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We consider the three dimensional free-boundary compressible elastodynamic system under the Rayleigh–Taylor sign condition. This describes the motion of an isentropic inviscid elastic medium with moving boundary. The deformation tensor is assumed to satisfy the neo-Hookean linear elasticity. The local well-posedness was proved by Trakhinin (J Differ Eq 264(3):1661–1715, 2018) by Nash–Moser iteration. In this paper, we give a new proof of the local well-posedness by the combination of classical energy method and hyperbolic approach. In the proof, we apply the tangential smoothing method to define the approximation system. The key observation is that the structure of the wave equation of pressure together with Christodoulou–Lindblad (Commun Pure Appl Math 53(12):1536–1602, 2000) elliptic estimates reduces the energy estimates to the control of tangentially-differentiated wave equations despite a potential loss of derivative in the source term. To the best of our knowledge, we first establish the nonlinear energy estimate without loss of regularity for free-boundary compressible elastodynamics. The energy estimate is also uniform in sound speed which yields the incompressible limit, that is, the solutions of the free-boundary compressible elastodynamic equations converge to the incompressible counterpart provided the convergence of initial datum. It is worth emphasizing that our method is completely applicable to compressible Euler equations. Our observation also shows that it is not necessary to include the full time derivatives in the boundary energy and analyze higher order wave equations as in Lindblad–Luo (Commun Pure Appl Math 71(7):1273–1333, 2018) and Luo (Ann. PDE 4(2):2506–2576, 2018) even if we require the energy is uniform in sound speed. Moreover, the enhanced regularity for compressible Euler equations obtained in Lindblad–Luo (Commun Pure Appl Math 71(7):1273–1333, 2018) and Luo (Ann. PDE 4(2):2506–2576, 2018) can still be recovered for a slightly compressible elastic medium by further delicate analysis of the Alinhac good unknowns, which is completely different from Euler equations. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the hydrodynamic limit to rarefaction waves of the compressible Euler system for the Landau equation with Coulomb potentials as the Knudsen number ε > 0 is vanishing. Precisely, whenever ε > 0 is small, for the Cauchy problem on the Landau equation with suitable initial data involving a scaling parameter a ∈ [ 2 3 , 1 ] , we construct the unique global-in-time uniform-in- ε solution around a local Maxwellian whose fluid quantities are the rarefaction wave of the corresponding Euler system. In the meantime, we establish the convergence of solutions to the Riemann rarefaction wave uniformly away from t = 0 at a rate ε 3 5 - 2 5 a | ln ε | as ε → 0 . The proof is based on the refined energy approach combining Guo (Commun Math Phys 231:391–434, 2002) and Liu et al. (Physica D 188:178–192, 2004) under the scaling transformation (t , x) → (ε - a t , ε - a x) . [ABSTRACT FROM AUTHOR]

In this paper we describe the fundamental solution of the equation that is obtained linearizing the Uehling–Uhlenbeck equation around the steady state of Kolmogorov type f( k) = k −7/6. Detailed estimates on its asymptotics are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we prove the local well-posedness for the ideal MHD equations in the Triebel–Lizorkin spaces and obtain a blow-up criterion of smooth solutions. Specifically, we fill a gap in a step of the proof of the local well-posedness part for the incompressible Euler equation in Chae (Comm Pure Appl Math 55:654–678 2002). [ABSTRACT FROM AUTHOR]

In this paper, we prove a stability result for measure perturbations of some class of stationary distributions of a Vlasov equation. We use this result to prove that the N particles approximation of these stationary distributions is uniformly valid on a time scale of order N 1/8, which is much longer than the usual log N scale. We also prove similar results for the approximation of the two-dimensional Euler equation by the vortex blob method. [ABSTRACT FROM AUTHOR]

This paper develops a new approach in the analysis of the Camassa–Holm equation. By introducing a new set of independent and dependent variables, the equation is transformed into a semilinear system, whose solutions are obtained as fixed points of a contractive transformation. These new variables resolve all singularities due to possible wave breaking. Returning to the original variables, we obtain a semigroup of global solutions, depending continuously on the initial data. Our solutions are conservative, in the sense that the total energy equals a constant, for almost every time. [ABSTRACT FROM AUTHOR]

We study the relaxation of Hamilton-Jacobi equations. The relaxation in our terminology is the following phenomenon: the pointwise supremum over a certain collection of subsolutions, in the almost everywhere sense, of a Hamilton-Jacobi equation yields a viscosity solution of the ``convexified'' Hamilton-Jacobi equation. This phenomenon has recently been observed in [13] in eikonal equations. We show in this paper that this relaxation is a common phenomenon for a wide range of Hamilton-Jacobi equations. [ABSTRACT FROM AUTHOR]

We introduce a new class of Wasserstein-type distances specifically designed to tackle questions concerning stability and convergence to equilibria for kinetic equations. Thanks to these new distances, we improve some classical estimates by Loeper (J Math Pures Appl (9) 86(1):68–79, 2006) and Dobrushin (Funktsional Anal i Prilozhen 13:48–58, 1979) on Vlasov-type equations, and we present an application to quasi-neutral limits. [ABSTRACT FROM AUTHOR]

Natural Sobolev-type estimates are proved for weak solutions of inhomogeneous parabolic equations in divergence form in a bounded cylinder O*=O×(0,T] which isd-Reifenberg flat in the space direction. The principal coefficients of the operator are assumed to be in BMO space with their BMO semi-norms small enough. [ABSTRACT FROM AUTHOR]

We present a systematic study of entire symmetric solutions $${u : \mathbb{R}^n \rightarrow\mathbb{R}^m}$$ of the vector Allen-Cahn equation where $${W:\mathbb{R}^m \rightarrow \mathbb{R}}$$ is smooth, symmetric, nonnegative with a finite number of zeros, and where $${ W_u= (\partial W / \partial u_1,\dots,\partial W / \partial u_m)^{\top}}$$ . We introduce a general notion of equivariance with respect to a homomorphism $${f:G\rightarrow\Gamma}$$ ( $${G, \Gamma}$$ reflection groups) and prove two abstract results, concerning the cases of G finite and G discrete, for the existence of equivariant solutions. Our approach is variational and based on a mapping property of the parabolic vector Allen-Cahn equation and on a pointwise estimate for vector minimizers. [ABSTRACT FROM AUTHOR]

We investigate the properties of certain elliptic systems leading, a priori, to solutions that belong to the space of Radon measures. We show that if the problem is equipped with a so-called asymptotic radial structure, then the solution can in fact be understood as a standard weak solution, with one proviso: analogously to the case of minimal surface equations, the attainment of the boundary value is penalized by a measure supported on (a subset of) the boundary, which, for the class of problems under consideration here, is the part of the boundary where a Neumann boundary condition is imposed. [ABSTRACT FROM AUTHOR]

We study the stability of stationary solutions of the two dimensional inviscid incompressible porous medium equation (IPM). We show that solutions which are near certain stable stationary solutions must converge as t → ∞ to a stationary solution of the IPM equation. It turns out that linearizing the IPM equation about certain stable stationary solutions gives a non-local partial damping mechanism. On the torus, the linearized problem has a very large set of stationary (undamped) modes. This makes the problem of long-time behavior more difficult since there is the possibility of a cascading non-linear growth along the stationary modes of the linearized problem. We solve this by, more or less, doing a second linearization around the undamped modes, exploiting a special non-linear structure, and showing that the stationary modes can be controlled. [ABSTRACT FROM AUTHOR]

We investigate the evolution of rigid bodies in a viscous incompressible fluid. The flow is governed by the 2D Navier–Stokes equations, set in a bounded domain with Dirichlet boundary conditions. The boundaries of the solids and the domain have Hölder regularity C1, α, 0 < α ≦ 1. First, we show the existence and uniqueness of strong solutions up to the collision. A key ingredient is a BMO bound on the velocity gradient, which substitutes to the standard H2 estimate for smoother domains. Then, we study the asymptotic behaviour of one C1, α body falling over a flat surface. We show that a collision is possible in finite time if and only if α < 1/2. [ABSTRACT FROM AUTHOR]

In recent years two nonlinear dispersive partial differential equations have attracted much attention due to their integrable structure. We prove that both equations arise in the modeling of the propagation of shallow water waves over a flat bed. The equations capture stronger nonlinear effects than the classical nonlinear dispersive Benjamin–Bona–Mahoney and Korteweg–de Vries equations. In particular, they accommodate wave breaking phenomena. [ABSTRACT FROM AUTHOR]

We study the limit as ε → 0 of the entropy solutions of the equation $${\partial_t u^\varepsilon + {\rm div}_x \left[A \left(\frac{x}{\varepsilon},u^\varepsilon \right)\right] =0}$$ . We prove that the sequence u ε two-scale converges toward a function u( t, x, y), and u is the unique solution of a limit evolution problem. The remarkable point is that the limit problem is not a scalar conservation law, but rather a kinetic equation in which the macroscopic and microscopic variables are mixed. We also prove a strong convergence result in $${L_{\rm loc}^1}$$ . [ABSTRACT FROM AUTHOR]

We study the existence and the asymptotic behavior of large amplitude high-frequency oscillating waves subjected to the two-dimensional Burger equation. This program is achieved by developing tools related to supercritical WKB analysis. By selecting solutions which are divergence free, we show that incompressible or compressible two-dimensional Euler equations are not locally closed for the weak L 2 topology. [ABSTRACT FROM AUTHOR]

We establish Liouville theorems for $$C_{loc}^{1,1}$$ , entire solutions and locally Lipschitz entire weak solutions to general degenerate conformally invariant fully nonlinear elliptic equations of second order. For applications to local gradient estimates of solutions of general conformally invariant fully nonlinear elliptic equations of second order, see [20]. [ABSTRACT FROM AUTHOR]

Kinetic transport equations with a given confining potential and non-linear relaxation type collision operators are considered. General (monotone) energy dependent equilibrium distributions are allowed with a chemical potential ensuring mass conservation. Existence and uniqueness of solutions is proved for initial data bounded by equilibrium distributions. The diffusive macroscopic limit is carried out using compensated compactness theory. The results are drift-diffusion equations with non linear diffusion. The most notable examples are of the form $$\partial_t \rho=\nabla\cdot (\nabla \rho^m + \rho\,\nabla V)$$ , ranging from porous medium equations to fast diffusion, with the exponent satisfying $$0 < m < 5/3$$ in $${\mathbb{R}}^3$$ . [ABSTRACT FROM AUTHOR]

We prove stability of steady flows of an ideal fluid in a bounded, simply connected, planar region, that are strict maximisers or minimisers of kinetic energy on an isovortical surface. The proof uses conservation of energy and transport of vorticity for solutions of the vorticity equation with initial data inL p forp>4/3. A related stability theorem using conservation of angular momentum in a circular domain is also proved. [ABSTRACT FROM AUTHOR]

This note generalizes the notion of dissipative solutions to non-isotropic degenerate parabolic balance laws introduced in [3]. The new definition allows us to use a larger class of test functions than the one used in [3] to study the equivalence between dissipative and entropy solutions. As a result, it is possible to study general relaxation-type approximations (limits). [ABSTRACT FROM AUTHOR]

The uniqueness of solutions of the Navier-Stokes equations in the whole space is established when the velocity field is bounded and the pressure field is a BMO-valued locally integrable-in-time function for bounded initial data. Here the velocity field may not decay at space infinity. Although there are a few results concerning uniqueness without the decay assumption, our result is new and applicable for solutions constructed by solving the integral equations. [ABSTRACT FROM AUTHOR]

We construct steady states of the Euler-Poisson system with a barotropic equation of state as minimizers of a suitably defined energy functional. Their minimizing property implies the non-linear stability of such states against general, i.e., not necessarily spherically symmetric, perturbations. The mathematical approach is based on previous stability results for the Vlasov-Poisson system by Y. Guo and G. Rein, exploiting the energy-Casimir technique. The analysis is conditional in the sense that it assumes the existence of solutions to the initial value problem for the Euler-Poisson system which preserve mass and energy. The relation between the Euler-Poisson and the Vlasov-Poisson system in this context is also explored. [ABSTRACT FROM AUTHOR]

We consider the semidiscrete upwind scheme (1) u(t, x)t + 1 / ε( f (u(t, x)) - f (u(t, x - ε))) = 0. We prove that if the initial data u of (1) has small total variation, then the solution uε (t) has uniformly bounded BV norm, independent of t, ε. Moreover by studying the equation for a perturbation of (1) we prove the Lipschitz-continuous dependence of uε (t) on the initial data. Using a technique similar to the vanishing-viscosity case, we show that as ε → 0 the solution uε (t) converges to a weak solution of the corresponding hyperbolic system, (2) ut + f (u)x = 0. Moreover this weak solution coincides with the trajectory of a Riemann semigroup, which is uniquely determined by the extension of Liu's Riemann solver to general hyperbolic systems. [ABSTRACT FROM AUTHOR]