Spinodal Decomposition and Coarsening Fronts in the Cahn–Hilliard Equation

We study spinodal decomposition and coarsening when initiated by localized disturbances in the Cahn–Hilliard equation. Spatio-temporal dynamics are governed by multi-stage invasion fronts. The first front invades a spinodal unstable equilibrium and creates a spatially periodic unstable pattern. Secondary fronts invade this unstable pattern and create a coarser pattern in the wake. We give linear predictions for speeds and wavenumbers in this process and show existence of corresponding nonlinear fronts. The existence proof is based on Conley index theory, a priori estimates, and Galerkin approximations. We also compare our results and predictions with direct numerical simulations and report on some interesting bifurcations.     

Spatial Analyticity on the Global Attractor for the Kuramoto–Sivashinsky Equation

For the Kuramoto–Sivashinsky equation with L-periodic boundary conditions we show that the radius of space analyticity on the global attractor is lower-semicontinuous function at the stationary solutions, and thereby deduce the existence of a neighborhood in the global attractor of the set of all stationary solutions in which the radius of analyticity is independent of the bifurcation parameter L. As an application of the result, we prove that the number of rapid spatial oscillations of functions belonging to this neighborhood is, up to a logarithmic correction, at most linear in L.     

Vortex Dynamics for the Nonlinear Maxwell–Klein–Gordon Equation

We derive the vortex dynamics for the nonlinear Maxwell–Klein–Gordon equation with the Ginzburg–Landau type potential. In particular, we consider the case when the external electric fields are of order \({O( | \log \epsilon |^{\frac{1}{2}})}\). We study the convergence of the space–time Jacobian \({\partial_t \psi \cdot i \nabla \psi}\) as an interaction term between the vortices and electric fields. An explicit form of the limiting vector measure is shown.     

Anisotropic Estimates for the Two-Dimensional Kuramoto–Sivashinsky Equation

We address the global solvability of the Kuramoto–Sivashinsky equation in a rectangular domain \([0,L_1]\times [0,L_2]\) . We give sufficient conditions on the width \(L_2\) of the domain, depending on the length \(L_1\) , so that the obtained solutions are global. Our proofs are based on anisotropic estimates.     

Bounded Domain Problem for the Modified Buckley–Leverett Equation

The focus of the present study is the modified Buckley–Leverett (MBL) equation describing two-phase flow in porous media. The MBL equation differs from the classical Buckley–Leverett (BL) equation by including a balanced diffusive–dispersive combination. The dispersive term is a third order mixed derivatives term, which models the dynamic effects in the pressure difference between the two phases. The classical BL equation gives a monotone water saturation profile for any Riemann problem; on the contrast, when the dispersive parameter is large enough, the MBL equation delivers a non-monotone water saturation profile for certain Riemann problems as suggested by the experimental observations. In this paper, we first show that for the MBL equation, the solution of the finite interval \([0,L]\) boundary value problem converges to that of the half line \([0,+\infty )\) boundary value problem exponentially as \(L\rightarrow +\infty \) . This result provides a justification for the use of the finite interval in numerical studies for the half line problem [Y. Wang and C.-Y. Kao, Central schemes for the modified Buckley–Leverett equation, J. Comput. Sci. 4(1–2), 12 – 23, 2013]. Furthermore, we numerically verify that the convergence rate is consistent with the theoretical derivation. Numerical results confirm the existence of non-monotone water saturation profiles consisting of constant states separated by shocks.     

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).     

A Birkhoff–Lewis Type Theorem for the Nonlinear Wave Equation

We give an extension of the celebrated Birkhoff–Lewis theorem to the nonlinear wave equation. Accordingly, we find infinitely many periodic orbits with longer and longer minimal periods accumulating at the origin, which is an elliptic equilibrium of the associated infinite-dimensional Hamiltonian system.     

Local Exact Controllability for the One-Dimensional Compressible Navier–Stokes Equation

In this paper we deal with the isentropic (compressible) Navier-Stokes equation in one space dimension and we adress the problem of the boundary controllability for this system. We prove that we can drive initial conditions which are sufficiently close to some constant states to those constant states. This is done under some natural hypotheses on the time of control and on the regularity on the initial conditions.     

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).     

Stability for Rayleigh–Benard Convective Solutions of the Boltzmann Equation

We consider the Boltzmann equation for a gas in a horizontal slab, subject to a gravitational force. The boundary conditions are of diffusive type, specifying the wall temperatures, so that the top temperature is lower than the bottom one (Benard setup). We consider a 2-dimensional convective stationary solution, which for small Knudsen numbers is close to the convective stationary solution of the Oberbeck–Boussinesq equations, near and above the bifurcation point, and prove its stability under 2-d small perturbations, for Rayleigh numbers above and close to the bifurcation point and for small Knudsen numbers.     

Role of the Pressure for Validity of the Energy Equality for Solutions of the Navier–Stokes Equation

We prove that a weak solution of the Navier–Stokes system satisfies the energy equality if the associated pressure is locally square integrable. A similar statement is shown to hold for the Euler system.     

On Decay Properties of Solutions to the IVP for the Benjamin–Ono Equation

In this work we investigate unique continuation properties of solutions to the initial value problem associated to the Benjamin–Ono equation in weighted Sobolev spaces $Z_{s,r}=H^s(\mathbb R )\cap L^2(|x|^{2r}dx)$ for $s\in \mathbb R $ , and $s\ge 1$ , $s\ge r$ . More precisely, we prove that the uniqueness property based on a decay requirement at three times can not be lowered to two times even by imposing stronger decay on the initial data.     

Uniform Regularity for the Navier–Stokes Equation with Navier Boundary Condition

We prove that there exists an interval of time which is uniform in the vanishing viscosity limit and for which the Navier–Stokes equation with the Navier boundary condition has a strong solution. This solution is uniformly bounded in a conormal Sobolev space and has only one normal derivative bounded in L ^∞. This allows us to obtain the vanishing viscosity limit to the incompressible Euler system from a strong compactness argument.     

Strong Unique Continuation for the Navier–Stokes Equation with Non-Analytic Forcing

We establish the strong unique continuation property for differences of solutions to the Navier–Stokes system with Gevrey forcing. For this purpose, we provide Carleman-type inequalities with the same singular weight for the Laplacian and the heat operator.     

Development of the Spectral Sturm–Liouville Method for the Solution of a Boundary-Value Problem for the Biharmonic Equation

We generalize the spectral Sturm–Liouville method for the solution of the biharmonic equation. The characteristic equation for the determination of eigenvalues is investigated and eigenfunctions are constructed. We determine the stress-strain state for a rectangular plate loaded by arbitrary forces on its sides. For an arbitrary external load, we obtain a relation for the stress-strain state in the form of a series in eigenfunctions. A method of integral moments for the determination of the coefficients of the series is proposed. The Saint-Venant principle is verified.     

A Navier–Stokes Approximation of the 3D Euler Equation with the Zero Flux on the Boundary

Under assumptions on smoothness of the initial velocity and the external body force, we prove that there exists T ₀ > 0, ν ₀ > 0 and a unique continuous family of strong solutions u ^ν (0 ≤ ν < ν ₀) of the Euler or Navier–Stokes initial-boundary value problem on the time interval (0, T ₀). In addition to the condition of the zero flux, the solutions of the Navier–Stokes equation satisfy certain natural boundary conditions imposed on curl u ^ν and curl ² u ^ν .      

Extensions to the Macroscopic Navier–Stokes Equation

A development is provided showing that for any phase, by not neglecting the macroscopic terms of the deviation from the intensive momentum and of the dispersive momentum, we obtain a macroscopic secondary momentum balance equation coupled with a macroscopic dominant momentum balance equation that is valid at a larger spatial scale. The macroscopic secondary momentum balance equation is in the form of a wave equation that propagates the deviation from the intensive momentum while concurrently, in the case of a Newtonian fluid and under certain assumptions, the macroscopic dominant momentum balance equation may be approximated by Darcys equation to address drag dominant flow. We then develop extensions to the dominant macroscopic Navier–Stokes (NS) equation for saturated porous matrices, to account for the pressure gradient at the microscopic solid-fluid interfaces. At the microscopic interfaces we introduce the exchange of inertia between the phases, accounting for the relative fluid square velocities and the rate of these velocities, interpreted as Forchheimer terms. Conditions are provided to approximate the extended dominant NS equation by Forchheimer quadratic momentum law or by Darcys linear momentum law. We also show that the dominant NS equation can conform into a nonlinear wave equation. The one-dimensional numerical solution of this nonlinear wave equation demonstrates good qualitative agreement with experiments for the case of a highly deformable elasto-plastic matrix.     

Homogenization and Enhancement for the G—Equation

We consider the so-called G-equation, a level set Hamilton–Jacobi equation used as a sharp interface model for flame propagation, perturbed by an oscillatory advection in a spatio-temporal periodic environment. Assuming that the advection has suitably small spatial divergence, we prove that, as the size of the oscillations diminishes, the solutions homogenize (average out) and converge to the solution of an effective anisotropic first-order (spatio-temporal homogeneous) level set equation. Moreover, we obtain a rate of convergence and show that, under certain conditions, the averaging enhances the velocity of the underlying front. We also prove that, at scale one, the level sets of the solutions of the oscillatory problem converge, at long times, to the Wulff shape associated with the effective Hamiltonian. Finally, we also consider advection depending on position at the integral scale.