Journal of Hyperbolic Differential Equations

In this paper we prove the uniform-in-time $L^2$ convergence for the Fourier–Galerkin approximation to Yudovich solutions of the 2D Euler equations. Precisely, we show that both the approximating velocity and the approximating vorticity converge strongly in $C(L^2)$. Moreover, for the convergence of the velocity we provide an explicit rate of convergence. The proofs are based on a relative entropy approach and the Osgood lemma. Related results under different assumptions on the vorticity are also proved.

This paper intends to provide a brief review of the current well-posedness theory for hyperbolic systems of conservation laws in one space dimension, also pointing out open problems and possible research directions.

In this note, we review and expand upon the material presented in the minicourse titled “Instability and Nonuniqueness in Fluid Dynamics,” which the author held during the “Meeting on Nonlinear Evolution PDEs, Fluid Dynamics, and Transport Equations” at the Ettore Majorana Centre in Erice in May 2023.

We consider a continuous solution $u$ of the balance law

This short note provides explicit solutions to the linearized Boussinesq equations around the stably stratified Couette flow posed on $𝕋\times ℝ$. We consider the long-time behavior of such solutions and prove pointwise inviscid damping of the perturbed density and velocity field for any positive Richardson number, with optimal rates. The explicit solution is obtained through the limiting absorption principle whereas the inviscid damping is proved using oscillatory integral methods.

We consider a class of nonlocal conservation laws with exponential kernel and prove that quantities involving the nonlocal term $W:={𝟙}_{(-\infty ,0]}(\cdot )exp(\cdot )\ast \rho$ satisfy an Oleĭnik-type entropy condition. More precisely, under different sets of assumptions on the velocity function $V$, we prove that $W$ satisfies a one-sided Lipschitz condition and that $V^{\prime }(W)W{\partial }_{x}W$ satisfies a one-sided bound, respectively. As a byproduct, we deduce that, as the exponential kernel is rescaled to converge to a Dirac delta distribution, the weak solution of the nonlocal problem converges to the unique entropy-admissible solution of the corresponding local conservation law, under the only assumption that the initial datum is essentially bounded and not necessarily of bounded variation.

We prove the existence of large amplitude bi-periodic traveling waves (stationary in a moving frame) of the two-dimensional non-resistive Magnetohydrodynamics (MHD) system with a traveling wave external force with large velocity speed $\lambda ({\omega }_1,{\omega }_2)$ and of amplitude of order $O({\lambda }^{1^{+}})$ where $\lambda \gg 1$ is a large parameter. For most values of $\omega =({\omega }_1,{\omega }_2)$ and for $\lambda \gg 1$ large enough, we construct bi-periodic traveling wave solutions of arbitrarily large amplitude as $\lambda \to +\infty$. More precisely, we show that the velocity field is of order $O({\lambda }^{0^{+}})$, whereas the magnetic field is close to a constant vector as $\lambda \to +\infty$. Due to the presence of small divisors, the proof is based on a nonlinear Nash–Moser scheme adapted to construct nonlinear waves of large amplitude. The main difficulty is that the linearized equation at any approximate solution is an unbounded perturbation of large size of a diagonal operator and hence the problem is not perturbative. The invertibility of the linearized operator is then performed by using tools from micro-local analysis and normal forms together with a sharp analysis of high- and low-frequency regimes with respect to the large parameter $\lambda \gg 1$. To the best of our knowledge, this is the first result in which global in time, large amplitude solutions are constructed for the 2D non-resistive MHD system with periodic boundary conditions and also the first existence result of large amplitude quasi-periodic solutions for a nonlinear PDE in higher space dimension.

We consider the two-dimensional (2D) Magneto-HydroDynamics (MHD) equations describing the evolution of a viscous incompressible electrically conducting fluid. In this paper, adopting the vorticity-current formulation and assuming that the initial fluid vorticity and magnetic current are in $L^1({ℝ}^2)$, we prove the local in-time existence and regularity of the solutions.

In this paper, we study Majda’s example for stability of mixed problems introduced in [A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences, Vol. 53 (Springer Verlag, 1984)]. After some transformation we analyze the stability of the problem by computing the roots of the Kreiss–Lopatinskiĭ determinant and recover the different cases as in Majda [Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences, Vol. 53 (Springer Verlag, 1984)]. Then, we focus on the weakly stable case and prove the a priori estimate of the solution. The proof follows by adapting the approach of Coulombel and Secchi [The stability of compressible vortex sheets in two space dimensions, Indiana Univ. Math. J. 53 (2004) 941–1012] for the linear stability of 2D compressible vortex sheets with the simplification introduced by Chen et al. [Linear stability of compressible vortex sheets in two-dimensional elastodynamics, Adv. Math. 311 (2017) 18–60]. Compared to [J.-F. Coulombel and P. Secchi, The stability of compressible vortex sheets in two space dimensions, Indiana Univ. Math. J. 53 (2004) 941–1012], we improve the a priori energy estimate in that the solution is estimated in suitable weighted Sobolev spaces, anisotropic in the frequency space, whose definition reflects the properties of the associated Lopatinskiĭ determinant. Moreover, we show that this a priori energy estimate is optimal. This very simple example appears useful for the comprehension of the method of proof of the a priori estimate, beyond the technicalities of [R. M. Chen, J. Hu and D. Wang, Linear stability of compressible vortex sheets in two-dimensional elastodynamics, Adv. Math. 311 (2017) 18–60; J.-F. Coulombel and P. Secchi, The stability of compressible vortex sheets in two space dimensions, Indiana Univ. Math. J. 53 (2004) 941–1012].

In the first part of this paper we present some exact solutions for special hyperbolic-parabolic systems with sustained oscillations induced by the initial data, most notably the compressible Navier–Stokes system with non-monotone pressure. This part complements [A. E. Tzavaras, Sustained oscillations in hyperbolic-parabolic systems, Arch. Ration. Mech. Anal. 248 (2024) 51] where such examples are extensively studied. The second part deals with the problem of homogenization for one-dimensional models describing phase transitions for viscoelastic materials. Ideas from the kinetic formulation of conservation laws are employed to derive effective equations that describe the propagation of oscillations.