This paper is concerned with the optimal boundary control of the Maxwell system. We consider a problem where the quadratic functional to be minimized penalizes the electromagnetic field at a given final time. Since the state is weighted in the energy space topology — a physically realistic choice —, the property that the optimal cost operator does satisfy the Riccati equation (RE) corresponding to the optimization problem is missed, just like in the case of other significant hyperbolic partial differential equations; however, we prove that this Riccati operator as well as the optimal solution can be recovered by means of approximating problems for which the optimal synthesis holds via proper differential Riccati equations. In the case of zero conductivity, an explicit representation of the optimal pair is valid which does not demand the well-posedness of the RE, instead.

In this paper, we investigate an abstract model associated with the Timoshenko system, incorporating fractional dissipative effects. The fractional dissipative effect is characterized by the powers of an arbitrary, strictly positive self-adjoint operator, with domain densely embedded in a Hilbert space. Our first main result demonstrates that the operator, derived by reformulating the abstract system as a first-order system, is the generator of a strongly continuous semigroup. The second main result is that the same semigroup can exhibit properties such as exponential stability, analyticity, or belong to a certain Gevrey class, with its precise behavior dependent on the assigned values of these powers. Furthermore, we establish that the Gevrey class obtained is sharp.

In this paper, we show the shock formation in the compressible Euler equations with time-dependent damping [Formula: see text] in three spatial dimensions without any symmetry conditions. We prove that the formation of shocks results from the collapse of the characteristic hypersurfaces and give a geometric description of the shock formation. As a shock forms, the first derivatives of the velocity and the density blow up. Furthermore, the lifespan of the solution [Formula: see text] is exponentially large (i.e. almost global existence) and can be computed explicitly. Due to the damping effect, the shock formation time [Formula: see text] will be shifted. This result, together with the author’s previous result [Z. Chen, Formation of shifted shock for the 3D compressible euler equations with damping, preprint (2022), arXiv:2210.13796.] provides a complete understanding of the shock formation mechanism for the 3D compressible Euler equations with damping [Formula: see text] for all [Formula: see text]. The methods in the paper can also be extended to show the shock formation to the Euler equations with general time-dependent damping.

The paper deals with the long-time behavior of an initial and boundary value problem for a Spherical Harmonics Expansion model coupled with the Poisson equation (SHE-Poisson system) associated with a well-prepared boundary. The long-time behavior of the one-dimensional SHE-Poisson system is studied by an entropy-entropy dissipation method for all scalar diffusion matrix [Formula: see text] with sub-critical and supercritical exponent [Formula: see text], followed by a strong convergence to the coupled equilibrium Gibbs–Poisson state. A rate of convergence is given; generalizing the result of [J. Haskovec, N. Masmoudi, C. Schmeiser and M. L. Tayeb, The spherical harmonics expansion model coupled to the Poisson equation, Kinet. Relat. Models 4 (2011) 1063–1079]. An extension of the long-time behavior to the multi-dimensional case in a weak [Formula: see text]-sense is proved using a relative entropy method.

We prove that the family of solutions to vanishing viscosity approximation for multidimensional scalar conservation laws with discontinuous non-aligned flux and zero initial data in the limit generates a singular measure supported along the discontinuity surface.

In this work, we consider a mathematical model of a viscoelastic incompressible fluid governed by the Navier–Stokes–Voigt equations in a three-dimensional thin domain [Formula: see text], with a damping term and Tresca friction law. First, we provide the problem statement and the weak variational formulation of the considered problem. We then study the asymptotic analysis of the problem when the dimension of the domain tend to zero. Through this analysis, we deduce the limit problem and explain the Reynolds equation that appears under these assumptions.

Peter D. Lax has passed away recently. His important contributions to partial differential equations, computational mathematics, and connection to science have benefitted us over the past decades. Several well-known mathematical terms are named after him, including the Lax Admissibility Condition for hyperbolic conservation laws, Lax Pairs for completely integrable systems, Lax Equivalence Theorem for numerical schemes, and Lax–Milgram Theorem in functional analysis. Peter’s unique sense of beauty and his philosophy of the universality of mathematics will continue to fascinate us. He is a member of the National Academy of Sciences, was awarded the National Medal of Science in 1986, the Wolf Prize in 1987, and the Abel Prize in 2005.

In this note, we study the well-posedness of two systems of Burgers type arising in nonlinear wave interactions. The first model describes the interaction of a Burger’s bore with the classical Korteweg–de Vries equation while the second exemplify the interaction of weak sound waves and entropy waves with small amplitudes. For the former, we show the local existence and uniqueness of solutions in Sobolev spaces and Wiener-type spaces. For the latter, we provide an elementary proof of finite time singularity.

We prove the linear orbital stability of spectrally stable stationary discrete shock profiles for conservative finite difference schemes applied to systems of conservation laws. The proof relies on an accurate description of the pointwise asymptotic behavior of the Green’s function associated with those discrete shock profiles, improving on the result of Lafitte-Godillon. The main novelty of this stability result is that it applies to a fairly large family of schemes that introduce some artificial possibly high-order viscosity. The result is obtained under a sharp spectral assumption rather than by imposing a smallness assumption on the shock amplitude.