Strichartz Estimates and Moment Bounds for the Relativistic Vlasov–Maxwell System

We consider the relativistic Vlasov–Maxwell system with data of unrestricted size and without compact support in momentum space. In the two-dimensional and the two-and-a-half-dimensional cases, Glassey–Schaeffer proved (Commun Math Phys 185:257–284, 1997; Arch Ration Mech Anal 141:331–354, 1998; Arch Ration Mech Anal. 141:355–374, 1998) that for regular initial data with compact momentum support this system has unique global in time classical solutions. In this work we do not assume compact momentum support for the initial data and instead require only that the data have polynomial decay in momentum space. In the two-dimensional and the two-and-a-half-dimensional cases, we prove the global existence, uniqueness and regularity for solutions arising from this class of initial data. To this end we use Strichartz estimates and prove that suitable moments of the solution remain bounded. Moreover, we obtain a slight improvement of the temporal growth of the \({L^\infty_x}\) norms of the electromagnetic fields compared to Glassey and Schaeffer (Commun Math Phys 185:257–284, 1997; Arch Ration Mech Anal 141:355–374, 1998). In the three-dimensional case, we apply Strichartz estimates and moment bounds to show that a regular solution can be extended as long as \({{\|p_0^{\theta} f \|_{L^{q}_{x}L^1_{p}}}}\) remains bounded for \({\theta > \frac{2}{q}}\), \({2 < q \leqq \infty}\). This improves previous results of Pallard (Indiana Univ Math J 54(5):1395–1409, 2005; Commun Math Sci 13(2):347–354, 2015).     

On Reducibility of 1d Wave Equation with Quasiperiodic in Time Potentials

In this paper we use a KAM theorem of Grébert and Thomann (Commun Math Phys 307:383–427, 2011) to prove the reducibility of the 1d wave equation with Dirichlet boundery conditions on \([0,\pi ]\) with a quasi-periodic in time potential under some symmetry assumptions. From Mathieu–Hill operator’s known results (Eastham in The spectral theory of periodic differential operators, Hafner, New York, 1974; Magnus and Winkler in Hill’s equation, Wiley-Interscience, London, 1969) and Bourgain’s techniques (Commun Math Phys 204:207–247, 1999), we prove that for any \(\epsilon \) small enough, there exist a \(0<m_{\epsilon }\le 1\) and one solution \(u_{\epsilon }(t,x)\) with

$$\begin{aligned} \Vert u_{\epsilon }(t_n,x)\Vert _{H^1({\mathbb {T}})}\rightarrow \infty , \qquad |t_n|\rightarrow \infty , \end{aligned}$$

where \(u_{\epsilon }(t,x)\) satisfies 1d wave equation

$$\begin{aligned} u_{tt}-u_{xx}+m_{\epsilon }u-\epsilon \cos 2t u=0, \end{aligned}$$

with Dirichlet boundery conditions on \([0,\pi ]\).

     

Partial Regularity of the Suitable Weak Solutions to the Multi-dimensional Incompressible Boussinesq Equations

This work is concerned with the partial regularity of the suitable weak solutions to the Boussinesq equations in \(\mathbb {R}^{n}\) where \(n=3,\,4\). By means of the De Giorgi iteration method developed in Vasseur (Nonlinear Differ Equ Appl 14(5–6):753–785, 2007), Wang, Wu (J Differ Equ 256(3):1224–1249, 2014), we obtain that \(n-2\) dimensional parabolic Hausdorff measure of the possible singular points set of the suitable weak solutions to this system is zero. Particularly, we obtain some interior regularity criteria only in terms of the scaled mixed norm of velocity for the suitable weak solutions to the Boussinesq equations, which implies that the potential singular points may only stem from the velocity field.     

On the Regularity Set and Angular Integrability for the Navier–Stokes Equation

We investigate the size of the regular set for suitable weak solutions of the Navier–Stokes equation, in the sense of Caffarelli–Kohn–Nirenberg (Commun Pure Appl Math 35:771–831, 1982). We consider initial data in weighted Lebesgue spaces with mixed radial-angular integrability, and we prove that the regular set increases if the data have higher angular integrability, invading the whole half space \({\{t > 0\}}\) in an appropriate limit. In particular, we obtain that if the \({L^{2}}\) norm with weight \({|x|^{-\frac12}}\) of the data tends to 0, the regular set invades \({\{t > 0\}}\); this result improves Theorem D of Caffarelli et al. (Commun Pure Appl Math 35:771–831, 1982).     

Lie symmetry analysis and invariant solutions of $$\varvec{}$$-dimensional Calogero–Bogoyavlenskii–Schiff equation

Lie group analysis is applied to carry out the similarity reductions of the \((3+1)\)-dimensional Calogero–Bogoyavlenskii–Schiff (CBS) equation. We obtain generators of infinitesimal transformations of the CBS equation and each of these generators depend on various parameters which give us a set of Lie algebras. For each of these Lie algebras, Lie symmetry method reduces the \((3+1)\)-dimensional CBS equation into a new \((2+1)\)-dimensional partial differential equation and to an ordinary differential equation. In addition, we obtain commutator table of Lie brackets and symmetry groups for the CBS equation. Finally, we obtain closed-form solutions of the CBS equation by using the invariance property of Lie group transformations.     

On a Nonlinear Model for Tumor Growth in a Cellular Medium

We investigate the dynamics of a nonlinear model for tumor growth within a cellular medium. In this setting the “tumor” is viewed as a multiphase flow consisting of cancerous cells in either proliferating phase or quiescent phase and a collection of cells accounting for the “waste” and/or dead cells in the presence of a nutrient. Here, the tumor is thought of as a growing continuum \(\Omega \) with boundary \(\partial \Omega \) both of which evolve in time. In particular, the evolution of the boundary \(\partial \Omega \) is prescibed by a given velocity \({{{\varvec{V}}}.}\) The key characteristic of the present model is that the total density of cancerous cells is allowed to vary, which is often the case within cellular media. We refer the reader to the articles (Enault in Mathematical study of models of tumor growth, 2010; Li and Lowengrub in J Theor Biol, 343:79–91, 2014) where compressible type tumor growth models are investigated. Global-in-time weak solutions are obtained using an approach based on penalization of the boundary behavior, diffusion, viscosity and pressure in the weak formulation, as well as convergence and compactness arguments in the spirit of Lions (Mathematical topics in fluid dynamics. Compressible models, 1998) [see also Donatelli and Trivisa (J Math Fluid Mech 16: 787–803, 2004), Feireisl (Dynamics of viscous compressible fluids, 2014)].     

Topological and Measure-Theoretical Entropies of Nonautonomous Dynamical Systems

We consider the dynamics of a nonautonomous dynamical system determined by a sequence of continuous self-maps \(f_n:X \rightarrow X,\) where \( n \in {\mathbb {N}},\) defined on a compact metric space X. Applying the theory of the Carathéodory structures, elaborated by Pesin (Dimension Theory in Dynamical Systems. Chicago Lectures in Mathematics. The University of Chicago Press, Chicago, 1997), we construct a Carathéodory structure whose capacity coincides with the topological entropy of the considered system. Generalizing the notion of local measure entropy, introduced by Brin and Katok (in: Palis (ed) Geometric Dynamics, Lecture Notes in Mathematics. Springer, Berlin 1983) for a single map, to a nonautonomous dynamical system we provide a lower and upper estimations of the topological entropy by local measure entropies. The theorems of the paper generalize results of Kawan (Nonautonomous Stoch Dyn Syst 1:26–52, 2013) and of Feng and Huang (J Funct Anal 263:2228–2254, 2012). Also, we construct a new entropy-like invariant such the entropy of a sequence \(\{f_n:X \rightarrow X\}_{n=1}^{\infty }\) of Lipschitz continuous maps with the same Lipschitz constant \(L >1,\) restricted to a subset \(Y\subset X,\) is upper bounded by Hausdorff dimension of Y multiplied by the logarithm of the Lipschitz constant L. This gives a generalizations of results of Dai et al. (Sci China Ser A 41:1068–1075, 1998) and Misiurewicz (Discret Contin Dyn Syst 10:827–833, 2004).     

Global Strong Well-Posedness of the Three Dimensional Primitive Equations in {L^p}-Spaces

In this article, an \({L^p}\)-approach to the primitive equations is developed. In particular, it is shown that the three dimensional primitive equations admit a unique, global strong solution for all initial data \({a \in [X_p,D(A_p)]_{1/p}}\) provided \({p \in [6/5,\infty)}\). To this end, the hydrostatic Stokes operator \({A_p}\) defined on \({X_p}\), the subspace of \({L^p}\) associated with the hydrostatic Helmholtz projection, is introduced and investigated. Choosing \({p}\) large, one obtains global well-posedness of the primitive equations for strong solutions for initial data \({a}\) having less differentiability properties than \({H^1}\), hereby generalizing in particular a result by Cao and Titi (Ann Math 166:245–267, 2007) to the case of non-smooth initial data.     

From the Highly Compressible Navier–Stokes Equations to Fast Diffusion and Porous Media Equations,Existence of Global Weak Solution for the Quasi-Solutions

We consider the compressible Navier–Stokes equations for viscous and barotropic fluids with density dependent viscosity. The aim is to investigate mathematical properties of solutions of the Navier–Stokes equations using solutions of the pressureless Navier–Stokes equations, that we call quasi solutions. This regime corresponds to the limit of highly compressible flows. In this paper we are interested in proving the announced result in Haspot (Proceedings of the 14th international conference on hyperbolic problems held in Padova, pp 667–674, 2014) concerning the existence of global weak solution for the quasi-solutions, we also observe that for some choice of initial data (irrotationnal) the quasi solutions verify the porous media, the heat equation or the fast diffusion equations in function of the structure of the viscosity coefficients. In particular it implies that it exists classical quasi-solutions in the sense that they are \({C^{\infty}}\) on \({(0,T)\times \mathbb{R}^{N}}\) for any \({T > 0}\). Finally we show the convergence of the global weak solution of compressible Navier–Stokes equations to the quasi solutions in the case of a vanishing pressure limit process. In particular for highly compressible equations the speed of propagation of the density is quasi finite when the viscosity corresponds to \({\mu(\rho)=\rho^{\alpha}}\) with \({\alpha > 1}\). Furthermore the density is not far from converging asymptotically in time to the Barrenblatt solution of mass the initial density \({\rho_{0}}\).     

Integrable $${\varvec{}}$$-dimensional systems with rational Lax pairs

The search for new integrable \((3+1)\)-dimensional partial differential systems is among the most important challenges in the modern integrability theory. It turns out that such a system can be associated with any pair of rational functions of one variable in general position, as established below using contact Lax pairs introduced in Sergyeyev (Lett Math Phys, 2017.  https://doi.org/10.1007/s11005-017-1013-4, arXiv:1401.2122).     

Multiple Periodic Solutions of an Equation with State-Dependent Delay

Given \({N \in \mathbb N}\) we prove the existence, for parameter values in a certain range, of N distinct periodic solutions of a state-dependent delay equation studied by Walther (Differ Integral Equ 15:923–944, 2002).     

Propagation of Elastic Waves in a Gas-Filled Poroelastic Medium: The Influence of the Boundary Conditions

We consider the propagation of elastic waves in gas-filled porous media at small but non-zero values of Knudsen numbers \( {\text{Kn}} \), where \( {\text{Kn}} = \lambda /l \), \( \lambda \) is the mean free path of gas molecules; \( l \) is the characteristic size of inclusion (the so-called slip regime). In this case, it is possible to apply the classic equations of hydrodynamics with modified boundary conditions at solid walls. We have assumed that the gas molecules distribution function is satisfied at the modified Maxwell boundary conditions (Struchtrup 2013; Mohammadzadeh and Struchtrup 2015). We have obtained the expressions for drag and added mass coefficients for the Biot equations of poroelasticity for a system of randomly oriented gas-filled cylindrical capillaries. Our calculations have shown that the drag and added mass coefficients depend considerably on the Knudsen number and the properties of the surface. The influence of the interfacial slip effect on the velocities of the compressional wave of the first kind and shear wave is small, but the velocity and attenuation of the compressional wave of the second kind are considerably influenced by this effect. The results obtained show the fundamental possibility of the determination of the accommodation coefficient by measuring the velocity of the compressional wave of the second kind for different values of the Knudsen number.     

Injectivity of the Cauchy-stress tensor along rank-one connected lines under strict rank-one convexity condition

In this note, we show that the Cauchy stress tensor \(\sigma\) in nonlinear elasticity is injective along rank-one connected lines provided that the constitutive law is strictly rank-one convex. This means that \(\sigma(F+\xi\otimes\eta)=\sigma(F)\) implies \(\xi \otimes\eta=0\) under strict rank-one convexity. As a consequence of this seemingly unnoticed observation, it follows that rank-one convexity and a homogeneous Cauchy stress imply that the left Cauchy-Green strain is homogeneous, as is shown in Mihai and Neff (Int. J. Non-Linear Mech., 2016, to appear).     

On the Global Regularity of the Two-Dimensional Density Patch for Inhomogeneous Incompressible Viscous Flow

Regarding P.-L. Lions’ open question in Oxford Lecture Series in Mathematics and its Applications, Vol. 3 (1996) concerning the propagation of regularity for the density patch, we establish the global existence of solutions to the two-dimensional inhomogeneous incompressible Navier–Stokes system with initial density given by \({(1 - \eta){\bf 1}_{{\Omega}_{0}} + {\bf 1}_{{\Omega}_{0}^{c}}}\) for some small enough constant \({\eta}\) and some \({W^{k+2,p}}\) domain \({\Omega_{0}}\), with initial vorticity belonging to \({L^{1} \cap L^{p}}\) and with appropriate tangential regularities. Furthermore, we prove that the regularity of the domain \({\Omega_0}\) is preserved by time evolution.     

Two-subsystem thermodynamics for the mechanics of aging amorphous solids

The effect of physical aging on the mechanics of amorphous solids as well as mechanical rejuvenation is modeled with nonequilibrium thermodynamics, using the concept of two thermal subsystems, namely a kinetic one and a configurational one. Earlier work (Semkiv and Hütter in J Non-Equilib Thermodyn 41(2):79–88, 2016) is extended to account for a fully general coupling of the two thermal subsystems. This coupling gives rise to hypoelastic-type contributions in the expression for the Cauchy stress tensor, that reduces to the more common hyperelastic case for sufficiently long aging. The general model, particularly the reversible and irreversible couplings between the thermal subsystems, is compared in detail with models in the literature (Boyce et al. in Mech Mater 7:15–33, 1988; Buckley et al. in J Mech Phys Solids 52:2355–2377, 2004; Klompen et al. in Macromolecules 38:6997–7008, 2005; Kamrin and Bouchbinder in J Mech Phys Solids 73:269–288 2014; Xiao and Nguyen in J Mech Phys Solids 82:62–81, 2015). It is found that only for the case of Kamrin and Bouchbinder (J Mech Phys Solids 73:269–288, 2014) there is a nontrivial coupling between the thermal subsystems in the reversible dynamics, for which the Jacobi identity is automatically satisfied. Moreover, in their work as well as in Boyce et al. (Mech Mater 7:15–33, 1988), viscoplastic deformation is driven by the deviatoric part of the Cauchy stress tensor, while for Buckley et al. (J Mech Phys Solids 52:2355–2377, 2004) and Xiao and Nguyen (J Mech Phys Solids 82:62–81, 2015) this is not the case.     

Further Dense Properties of the Space of Circle Diffeomorphisms with a Liouville Rotation Number

In continuation of Matsumoto’s paper (Nonlinearity 25:1495–1511, 2012) we show that various subspaces are \(C^{\infty }\)-dense in the space of orientation-preserving \(C^{\infty }\)-diffeomorphisms of the circle with rotation number \(\alpha \), where \(\alpha \in {\mathbb {S}}^1\) is any prescribed Liouville number. In particular, for every odometer \({\mathcal {O}}\) of product type we prove the denseness of the subspace of diffeomorphisms which are orbit-equivalent to \({\mathcal {O}}\).     

On Nonperiodic Euler Flows with Hölder Regularity

In (Isett, Regularity in time along the coarse scale flow for the Euler equations, 2013), the first author proposed a strengthening of Onsager’s conjecture on the failure of energy conservation for incompressible Euler flows with Hölder regularity not exceeding \({1/3}\). This stronger form of the conjecture implies that anomalous dissipation will fail for a generic Euler flow with regularity below the Onsager critical space \({L_t^\infty B_{3,\infty}^{1/3}}\) due to low regularity of the energy profile. This paper is the first and main paper in a series of two, the results of which may be viewed as first steps towards establishing the conjectured failure of energy regularity for generic solutions with Hölder exponent less than \({1/5}\). The main result of the present paper shows that any given smooth Euler flow can be perturbed in \({C^{1/5-\epsilon}_{t,x}}\) on any pre-compact subset of \({\mathbb{R}\times \mathbb{R}^3}\) to violate energy conservation. Furthermore, the perturbed solution is no smoother than \({C^{1/5-\epsilon}_{t,x}}\). As a corollary of this theorem, we show the existence of nonzero \({C^{1/5-\epsilon}_{t,x}}\) solutions to Euler with compact space-time support, generalizing previous work of the first author (Isett, Hölder continuous Euler flows in three dimensions with compact support in time, 2012) to the nonperiodic setting.     

On the Regularity for the Navier-Slip Thin-Film Equation in the Perfect Wetting Regime

In this paper, we consider a compressible two-fluid model with constant viscosity coefficients and unequal pressure functions \({P^+\neq P^-}\). As mentioned in the seminal work by Bresch, Desjardins, et al. (Arch Rational Mech Anal 196:599–629, 2010) for the compressible two-fluid model, where \({P^+=P^-}\) (common pressure) is used and capillarity effects are accounted for in terms of a third-order derivative of density, the case of constant viscosity coefficients cannot be handled in their settings. Besides, their analysis relies on a special choice for the density-dependent viscosity [refer also to another reference (Commun Math Phys 309:737–755, 2012) by Bresch, Huang and Li for a study of the same model in one dimension but without capillarity effects]. In this work, we obtain the global solution and its optimal decay rate (in time) with constant viscosity coefficients and some smallness assumptions. In particular, capillary pressure is taken into account in the sense that \({\Delta P=P^+ - P^-=f\neq 0}\) where the difference function \({f}\) is assumed to be a strictly decreasing function near the equilibrium relative to the fluid corresponding to \({P^-}\). This assumption plays an key role in the analysis and appears to have an essential stabilization effect on the model in question.     

An Analytical Solution for Capillary Gravity Drainage with Dominant Viscous Forces

The development of the capillary fringe during gravity drainage has a significant influence on saturation and pressure distributions in porous formations (Sarkarfarshi et al. in Int J Greenh Gas Control 23:61–71, 2014). This paper introduces an analytical solution for gravity drainage in an axisymmetric geometry with significant capillary pressure. The drainage process results from the injection of a lighter and less viscous injectant into a porous medium saturated with a heavier and more viscous pore fluid. If the viscous force dominates the capillary and the buoyancy forces, then the flow regime is approximated by differential equations and the admissible solution comprises a front shock wave and a trailing simple wave. In contrast to existing analytical solutions for capillary gravity drainage problems (e.g., Nordbotten and Dahle in 47(2) 2011; Golding et al. in J Fluid Mech 678:248–270 2011), this solution targets the saturation distribution during injection at an earlier point in time. Another contribution of this analytical solution is the incorporation of a completely drained flow regime close to the injection well. The analytical solution demonstrates the strong dependency of the saturation distribution upon relative permeability functions, gas entry capillary pressure, and residual saturation. The analytical results are compared to results from a commercial reservoir engineering software package (\(\hbox {CMG } \hbox {STARS}^{\mathrm{TM}}\)).     

Sharp Weighted Korn and Korn-Like Inequalities and an Application to Washers

In this paper we prove asymptotically sharp weighted “first-and-a-half” \(2D\) Korn and Korn-like inequalities with a singular weight occurring from Cartesian to cylindrical change of variables. We prove some Hardy and the so-called “harmonic function gradient separation” inequalities with the same singular weight. Then we apply the obtained \(2D\) inequalities to prove similar inequalities for washers with thickness \(h\) subject to vanishing Dirichlet boundary conditions on the inner and outer thin faces of the washer. A washer can be regarded in two ways: As the limit case of a conical shell when the slope goes to zero, or as a very short hollow cylinder. While the optimal Korn constant in the first Korn inequality for a conical shell with thickness \(h\) and with a positive slope scales like \(h^{1.5}\), e.g., (Grabovsky and Harutyunyan in arXiv:1602.03601, 2016), the optimal Korn constant in the first Korn inequality for a washer scales like \(h^{2}\) and depends only on the outer radius of the washer as we show in the present work. The Korn constant in the first and a half inequality scales like \(h\) and depends only on \(h\). The optimal Korn constant is realized by a Kirchhoff Ansatz. This results can be applied to calculate the critical buckling load of a washer under in plane loads, e.g., (Antman and Stepanov in J. Elast. 124(2):243–278, 2016).