown in Kiselev et al. (Commun Pure Appl Math 70(7):1253–1315, 2017) and Kiselev et al. (Ann Math 3:909–948, 2016). Second, we establish that patches remain regular for \(0<\alpha <2\) as long as the arc-chord condition and the regularity of order \(C^{1+\delta }\) for \(\delta >\alpha /2\) are time integrable. This finite-time singularity criterion holds for lower regularity than the regularity shown in numerical simulations in Córdoba et al. (Proc Natl Acad Sci USA 102:5949–5952, 2005) and Scott and Dritschel (Phys Rev Lett 112:144505, 2014) for surface quasi-geostrophic patches, where the curvature of the contour blows up numerically. This is the first proof of a finite-time singularity criterion lower than or equal to the regularity in the numerics. Finally, we also improve results in Gancedo (Adv Math 217(6):2569–2598, 2008) and in Chae et al. (Commun Pure Appl Math 65(8):1037–1066, 2012), giving local existence for patches in \(H^{2}\) for \(0<\alpha < 1\) and in \(H^3\) for \(1<\alpha <2\) . PubDate: 2021-03-20