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This article is devoted to the Relativistic Vlasov-Maxwell system in space dimension three. We prove the local smooth solvability for weak topologies (and its long time version for small data). This result is derived from a representation formula decoding how the momentum spreads, and showing that the domain of influence in momentum is controlled by mild information. We do so by developing a Radon Fourier analysis on the RVM system, leading to the study of a class of singular weighted integrals. In the end, we implement our method to construct smooth solutions to the RVM system in the regime of dense, hot and strongly magnetized plasmas. This is done by investigating the stability properties near a class of approximate solutions., Comment: 36 pages

We examine the stability of vortex-like configuration of magnetization in magnetic materials, so-called the magnetic skyrmion. These correspond to critical points of the Landau-Lifshitz energy with the Dzyaloshinskii-Moriya (DM) interactions. In an earlier work of the D\"oring and Melcher, it is known that the skyrmion is a ground state when the coefficient of the DM term is small. In this paper, we prove that there is an explicit critical value of the coefficient above which the skyrmion is unstable, while stable below this threshold. Moreover, we show that in the unstable regime, the infimum of energy is not bounded below, by giving an explicit counterexample with a sort of helical configuration. This mathematically explains the occurrence of phase transition observed in some experiments., Comment: 12 pages

We obtain probabilistic local well-posedness in quasilinear regimes for the Schr\"odinger half-wave equation with a cubic nonlinearity. We need to use a refined ansatz because of the lack of probabilistic smoothing in the Picard's iterations, which is due to the high-low-low frequency interactions. The proof is an adaptation of the method of Bringmann on the derivative nonlinear wave equation to Schr\"odinger-type equations. In addition, we discuss ill-posedness results for this equation.

The primitive equations (PEs) model large scale dynamics of the oceans and the atmosphere. While it is by now well-known that the three-dimensional viscous PEs is globally well-posed in Sobolev spaces, and that there are solutions to the inviscid PEs (also called the hydrostatic Euler equations) that develop singularities in finite time, the qualitative description of the blowup still remains undiscovered. In this paper, we provide a full description of two blowup mechanisms, for a reduced PDE that is satisfied by a class of particular solutions to the PEs. In the first one a shock forms, and pressure effects are subleading, but in a critical way: they localize the singularity closer and closer to the boundary near the blow-up time (with a logarithmic in time law). This first mechanism involves a smooth blow-up profile and is stable among smooth enough solutions. In the second one the pressure effects are fully negligible; this dynamics involves a two-parameters family of non-smooth profiles, and is stable only by smoother perturbations., Comment: 29 pages

In this paper, we consider minimization problems related to the combined power-type nonlinear scalar field equations involving the Sobolev critical exponent in three space dimensions. In four and higher space dimensions, it is known that for any frequency and any power of the subcritical nonlinearity, there exists a ground state. In contrast to those cases, when the space dimension is three and the subcritical power is three or less, we can show that there exists a threshold frequency, above which no ground state exists, and below which the ground state exists. Furthermore, we prove the difference between two typical variational problems used to characterize the ground states.

This article is devoted to the kinetic description in phase space of magnetically confined plasmas. It addresses the problem of stability near equilibria of the Relativistic Vlasov Maxwell system. We work under the Glassey-Strauss compactly supported momentum assumption on the density function $f(t,\cdot)$. Magnetically confined plasmas are characterized by the presence of a strong external magnetic field $ x \mapsto \epsilon^{-1} \mathbf{B}_e(x)$, where $\epsilon$ is a small parameter related to the inverse gyrofrequency of electrons. In comparison, the self consistent internal electromagnetic fields $(E,B) $ are supposed to be small. In the non-magnetized setting, local $ C^1 $-solutions do exist but do not exclude the possibility of blow up in finite time for large data. Consequently, in the strongly magnetized case, since $ \epsilon^{-1} $ is large, standard results predict that the lifetime $T_\epsilon$ of solutions may shrink to zero when $ \epsilon $ goes to $ 0 $. In this article, through field straightening, and a time averaging procedure we show a uniform lower bound ($0

In this paper, we study the bifurcation problem from a line soliton for a stationary nonlinear Schr\"{o}dinger equation on the product space $\mathbb{R} \times \mathbb{T}$. We extend earlier results to a larger class of the nonlinearity in the equation. The salient point of our analysis relies on a lower bound of solution to the ``auxiliary equation'' and then on the application of the Crandall-Rabinowitz argument

In this paper, we study the transverse stability of the line Schr\"{o}dinger soliton under a full wave guide Schr\"{o}dinger flow on a cylindrical domain $\mathbb R\times\mathbb T$. When the nonlinearity is of power type $|\psi|^{p-1}\psi$ with $p>1$, we show that there exists a critical frequency $\omega_{p} >0$ such that the line standing wave is stable for $0<\omega < \omega_{p}$ and unstable for $\omega > \omega_{p}$. Furthermore, we characterize the ground state of the wave guide Schr\"{o}dinger equation. More precisely, we prove that there exists $\omega_{*} \in (0, \omega_{p}]$ such that the ground states coincide with the line standing waves for $\omega \in (0, \omega_{*}]$ and are different from the line standing waves for $\omega \in (\omega_{*}, \infty)$., Comment: To appear in JDDE

We characterize the fate of the solutions of Hill's type lunar problem using the ideas of ground states from PDE. In particular, the relative equilibrium will be defined as the ground state, which satisfies some crucial energetic variational properties in our analysis. We study the dynamics of the solutions below, at, and (slightly) above the ground state energy threshold., Comment: 36 pages, 7 figures

We study the effect of the rotation on the life-span of solutions to the $3D$ hydrostatic Euler equations with rotation and the inviscid Primitive equations (PEs) on the torus. The space of analytic functions appears to be the natural space to study the initial value problem for the inviscid PEs with general initial data, as they have been recently shown to exhibit Kelvin-Helmholtz type instability. First, for a short interval of time that is independent of the rate of rotation $|\Omega|$, we establish the local well-posedness of the inviscid PEs in the space of analytic functions. In addition, thanks to a fine analysis of the barotropic and baroclinic modes decomposition, we establish two results about the long time existence of solutions. (i) Independently of $|\Omega|$, we show that the life-span of the solution tends to infinity as the analytic norm of the initial baroclinic mode goes to zero. Moreover, we show in this case that the solution of the $3D$ inviscid PEs converges to the solution of the limit system, which is governed by the $2D$ Euler equations. (ii) We show that the life-span of the solution can be prolonged unboundedly with $|\Omega|\rightarrow \infty$, which is the main result of this paper. This is established for "well-prepared" initial data, namely, when only the Sobolev norm (but not the analytic norm) of the baroclinic mode is small enough, depending on $|\Omega|$. Furthermore, for large $|\Omega|$ and "well-prepared" initial data, we show that the solution to the $3D$ inviscid PEs is approximated by the solution to a simple limit resonant system with the same initial data.

Large scale dynamics of the oceans and the atmosphere are governed by the primitive equations (PEs). It is well-known that the three-dimensional viscous PEs is globally well-posed in Sobolev spaces. On the other hand, the inviscid PEs without rotation is known to be ill-posed in Sobolev spaces, and its smooth solutions can form singularity in finite time. In this paper, we extend the above results in the presence of rotation. First, we construct finite-time blowup solutions to the inviscid PEs with rotation, and establish that the inviscid PEs with rotation is ill-posed in Sobolev spaces in the sense that its perturbation around a certain steady state background flow is both linearly and nonlinearly ill-posed in Sobolev spaces. Its linear instability is of the Kelvin-Helmholtz type similar to the one appears in the context of vortex sheets problem. This implies that the inviscid PEs is also linearly ill-posed in Gevrey class of order $s > 1$, and suggests that a suitable space for the well-posedness is Gevrey class of order $s = 1$, which is exactly the space of analytic functions., Comment: 15 pages

We consider the planar Hill's lunar problem with a homogeneous gravitational potential. The investigation of the system is twofold. First, the starting conditions of the trajectories are classified into three classes, that is bounded, escaping, and collisional. Second, we study the no-return property of the Lagrange point $L_2$ and we observe that the escaping trajectories are scattered exponentially. Moreover, it is seen that in the supercritical case, with $\alpha \geq 2$, the basin boundaries are smooth. On the other hand, in the subcritical case, with $\alpha < 2$ the boundaries between the different types of basins exhibit fractal properties., Comment: 12 pages, 10 figures

We investigate the existence of ground state solutions of a Gross-Pitaevskii equation modeling the dynamics of pumped Bose Einstein condensates (BEC). The main interest in such BEC comes from its important nature as macroscopic quantum system, constituting an excellent alternative to the classical condensates which are hard to realize because of the very low temperature required. Nevertheless, the Gross Pitaevskii equation governing the new condensates presents some mathematical challenges due to the presence of the pumping and damping terms. Following a self-contained approach, we prove the existence of ground state solutions of this equation under suitable assumptions: This is equivalent to say that condensation occurs in these situations. We also solve the Cauchy problem of the nonlinear Schroedinger equation and prove some corresponding laws., Comment: 25 pages

We construct a singular solution of a stationary nonlinear Schr\"{o}dinger equation on $\mathbb{R}^2$ with square-exponential nonlinearity having linear behavior around zero. In view of Trudinger-Moser inequality, this type of nonlinearity has an energy-critical growth. We use this singular solution to prove non-uniqueness of strong solutions for the Cauchy problem of the corresponding semilinear heat equation. The proof relies on explicit computation showing a regularizing effect of the heat equation in an appropriate functional space.

We study existence of maximizer for the Trudinger-Moser inequality with general nonlinearity of the critical growth on $R^2$, as well as on the disk. We derive a very sharp threshold nonlinearity between the existence and the non-existence in each case, in asymptotic expansions with respect to growth and decay of the function. The expansions are explicit, using Ap\'ery's constant. We also obtain an asymptotic expansion for the exponential radial Sobolev inequality on $R^2$., Comment: 41 pages

We use the idea of ground states and excited states in nonlinear dispersive equations (e.g. Klein-Gordon and Schr\"odinger equations) to characterize solutions in the N-body problem with strong force under some energy constraints. Indeed, relative equilibria of the N-body problem play a similar role as solitons in PDE. We introduce the ground state and excited energy for the N-body problem. {We are able to give a conditional dichotomy of the global existence and singularity below the excited energy in Theorem \ref{thm:dichotomy}, the proof of which seems original and simple. This dichotomy is given by the sign of a threshold function $K_\omega$}. The characterization for the two-body problem in this new perspective is non-conditional and it resembles the results in PDE nicely. For $N\geq3$, we will give some refinements of the characterization, in particular, we examine the situation where there are infinitely transitions for the sign of $K_\omega$., Comment: 38 pages, 13 figures

We consider combined power-type nonlinear scalar field equations with the Sobolev critical exponent. In \cite{AIKN3}, it was shown that if the frequency parameter is sufficiently small, then the positive ground state is nondegenerate and linearly unstable, together with an application to a study of global dynamics for nonlinear Schr\"odinger equations. In this paper, we prove the nondegeneracy and linear instability of the ground state frequency for sufficiently large frequency parameters. Moreover, we show that the derivative of the mass of ground state with respect to the frequency is negative.

In this paper, we study the solitary wave and the Cauchy problem for Half-wave-Schr\"{o}dinger equations in the plane. First, we show the existence and orbital stability of the ground states. Secondly, we prove that traveling waves exist and converge to zero as the velocity tends to $1$. Finally, we solve the Cauchy problem for initial data in $L^{2}_{x}H^{s}_{y}(\mathbb{R}^{2})$, with $s>\frac{1}{2}$.

We consider the two dimensional unsteady Prandtl system. For a special class of outer Euler flows and solutions of the Prandtl system, the trace of the tangential derivative of the tangential velocity along the transversal axis solves a closed one dimensional equation. First, we give a precise description of singular solutions for this reduced problem. A stable blow-up pattern is found, in which the blow-up point is ejected to infinity in finite time, and the solutions form a plateau with growing length. Second, in the case where, for a general analytic solution, this trace of the derivative on the axis follows the stable blow-up pattern, we show persistence of analyticity around the axis up to the blow-up time, and establish a universal lower bound of $(T-t)^{7/4}$ for its radius of analyticity., Comment: 79 pages, second version adds a new result proving a general quantitative persistence of analyticity in a two-dimensional neighbourhood of the vertical axis

The study of the uniqueness and nondegeneracy of ground state solutions to semilinear elliptic equations is of great importance because of the resulting energy landscape and its implications for the various dynamics. In [AIKN3], semilinear elliptic equations with combined power-type nonlinearities involving the Sobolev critical exponent are studied. There, it is shown that if the dimension is four or higher, and the frequency is sufficiently small, then the positive radial ground state is unique and nondegenerate. In this paper, we extend these results to the case of high frequencies when the dimension is five and higher. After suitably rescaling the equation, we demonstrate that the main behavior of the solutions is given by the Sobolev critical part for which the ground states are explicit, and their degeneracy is well characterized. Our result is a key step towards the study of the different dynamics of solutions of the corresponding nonlinear Schr\"odinger and Klein-Gordon equations with energies above the energy of the ground state. Our restriction on the dimension is mainly due to the existence of resonances in dimension three and four., Comment: Revised version. See the first version for a proof of Proposition B.1. To appear in Calculus of Variations and Partial Differential Equations

We study the stability of the Kolmogorov flows which are stationary solutions to the two-dimensional Navier-Stokes equations in the presence of the shear external force. We establish the linear stability estimate when the viscosity coefficient $\nu$ is sufficiently small, where the enhanced dissipation is rigorously verified in the time scale $O(\nu^{-\frac12})$ for solutions to the linearized problem, which has been numerically conjectured and is much shorter than the usual viscous time scale $O(\nu^{-1})$. Our approach is based on the detailed analysis for the resolvent problem. We also provide the abstract framework which is applicable to the resolvent estimate for the Kolmogorov flows., Comment: 69 pages, to appear in Annals of PDEs

Consider the thin-film equation $h_t + \left(h h_{yyy}\right)_y = 0$ with a zero contact angle at the free boundary, that is, at the triple junction where liquid, gas, and solid meet. Previous results on stability and well-posedness of this equation have focused on perturbations of equilibrium-stationary or self-similar profiles, the latter eventually wetting the whole surface. These solutions have their counterparts for the second-order porous-medium equation $h_t - (h^m)_{yy} = 0$, where $m > 1$ is a free parameter. Both porous-medium and thin-film equation degenerate as $h \searrow 0$, but the porous-medium equation additionally fulfills a comparison principle while the thin-film equation does not. In this note, we consider traveling waves $h = \frac V 6 x^3 + \nu x^2$ for $x \ge 0$, where $x = y-V t$ and $V, \nu \ge 0$ are free parameters. These traveling waves are receding and therefore describe de-wetting, a phenomenon genuinely linked to the fourth-order nature of the thin-film equation and not encountered in the porous-medium case as it violates the comparison principle. The linear stability analysis leads to a linear fourth-order degenerate-parabolic operator for which we prove maximal-regularity estimates to arbitrary orders of the expansion in $x$ in a right-neighborhood of the contact line $x = 0$. This leads to a well-posedness and stability result for the corresponding nonlinear equation. As the linearized evolution has different scaling as $x \searrow 0$ and $x \to \infty$, the analysis is more intricate than in related previous works. We anticipate that our approach is a natural step towards investigating other situations in which the comparison principle is violated, such as droplet rupture., Comment: 54 pages, revised version, minor changes and corrections, changed citation style to numbers

We consider the energy supercritical wave maps from $\mathbb{R}^d$ into the $d$-sphere $\mathbb{S}^d$ with $d \geq 7$. Under an additional assumption of 1-corotational symmetry, the problem reduces to the one dimensional semilinear wave equation $$\partial_t^2 u = \partial^2_r u + \frac{(d-1)}{r}\partial_r u - \frac{(d-1)}{2r^2}\sin(2u).$$ We construct for this equation a family of $\mathcal{C}^{\infty}$ solutions which blow up in finite time via concentration of the universal profile $$u(r,t) \sim Q\left(\frac{r}{\lambda(t)}\right),$$ where $Q$ is the stationary solution of the equation and the speed is given by the quantized rates $$\lambda(t) \sim c_u(T-t)^\frac{\ell}{\gamma}, \quad \ell \in \mathbb{N}^*, \;\; \ell > \gamma = \gamma(d) \in (1,2].$$ The construction relies on two arguments: the reduction of the problem to a finite-dimensional one thanks to a robust universal energy method and modulation techniques developed by Merle, Rapha\"el and Rodnianski for the energy supercritical nonlinear Schr\"odinger equation, then we proceed by contradiction to solve the finite-dimensional problem and conclude using the Brouwer fixed point theorem., Comment: We added Remark 1.7 to comment about the continuum and discrete quantization of blowup rates. We also added references and corrected many typos. arXiv admin note: text overlap with arXiv:1611.08877

We consider the energy supercritical harmonic heat flow from $\mathbb{R}^d$ into the $d$-sphere $\mathbb{S}^d$ with $d \geq 7$. Under an additional assumption of 1-corotational symmetry, the problem reduces to the one dimensional semilinear heat equation $$\partial_t u = \partial^2_r u + \frac{(d-1)}{r}\partial_r u - \frac{(d-1)}{2r^2}\sin(2u).$$ We construct for this equation a family of $\mathcal{C}^{\infty}$ solutions which blow up in finite time via concentration of the universal profile $$u(r,t) \sim Q\left(\frac{r}{\lambda(t)}\right),$$ where $Q$ is the stationary solution of the equation and the speed is given by the quantized rates $$\lambda(t) \sim c_u(T-t)^\frac{\ell}{\gamma}, \quad \ell \in \mathbb{N}^*, \;\; 2\ell > \gamma = \gamma(d) \in (1,2].$$ The construction relies on two arguments: the reduction of the problem to a finite-dimensional one thanks to a robust universal energy method and modulation techniques developed by Merle, Rapha\"el and Rodnianski [Camb. Jour. Math, 3(4):439-617, 2015] for the energy supercritical nonlinear Schr\"odinger equation and by Rapha\"el and Schweyer [Anal. PDE, 7(8):1713-1805, 2014] for the energy critical harmonic heat flow, then we proceed by contradiction to solve the finite-dimensional problem and conclude using the Brouwer fixed point theorem. Moreover, our constructed solutions are in fact $(\ell - 1)$ codimension stable under perturbations of the initial data. As a consequence, the case $\ell = 1$ corresponds to a stable type II blowup regime., Comment: 68 pages

For the 3D Navier-Stokes-Maxwell problem on the whole space and in the presence of external time-periodic forces, first we study the existence of time-periodic small solutions, and then we prove their asymptotic stability. We use new type of spaces to account for averaged decay in time., Comment: 40 pages

We consider the nonlinear Schrodinger equations with combined type local interactions with energy critical growth, and we study the solutions slightly above the ground state threshold at low frequencies, so that we obtain a so called nine set theory developed by Nakanishi and Schlag., Comment: To appear in Memoirs of AMS, 123 pages, 2 figures

We ask whether Hamiltonian vector fields defined on spaces of constant Gaussian curvature $\kappa$ (spheres, for $\kappa>0$, and hyperbolic spheres, for $\kappa<0$), pass continuously through the value $\kappa=0$ if the potential functions $U_\kappa, \kappa\in\mathbb R$, that define them satisfy the property $\lim_{\kappa\to 0}U_\kappa=U_0$, where $U_0$ corresponds to the Euclidean case. We prove that the answer to this question is positive, both in the 2- and 3-dimensional cases, which are of physical interest, and then apply our conclusions to the gravitational $N$-body problem., Comment: 12 pages

We obtain a natural extension of the Vlasov-Poisson system for stellar dynamics to spaces of constant Gaussian curvature $\kappa\ne 0$: the unit sphere $\mathbb S^2$, for $\kappa>0$, and the unit hyperbolic sphere $\mathbb H^2$, for $\kappa<0$. These equations can be easily generalized to higher dimensions. When the particles move on a geodesic, the system reduces to a 1-dimensional problem that is more singular than the classical analogue of the Vlasov-Poisson system. In the analysis of this reduced model, we study the well-posedness of the problem and derive Penrose-type conditions for linear stability around homogeneous solutions in the sense of Landau damping., Comment: 34 pages, 2 figures

This article resolves some errors in the paper "Scattering threshold for the focusing nonlinear Klein-Gordon equation", Analysis & PDE 4 (2011) no. 3, 405-460. The errors are in the energy-critical cases in two and higher dimensions., Comment: 11 pages

In earlier works, we have shown the uniform decay of the local energy of the damped wave equation in exterior domain when the damper is spatially localized near captive rays. In order to have uniform decay of the total energy, the damper has also to act at space infinity. In this work, we establish uniform decay of both the local and global energies. The rates of decay turns out to be the same as those for the heat equation, which shows that an effective damper at space infinity strengthens the parabolic structure in the equation., Comment: 17 pages. To appear in Journal of Differential Equations

We study a two fluid system which models the motion of a charged fluid with Rayleigh friction, and in the presence of an electro-magnetic field satisfying Maxwell's equations. We study the well-posdness of the system in both space dimensions two and three. Regardless of the size of the initial data, we first prove the global well-posedness of the Cauchy problem when the space dimension is two. However, in space dimension three, we construct global weak-solutions \`a la Leray, and we prove the local well-posedness of Kato-type solutions. These solutions turn out to be global when the initial data are sufficiently small. Our results extend Giga-Yoshida (1984) ones to the space dimension two, and improve them in terms of requiring less regularity on the velocity fields.

In this article, we establish in the radial framework the $H^1$-scattering for the critical 2-D nonlinear Schr\"odinger equation with exponential growth. Our strategy relies on both the a priori estimate derived in \cite{CGT, PV} and the characterization of the lack of compactness of the Sobolev embedding of $H_{rad}^1(\R^2)$ into the critical Orlicz space ${\cL}(\R^2)$ settled in \cite{BMM}. The radial setting, and particularly the fact that we deal with bounded functions far away from the origin, occurs in a crucial way in our approach.

In an earlier work we have shown the global (for all initial data and all time) well-posedness of strong solutions to the three-dimensional viscous primitive equations of large scale oceanic and atmospheric dynamics. In this paper we show that for certain class of initial data the corresponding smooth solutions of the inviscid (non-viscous) primitive equations blow up in finite time. Specifically, we consider the three-dimensional inviscid primitive equations in a three-dimensional infinite horizontal channel, subject to periodic boundary conditions in the horizontal directions, and with no-normal flow boundary conditions on the solid, top and bottom, boundaries. For certain class of initial data we reduce this system into the two-dimensional system of primitive equations in an infinite horizontal strip with the same type of boundary conditions; and then show that for specific sub-class of initial data the corresponding smooth solutions of the reduced inviscid two-dimensional system develop singularities in finite time.

We study the local and global wellposedness of a full system of Magneto-Hydro-Dynamic equations. The system is a coupling of the forced (Lorentz force) incompressible Navier-Stokes equations with the Maxwell equations through Ohm's law for the current. We show the local existence of mild solutions for arbitrarily large data in a space similar to the scale invariant spaces classically used for Navier-Stokes. These solutions are global if the initial data are small enough., Comment: 16 pages

We study the energy-critical focusing nonlinear Schr\"odinger equation with an energy- subcritical perturbation. We show the existence of a ground state in the four or higher dimensions. Moreover, we give a sufficient and necessary condition for a solution to scatter, in the spirit of Kenig-Merle [16]., Comment: 2 figures, 48 pages