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We construct solutions to the two dimensional parabolic-elliptic Keller-Segel model for chemotaxis that blow up in finite time $T$. The solution is decomposed as the sum of a stationary state concentrated at scale $\lambda$ and of a…
We consider the free boundary problem for the plasma-vacuum interface in ideal compressible magnetohydrodynamics (MHD). In the plasma region the flow is governed by the usual compressible MHD equations, while in the vacuum region we…
Numerical evidence for a finite-time singularity in ideal 3D magnetohydrodynamics (MHD) is presented. The simulations start from two interlocking magnetic flux rings with no initial velocity. The magnetic curvature force causes the flux…
For a general formulation of linearised hybrid inverse problems in impedance tomography, the qualitative properties of the solutions are analysed. Using an appropriate scalar pseudo-differential formulation, the problems are shown to permit…
Classical plasma with any degree degeneration of electronic gas is considered. In plasma two external electromagnetic field are propagation. It is required to find the plasma response on these fields. From kinetic Vlasov equation for…
We establish the local existence and uniqueness of multi-dimensional contact discontinuities for the ideal compressible magnetohydrodynamics (MHD) in Sobolev spaces, which are most typical interfacial waves for astrophysical plasmas and…
The Cauchy problem for a quasilinear system of hyperbolic equations describing plane one-dimensional relativistic oscillations of electrons in a cold plasma is considered. For some simplified formulation of the problem, a criterion for the…
We consider the linear stability problem for a 3D cylindrically symmetric equilibrium of the relativistic Vlasov-Maxwell system that describes a collisionless plasma. For an equilibrium whose distribution function decreases monotonically…
We review our recent work leading to steady-state solutions of the semiclassical (Maxwell-Bloch) equations of a laser. These are coupled non-linear partial differential equations in space and time which have previously been solved either by…
We construct splash-squeeze singularities for the free boundary ideal incompressible plasma-vacuum system, in which two arcs of the plasma boundary come together to form a smooth, glancing self-intersection. As the interface…
Context: We integrate the 2D MHD ideal equations of a straight slab to simulate observational results associated with fundamental sausage trapped modes. Aims: Starting from a non-equilibrium state with a dense chromospheric layer, we…
In nonrelativistic quantum mechanics the spontaneous generation of singularities in smooth and finite wave functions, is a well understood phenomenon also occurring for free particles. We use the familiar analogy between the two-dimensional…
We explore analytically the structure of relativistic shock and solitary wave solutions in collisionless plasmas. In the wave frame of reference, a cold plasma is flowing from one end and impacting on a low velocity plasma. First we show…
In this investigation we perform a systematic computational search for potential singularities in 3D Navier-Stokes flows based on the Ladyzhenskaya-Prodi-Serrin conditions. They assert that if the quantity $\int_0^T \| \mathbf{u}(t)…
We consider the free boundary problem for the plasma-vacuum interface in ideal compressible magnetohydrodynamics (MHD). In the plasma region, the flow is governed by the usual compressible MHD equations, while in the vacuum region we…
We consider a problem relating to magnetic confinement devices known as stellarators. Plasma is confined by magnetic fields generated by current-carrying coils, and here we investigate how closely to the plasma they need to be positioned.…
We show the absence of continuous symmetry breaking in 2D lattice systems without any smoothness assumptions on the interaction. We treat certain cases of interactions with integrable singularities. We also present cases of singular…
A didactic and systematic derivation of Noether point symmetries and conserved currents is put forward in special relativistic field theories, without a priori assumptions about the transformation laws. Given the Lagrangian density, the…
We present a new formulation based on the classical Dirichlet-Neumann formulation for interface coupling problems in linearized elasticity. By using Taylor series expansions, we derive a new set of interface conditions that allow our…
In this paper, using the similarity method, we construct particular solutions with singularities for degenerate high-order equations. The considered equations have singularities of the first and second kind. Particular solutions are…