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In the paper we study the Boltzmann equation in the diffusive limit in a channel domain $\mathbb{T}^2\times (-1,1)$ for the 3D kinetic Couette flow. Our results demonstrate that the first-order approximation of the solutions is governed by the perturbed incompressible Navier-Stokes-Fourier system around the fluid Couette flow. Moverover, in the absence of external forces, the 3D kinetic Couette flow asymptotically converges over time to the 1D steady planar kinetic Couette flow. Our proof relies on (i) the Fourier transform on $\mathbb{T}^2$ to essentially reduce the 3D problem to a one-dimensional one, (ii) anisotropic Chemin-Lerner type function spaces, incorporating the Wiener algebra, to control nonlinear terms and address the singularity associated with a small Knudsen number in the diffusive limit, and (iii) Caflisch's decomposition, combined with the $L^2\cap L^\infty$ interplay technique, to manage the growth of large velocities., Comment: 67 pages

In this paper we consider gravity-capillarity Muskat bubbles in 2D. We obtain a new approach to improve our result in [25]. Due to a new bubble-adapted formulation, the improvement is two fold. We significantly condense the proof and we now obtain the global well-posedness result for Muskat bubbles in critical regularity., Comment: This paper extends our results in [25] and contains structural overlap

This paper introduces the 3D Peskin problem: a two-dimensional elastic membrane immersed in a three-dimensional steady Stokes flow. We obtain the equations that model this free boundary problem and show that they admit a boundary integral reduction, providing an evolution equation for the elastic interface. We consider general nonlinear elastic laws, i.e., the fully nonlinear Peskin problem, and prove that the problem is well-posed in low-regularity H\"older spaces. Moreover, we prove that the elastic membrane becomes smooth instantly in time., Comment: 93 pages

We study the problem where a one-dimensional elastic string is immersed in a two-dimensional steady Stokes fluid. This is known as the Stokes immersed boundary problem and also as the Peskin problem. We consider the case with equal viscosities and with a fully non-linear tension law; this model has been called the fully nonlinear Peskin problem. In this case we prove local in time well-posedness for arbitrary initial data in the scaling critical Besov space $\dot{B}^{3/2}_{2,1}(\mathbb{T} ; \mathbb{R}^2)$. We additionally prove the optimal higher order smoothing effects for the solution. To prove this result we derive a new formulation of the boundary integral equation that describes the parametrization of the string, and we crucially utilize a new cancellation structure., Comment: 75 pages

Although the nuclear fusion process has received a great deal of attention in recent years, the amount of mathematical analysis that supports the stability of the system seems to be relatively insufficient. This paper deals with the mathematical analysis of the magnetic confinement of the plasma via kinetic equations. We prove the global wellposedness of the Vlasov-Maxwell system in a two-dimensional annulus when a huge (but finite-in-time) external magnetic potential is imposed near the boundary. We assume that the solution is axisymmetric. The authors hope that this work is a step towards a more generalized work on the three-dimensional Tokamak structure. The highlight of this work is the physical assumptions on the external magnetic potential well which remains finite within a finite time interval and from that, we prove that the plasma never touches the boundary. In addition, we provide a sufficient condition on the magnitude of the external magnetic potential to guarantee that the plasma is confined in an annulus of the desired thickness which is slightly larger than the initial support. Our method uses the cylindrical coordinate forms of the Vlasov-Maxwell system., Comment: Accepted in a KRM special volume in the memory of Bob Glassey; 38 pages

This paper is concerned with the relativistic Boltzmann equation without angular cutoff. We establish the global-in-time existence, uniqueness, and asymptotic stability for solutions nearby the relativistic Maxwellian. We work in the case of a spatially periodic box. We assume the generic hard-interaction and soft-interaction conditions on the collision kernel that were derived by Dudy\'nski and Ekiel-Je$\dot{\text{z}}$ewska (Comm. Math. Phys. \textbf{115}(4):607--629, 1985), and our assumptions include the case of Israel particles (J. Math. Phys. \textbf{4}:1163--1181, 1963). In this physical situation, the angular function in the collision kernel is not locally integrable, and the collision operator behaves like a fractional diffusion operator. The coercivity estimates that are needed rely crucially on the sharp asymptotics for the frequency multiplier that has not been previously established. We further derive the relativistic analogue of the Carleman dual representation for the Boltzmann collision operator. This resolves the open question of perturbative global existence and uniqueness without the Grad's angular cut-off assumption., Comment: Accepted for publication in Ann. PDE, 133 pages, combines arXiv:2102.08846 and the previous version of this paper into one

This paper is concerned with the relativistic Boltzmann equation without angular cutoff. The non-cutoff theory for the relativistic Boltzmann equation has been rarely studied even under a smallness assumption on the initial data due to the lack of understanding of the spectrum and the need for coercivity estimates on the linearized collision operator. Namely, it is crucial to obtain the sharp asymptotics for the frequency multiplier to obtain this coercivity that has never been established before. In this paper, we prove the sharp asymptotics for the frequency multiplier for a general relativistic scattering kernel without angular cutoff. As a consequence of our calculations, we further explain how the well-known change of variables $p' \to p$ is not well defined in the special relativistic context., Comment: This work has been combined with another paper into arXiv:2103.15885 as a new version. Accepted for publication in Ann. PDE

The Peskin problem models the dynamics of a closed elastic filament immersed in an incompressible fluid. In this paper, we consider the case when the inner and outer viscosities are possibly different. This viscosity contrast adds further non-local effects to the system through the implicit non-local relation between the net force and the free interface. We prove the first global well-posedness result for the Peskin problem in this setting. The result applies for medium size initial interfaces in critical spaces and shows instant analytic smoothing. We carefully calculate the medium size constraint on the initial data. These results are new even without viscosity contrast., Comment: 55 pages (updated)

This article considers a long-outstanding open question regarding the Jacobian determinant for the relativistic Boltzmann equation in the center-of-momentum coordinates. For the Newtonian Boltzmann equation, the center-of-momentum coordinates have played a large role in the study of the Newtonian non-cutoff Boltzmann equation, in particular, we mention the widely used cancellation lemma by Alexandre et al. (Arch. Ration. Mech. Anal. 152(4):327-355, 2000). In this article we calculate specifically the very complicated Jacobian determinant, in ten variables, for the relativistic collision map from the momentum $p$ to the post-collisional momentum $p'$; specifically we calculate the determinant for $p\mapsto u = \theta p'+\left(1-\theta\right)p$ for $\theta \in [0,1]$. Afterwards we give an upper-bound for this determinant that has no singularity in both $p$ and $q$ variables. Next we give an example where we prove that the Jacobian goes to zero in a specific pointwise limit. We further explain the results of our numerical study which shows that the Jacobian determinant has a very large number of distinct points at which it is machine zero. This generalizes the work of Glassey-Strauss (Transport Theory Statist. Phys. 20(1):55-68, 1991) and Guo-Strain (Comm. Math. Phys. 310(3):649-673, 2012). These conclusions make it difficult to envision a direct relativistic analog of the Newtonian cancellation lemma in the center-of-momentum coordinates., Comment: 27 pages, 2 figures

In this paper, we prove the propagation of uniform upper bounds for the spatially homogeneous relativistic Boltzmann equation. These $L^\infty$ bounds have been known to be a challenging open problem in relativistic kinetic theory. To accomplish this, we establish two types of estimates for the gain part of the collision operator: first, we prove a potential type estimate and a relativistic hyper-surface integral estimate. We then combine those estimates using the relativistic counter-part of the Carleman representation to derive uniform control of the gain term for the relativistic collision operator. This allows us to prove the desired propagation of the uniform bounds of the solution. We further present two applications of the propagation of the uniform upper bounds: first we give another proof of the Boltzmann $H$-theorem, and second we prove the asymptotic convergence of solutions to the relativistic Maxwellian equilibrium., Comment: 33 pages, revised based upon recommendations of the referees

This paper proves the existence of small-amplitude global-in-time unique mild solutions to both the Landau equation including the Coulomb potential and the Boltzmann equation without angular cutoff. Since the well-known works (Guo, 2002) and (Gressman-Strain-2011, AMUXY-2012) on the construction of classical solutions in smooth Sobolev spaces which in particular are regular in the spatial variables, it still remains an open problem to obtain global solutions in an $L^\infty_{x,v}$ framework, similar to that in (Guo-2010), for the Boltzmann equation with cutoff in general bounded domains. One main difficulty arises from the interaction between the transport operator and the velocity-diffusion-type collision operator in the non-cutoff Boltzmann and Landau equations; another major difficulty is the potential formation of singularities for solutions to the boundary value problem. In the present work we introduce a new function space with low regularity in the spatial variable to treat the problem in cases when the spatial domain is either a torus, or a finite channel with boundary. For the latter case, either the inflow boundary condition or the specular reflection boundary condition is considered. An important property of the function space is that the $L^\infty_T L^2_v$ norm, in velocity and time, of the distribution function is in the Wiener algebra $A(\Omega)$ in the spatial variables. Besides the construction of global solutions in these function spaces, we additionally study the large-time behavior of solutions for both hard and soft potentials, and we further justify the property of propagation of regularity of solutions in the spatial variables., Comment: 65 pages

We prove the uniqueness of weak solutions to the spatially homogeneous special relativistic Landau equation under the conditional assumption that the solution satisfies $(p^0)^7 F(t,p) \in L^1 ([0,T]; L^\infty)$. The existence of standard weak solutions to the relativistic Landau equation has been shown recently in (arXiv:1806.08720), Comment: 37 pages, published version

In this paper we study the Cauchy problem for the spatially homogeneous relativistic Landau equation with Coulomb interactions. Despite it's physical importance, this equation has not received a lot of mathematical attention we think due to the extreme complexity of the relativistic structure of the kernel of the collision operator. In this paper we first largely decompose the structure of the relativistic Landau collision operator. After that we prove the global Entropy dissipation estimate. Then we prove the propagation of any polynomial moment for a weak solution. Lastly we prove the existence of a true weak solution for a large class of initial data., Comment: 48 pages

In this paper we prove the global existence, uniqueness, optimal large time decay rates, and uniform gain of analyticity for the exponential PDE $h_t=\Delta e^{-\Delta h}$ in the whole space $\mathbb{R}^d_x$. We assume the initial data is of medium size in the critical Wiener algebra $\Delta h \in A(\mathbb{R}^d)$. This exponential PDE was derived in (Krug, Dobbs, and Majaniemi in 1995) and more recently in (Marzuola and Weare 2013)., Comment: 25 pages

The Muskat problem models the filtration of two incompressible immiscible fluids of different characteristics in porous media. In this paper, we consider both the 2D and 3D setting of two fluids of different constant densities and different constant viscosities. In this situation, the related contour equations are non-local, not only in the evolution system, but also in the implicit relation between the amplitude of the vorticity and the free interface. Among other extra difficulties, no maximum principles are available for the amplitude and the slopes of the interface in $L^\infty$. We prove global in time existence results for medium size initial stable data in critical spaces. We also enhance previous methods by showing smoothing (instant analyticity), improving the medium size constant in 3D, together with sharp decay rates of analytic norms. The found technique is twofold, giving ill-posedness in unstable situations for very low regular solutions., Comment: New Version; with 2D results

We prove time decay of solutions to the Muskat equation in 2D and in 3D. In \cite{JEMS} and \cite{CCGRPS}, the authors introduce the norms $\|f\|_{s}(t)= \int_{\mathbb{R}^{2}} |\xi|^{s}|\hat{f}(\xi)| \ d\xi$ in order to prove global existence of solutions to the Muskat problem. In this paper, for the 3D Muskat problem, given initial data $f_{0}\in H^{l}(\mathbb{R}^{2})$ for some $l\geq 3$ such that $\|f_{0}\|_{1} < k_{0}$ for a constant $k_{0} \approx 1/5$, we prove uniform in time bounds of $\|f\|_{s}(t)$ for $-d < s < l-1$ and assuming $\|f_{0}\|_{\nu} < \infty$ we prove time decay estimates of the form $\|f\|_{s}(t) \lesssim (1+t)^{-s+\nu}$ for $0 \leq s \leq l-1$ and $-d \leq \nu < s$. These large time decay rates are the same as the optimal rate for the linear Muskat equation. We also prove analogous results in 2D., Comment: 21 pages, made minor changes to the abstract, notation, strategy of proof and reference sections

In this article we will introduce a new model to describe the leading order behavior of an ideal and axisymmetric fluid moving in a very narrow domain. After providing a formal derivation of the model, we will prove the well-posedness and provide a rigorous mathematical justification for the formal derivation under a new sign condition. Finally, a blowup result regarding this model will be discussed as well., Comment: 33 pages

We consider the $3$-dimensional relativistic Vlasov-Maxwell system with data without compact support in momentum space. We prove two continuation criteria for solutions to this system. First, we show that a regular solution can be continued if the integral of the electromagnetic field along any characteristic is assumed to be bounded. This can be viewed as a generalization of the classical result of Glassey-Strauss (1986) to data with non-compact momentum support. Second, we extend the methods in our companion paper (Part I) to show that a regular solution can be extended as long as $\|\pel_0^{\th} f \|_{L^q_xL^1_{\pel}}$ remains bounded for $\th>\frac 2q$, $2

The global existence of solutions to the relativistic Vlasov-Maxwell system given sufficiently regular finite energy initial data is a longstanding open problem. The main result of Glassey-Strauss (1986) shows that a solution $(f, E, B)$ remains $C^1$ as long as the momentum support of $f$ remains bounded. Alternate proofs were later given by Bouchut-Golse-Pallard (2003) and Klainerman-Staffilani (2002). We show that only the boundedness of the momentum support of $f$ after projecting to any two dimensional plane is needed for $(f, E, B)$ to remain $C^1$., Comment: 21 pages, to appear in Commun. Math. Phys

This paper considers the three dimensional Muskat problem in the stable regime. We obtain a conservation law which provides an $L^2$ maximum principle for the fluid interface. We also show global in time existence for strong and weak solutions with initial data controlled by explicit constants. Furthermore we refine the estimates from our paper \cite{PDPB} to obtain global existence and uniqueness for strong solutions with larger initial data than we previously had in 2D. Finally we provide global in time results in critical spaces, giving solutions with bounded slope and time integrable bounded curvature., Comment: Added results on global existence in critical spaces. Included endpoint regularity theorem

In this paper for either the sharp front Surface Quasi-Geostrophic equation or the Muskat problem we rule out the "splash singularity" blow-up scenario; in other words we prove that the contours evolving from either of these systems can not intersect at a single point while the free boundary remains smooth. Splash singularities have been shown to hold for the free boundary incompressible Euler equation in the form of the water waves contour evolution problem \cite{ADCPJ}. Our result confirms the numerical simulations in \cite{CFMR} where it is shown that the curvature blows up due to the contours collapsing at a point. Here we prove that maintaining control of the curvature will remove the possibility of pointwise interphase collapse. Another conclusion that we provide is a better understanding of the work \cite{DP3} in which squirt singularities are ruled out; here a positive volume of fluid between the contours can not be ejected in finite time., Comment: 12 pages

We prove that $k$-th order derivatives of perturbative classical solutions to the hard and soft potential Boltzmann equation (without the angular cut-off assumption) in the whole space, ${\mathbb R}^{n}_x$ with $n \ge 3$, converge in large-time to the global Maxwellian with the optimal decay rate of $O(t^{-1/2(k+\varrho+\frac{n}{2}-\frac{n}{r})})$ in the $L^r_x(L^2_{v})$-norm for any $2\leq r\leq \infty$. These results hold for any $\varrho \in [0, n/2]$ as long as initially $\| f_0|_{\dot{B}^{-\varrho,\infty}_2 L^2_{v}} < \infty$. In the hard potential case, we prove faster decay results in the sense that if $|\mathbf{P} f_0\|_{\dot{B}^{-\varrho,\infty}_2 L^2_{v}} < \infty$ and $|({\mathbf{I} - \mathbf{P}}) f_0|_{\dot{B}^{-\varrho+1,\infty}_2 L^2_{v}} < \infty$ for $\varrho \in (n/2, (n+2)/2]$ then the solution decays to zero in $L^2_v(L^2_x)$ with the optimal large time decay rate of $O(t^{-1/2\varrho})$., Comment: 47 pages. Updated paper to incorporate referee suggestions. Substantially shortened. Previous version may also be useful since it contains additional details in some places. Accepted for publication in Advances in Mathematics

In this article we prove a collection of new non-linear and non-local integral inequalities. As an example for $u\ge 0$ and $p\in (0,\infty)$ we obtain $$ \int_{\threed} dx ~ u^{p+1}(x) \le (\frac{p+1}{p})^2 \int_{\threed} dx ~ \{(-\triangle)^{-1} u(x) \} \nsm \nabla u^{\frac{p}{2}}(x)\nsm^2. $$ We use these inequalities to deduce global existence of solutions to a non-local heat equation with a quadratic non-linearity for large radial monotonic positive initial conditions. Specifically, we improve \cite{ksLM} to include all $\alpha\in (0, 74/75)$., Comment: 6 pages, to appear in Advances in Mathematics

For the Landau-Poisson system with Coulomb interaction in $\R^3_x$, we prove the global existence, uniqueness, and large time convergence rates to the Maxwellian equilibrium for solutions which start out sufficiently close., Comment: 50 pages

For the relativistic Boltzmann equation in $\R^3_x$, this work proves the global existence, uniqueness, positivity, and optimal time convergence rates to the relativistic Maxwellian for solutions which start out sufficiently close under the general physical soft potential assumption proposed in 1988 by Dudy{\'n}ski and Ekiel-Je{\.z}ewska., Comment: 34 pages

Consider steady-state weak solutions to the incompressible Navier-Stokes equations in six spatial dimensions. We prove that the 2D Hausdorff measure of the set of singular points is equal to zero. This problem was mentioned in 1988 by Struwe [24], during his study of the five dimensional case., Comment: 19 pages, revised version, to appear in Indiana Univ. Math. J

In this paper we prove the global in time well-posedness of the following non-local diffusion equation with $\alpha \in[0,2/3)$: $$ \partial_t u = {(-\triangle)^{-1}u} \triangle u + \alpha u^2, \quad u(t=0) = u_0. $$ The initial condition $u_0$ is positive, radial, and non-increasing with $u_0\in L^1\cap L^{2+\delta}(\threed)$ for some small $\delta >0$. There is no size restriction on $u_0$. This model problem appears of interest due to its structural similarity with Landau's equation from plasma physics, and moreover its radically different behavior from the semi-linear Heat equation: $u_t = \triangle u + \alpha u^2$., Comment: 38 pages, made changes according to the referee reports, in press at Comm. PDE

In the study of solutions to the relativistic Boltzmann equation, their regularity with respect to the momentum variables has been an outstanding question, even local in time, due to the initially unexpected growth in the post-collisional momentum variables which was discovered in 1991 by Glassey & Strauss \cite{MR1105532}. We establish momentum regularity within energy spaces via a new splitting technique and interplay between the Glassey-Strauss frame and the center of mass frame of the relativistic collision operator. In a periodic box, these new momentum regularity estimates lead to a proof of global existence of classical solutions to the two-species relativistic Vlasov-Boltzmann-Maxwell system for charged particles near Maxwellian with hard ball interaction., Comment: 23 pages; made revisions which were suggested by the referee; to appear in Comm. Math. Phys

In this paper we study the large-time behavior of perturbative classical solutions to the hard and soft potential Boltzmann equation without the angular cut-off assumption in the whole space $\threed_x$ with $\DgE$. We use the existence theory of global in time nearby Maxwellian solutions from \cite{gsNonCutA,gsNonCut0}. It has been a longstanding open problem to determine the large time decay rates for the soft potential Boltzmann equation in the whole space, with or without the angular cut-off assumption \cite{MR677262,MR2847536}. For perturbative initial data, we prove that solutions converge to the global Maxwellian with the optimal large-time decay rate of $O(t^{-\frac{\Ndim}{2}+\frac{\Ndim}{2r}})$ in the $L^2_\vel(L^r_x)$-norm for any $2\leq r\leq \infty$., Comment: 31 pages, final version to appear in KRM

This work proves the global stability of the Boltzmann equation (1872) with the physical collision kernels derived by Maxwell in 1866 for the full range of inverse-power intermolecular potentials, $r^{-(p-1)}$ with $p>2$, for initial perturbations of the Maxwellian equilibrium states, as announced in \cite{gsNonCutA}. We more generally cover collision kernels with parameters $s\in (0,1)$ and $\gamma$ satisfying $\gamma > -n$ in arbitrary dimensions $\mathbb{T}^n \times \mathbb{R}^n$ with $n\ge 2$. Moreover, we prove rapid convergence as predicted by the celebrated Boltzmann $H$-theorem. When $\gamma \ge -2s$, we have exponential time decay to the Maxwellian equilibrium states. When $\gamma <-2s$, our solutions decay polynomially fast in time with any rate. These results are completely constructive. Additionally, we prove sharp constructive upper and lower bounds for the linearized collision operator in terms of a geometric fractional Sobolev norm; we thus observe that a spectral gap exists only when $\gamma \ge -2s$, as conjectured in Mouhot-Strain \cite{MR2322149}. It will be observed that this fundamental equation, derived by both Boltzmann and Maxwell, grants a basic example where a range of geometric fractional derivatives occur in a physical model of the natural world. Our methods provide a new understanding of the grazing collisions in the Boltzmann theory., Comment: This paper is a combination, simplification, and extension of two separate preprints originally posted on the arXiv as (arXiv:0912.0888v1) and (arXiv:1002.3639v1). It was revised in July 2010 for the referee. In particular we now estimate all $\gamma > -n$. Please cite this version. 77 pages

It is often the case in mathematical analysis that solving an open problem can be facilitated by finding a new set of coordinates which may illumniate the known difficulties. In this article, we illustrate how to derive an assortment coordinates in which to represent the relativistic Boltzmann collision operator. We show the equivalence between some known representations, and others which seem to be new. One of these representations has been used recently to solve several open problems., Comment: 16 pages, to appear in the issue of K.R.M. dedicated to the memory of Carlo Cercignani

We study the local-in-time hydrodynamic limit of the relativistic Boltzmann equation using a Hilbert expansion. More specifically, we prove the existence of local solutions to the relativistic Boltzmann equation that are nearby the local relativistic Maxwellian constructed from a class of solutions to the relativistic Euler equations that includes a large subclass of near-constant, non-vacuum fluid states. In particular, for small Knudsen number, these solutions to the relativistic Boltzmann equation have dynamics that are effectively captured by corresponding solutions to the relativistic Euler equations., Comment: 50 pages

The Muskat problem models the dynamics of the interface between two incompressible immiscible fluids with different constant densities. In this work we prove three results. First we prove an $L^2(\R)$ maximum principle, in the form of a new ``log'' conservation law \eqref{ln} which is satisfied by the equation \eqref{ec1d} for the interface. Our second result is a proof of global existence of Lipschitz continuous solutions for initial data that satisfy $\|f_0\|_{L^\infty}<\infty$ and $\|\partial_x f_0\|_{L^\infty}<1$. We take advantage of the fact that the bound $\|\partial_x f_0\|_{L^\infty}<1$ is propagated by solutions, which grants strong compactness properties in comparison to the log conservation law. Lastly, we prove a global existence result for unique strong solutions if the initial data is smaller than an explicitly computable constant, for instance $\| f\|_1 \le 1/5$. Previous results of this sort used a small constant $\epsilon \ll1$ which was not explicit., Comment: 31 pages

This article provides sharp constructive upper and lower bound estimates for the non-linear Boltzmann collision operator with the full range of physical non cut-off collision kernels ($\gamma > -n$ and $s\in (0,1)$) in the trilinear $L^2(\R^n)$ energy $<\mathcal{Q}(g,f),f>$. These new estimates prove that, for a very general class of $g(v)$, the global diffusive behavior (on $f$) in the energy space is that of the geometric fractional derivative semi-norm identified in the linearized context in our earlier works [2009, 2010, 2010 arXiv:1011.5441v1]. We further prove new global entropy production estimates with the same anisotropic semi-norm. This resolves the longstanding, widespread heuristic conjecture about the sharp diffusive nature of the non cut-off Boltzmann collision operator in the energy space $L^2(\R^n)$., Comment: 29 pages, updated file based on referee report; Advances in Mathematics (2011)

In this paper we study the large-time behavior of classical solutions to the two-species Vlasov-Maxwell-Boltzmann system in the whole space $\R^3$. The existence of global in time nearby Maxwellian solutions is known from [34] in 2006. However the asymptotic behavior of these solutions has been a challenging open problem. Building on our previous work [10] on time decay for the simpler Vlasov-Poisson-Boltzmann system, we prove that these solutions converge to the global Maxwellian with the optimal decay rate of $O(t^{-3/2+\frac{3}{2r}})$ in $L^2_\xi(L^r_x)$-norm for any $2\leq r\leq \infty$ if initial perturbation is smooth enough and decays in space-velocity fast enough at infinity. Moreover, some explicit rates for the electromagnetic field tending to zero are also provided., Comment: 42 pages, updated the file according to the many helpful and valuable comments of the referee

We study the Cauchy Problem for the relativistic Boltzmann equation with near Vacuum initial data. Unique global in time "mild" solutions are obtained uniformly in the speed of light parameter $c \ge 1$. We furthermore prove that solutions to the relativistic Boltzmann equation converge to solutions of the Newtonian Boltzmann equation in the limit as $c\to\infty$ on arbitrary time intervals $[0,T]$, with convergence rate $1/c^{2-\epsilon}$ for any $\epsilon \in(0,2)$. This may be the first proof of unique global in time validity of the Newtonian limit for a Kinetic equation., Comment: 35 pages

In this paper it is shown that unique solutions to the relativistic Boltzmann equation exist for all time and decay with any polynomial rate towards their steady state relativistic Maxwellian provided that the initial data starts out sufficiently close in $L^\infty_\ell$. If the initial data are continuous then so is the corresponding solution. We work in the case of a spatially periodic box. Conditions on the collision kernel are generic in the sense of (Dudy{\'n}ski and Ekiel-Je{\.z}ewska, Comm. Math. Phys., 1988); this resolves the open question of global existence for the soft potentials., Comment: 64 pages

In this work we prove global stability for the Boltzmann equation (1872) with the physical collision kernels derived by Maxwell in 1866 for the full range of inverse power intermolecular potentials, $r^{-(p-1)}$ with $p>2$. This completes the work which we began in (arXiv:0912.0888v1). We more generally cover collision kernels with parameters $s\in (0,1)$ and $\gamma$ satisfying $\gamma > -(n-2)-2s$ in arbitrary dimensions $\mathbb{T}^n \times \mathbb{R}^n$ with $n\ge 2$. Moreover, we prove rapid convergence as predicted by the Boltzmann H-Theorem. When $\gamma + 2s \ge 0$, we have exponential time decay to the Maxwellian equilibrium states. When $\gamma + 2s < 0$, our solutions decay polynomially fast in time with any rate. These results are constructive. Additionally, we prove sharp constructive upper and lower bounds for the linearized collision operator in terms of a geometric fractional Sobolev norm; we thus observe that a spectral gap exists only when $\gamma + 2s \ge 0$, as conjectured in Mouhot-Strain (2007)., Comment: This file has not changed, but this work has been combined with (arXiv:0912.0888v1), simplified and extended into a new preprint, please see the updated version: arXiv:1011.5441v1

The Vlasov-Poisson-Boltzmann System governs the time evolution of the distribution function for the dilute charged particles in the presence of a self-consistent electric potential force through the Poisson equation. In this paper, we are concerned with the rate of convergence of solutions to equilibrium for this system over ${\mathbb{R}}^3$. It is shown that the electric field which is indeed responsible for the lowest-order part in the energy space reduces the speed of convergence and hence the dispersion of this system over the full space is slower than that of the Boltzmann equation without forces, where the exact difference between both power indices in the algebraic rates of convergence is 1/4. For the proof, in the linearized case with a given non-homogeneous source, Fourier analysis is employed to obtain time-decay properties of the solution operator. In the nonlinear case, the combination of the linearized results and the nonlinear energy estimates with the help of the proper Lyapunov-type inequalities leads to the optimal time-decay rate of perturbed solutions under some conditions on initial data., Comment: 37 pages

Consider axisymmetric strong solutions of the incompressible Navier-Stokes equations in $\R^3$ with non-trivial swirl. Let $z$ denote the axis of symmetry and $r$ measure the distance to the z-axis. Suppose the solution satisfies either $|v (x,t)| \le C_*{|t|^{-1/2}} $ or, for some $\e > 0$, $|v (x,t)| \le C_* r^{-1+\epsilon} |t|^{-\epsilon /2}$ for $-T_0\le t < 0$ and $0

Consider axisymmetric strong solutions of the incompressible Navier-Stokes equations in $\R^3$ with non-trivial swirl. Such solutions are not known to be globally defined, but it is shown in \cite{MR673830} that they could only blow up on the axis of symmetry. Let $z$ denote the axis of symmetry and $r$ measure the distance to the z-axis. Suppose the solution satisfies the pointwise scale invariant bound $|v (x,t)| \le C_*{(r^2 -t)^{-1/2}} $ for $-T_0\le t < 0$ and $0

In this paper we prove new constructive coercivity estimates for the Boltzmann collision operator without cutoff, that is for long-range interactions. In particular we give a generalized sufficient condition for the existence of a spectral gap which involves both the growth behavior of the collision kernel at large relative velocities and its singular behavior at grazing and frontal collisions. It provides in particular existence of a spectral gap and estimates on it for interactions deriving from the hard potentials $\phi(r) = r^{-(s−1)}$, $s \ge 5$ or the so-called moderately soft potentials $\phi(r) = r^{−(s−1)}$, $3 < s < 5$, (without angular cutoff). In particular this paper recovers (by constructive means), improves and extends previous results of Pao [46]. We also obtain constructive coercivity estimates for the Landau collision operator for the optimal coercivity norm pointed out in [34] and we formulate a conjecture about a unified necessary and sufficient condition for the existence of a spectral gap for Boltzmann and Landau linearized collision operators., Comment: 29 pages

The Balescu-Lenard equation from plasma physics is widely considered to include a highly accurate correction to Landau's fundamental collision operator. Yet so far it has seen very little mathematical study. We perform an extensive linearized analysis of this equation, which includes determining the asymptotic behavior of the new components of the linearized operator and establishing time decay rates for the linearized equation., Comment: 31 pages

The Vlasov-Maxwell-Boltzmann system is a fundamental model to describe the dynamics of dilute charged particles, where particles interact via collisions and through their self-consistent electromagnetic field. We prove the existence of global in time classical solutions to the Cauchy problem near Maxwellians., Comment: 23 pages