Perturbation of essential spectra of evolution operators and the Vlasov-Poisson-Boltzmann system
Robert T. Glassey Walter A. Strauss
Discrete & Continuous Dynamical Systems - A 1999, 5(3): 457-472 doi: 10.3934/dcds.1999.5.457
Consider a propagator defined on a Banach space whose norm satisfies an appropriate exponential bound. To this operator is added a bounded operator which is relatively smoothing in the sense of Vidav. The location of the essential spectrum of the perturbed propagator is then estimated. An application to kinetic theory is given for a system of particles that interact both through collisions and through their charges.
keywords: rare gases stability Vlasov–Poisson–Boltzmann System plasmas perturbation of propagators spectral theory.
Vorticity jumps in steady water waves
Walter A. Strauss
Discrete & Continuous Dynamical Systems - B 2012, 17(4): 1101-1112 doi: 10.3934/dcdsb.2012.17.1101
There exists a large family of water waves with jump discontinuities in the vorticity. These waves travel at a constant speed. They are two-dimensional, periodic, symmetric, and subject to the influence of gravity. Some of them have large amplitudes. Their existence is proven using local and global bifurcation theory, together with elliptic theory of weak solutions with nonlinear boundary conditions.
keywords: vorticity nonlinear elliptic. Water waves
Existence and blow up of small amplitude nonlinear waves with a negative potential
Walter A. Strauss Kimitoshi Tsutaya
Discrete & Continuous Dynamical Systems - A 1997, 3(2): 175-188 doi: 10.3934/dcds.1997.3.175
Consider a nonlinear wave equation in three space dimensions with zero mass together with a negative potential. If the potential is sufficiently short-range, then it does not alter the global existence of small-amplitude solutions. On the other hand, if the potential is sufficiently large, it will force some solutions to blow up in a finite time.
keywords: Nonlinear wave equation negative potential.
Analyticity of the nonlinear scattering operator
Benoît Pausader Walter A. Strauss
Discrete & Continuous Dynamical Systems - A 2009, 25(2): 617-626 doi: 10.3934/dcds.2009.25.617
We present a new and simpler proof that the nonlinear scattering operator $\S$ is analytic on energy space. We apply it in particular to a fourth-order nonlinear wave equation in Rn. In addition, we prove that $\S$ determines the scatterer uniquely and that for small powers there is no scattering.
keywords: NLS nonlinear beam equation analytic operator. Nonlinear scattering NLKG
Regular solutions of the Vlasov-Poisson-Fokker-Planck system
Kosuke Ono Walter A. Strauss
Discrete & Continuous Dynamical Systems - A 2000, 6(4): 751-772 doi: 10.3934/dcds.2000.6.751
We study the Vlasov-Poisson-Fokker-Planck system. For arbitrary data we prove the global well-posedness and gain of regularity of solutions under improved assumptions. We also prove that if the initial data are sufficiently small, the solutions satisfy optimal rates of asymptotic decay.
keywords: Fokker-Planck equation regularity asymptotic behavior. Kinetic theory Vlasov-Poisson-Fokker-Planck system plasma physics
Global magnetic confinement for the 1.5D Vlasov-Maxwell system
Toan T. Nguyen Truyen V. Nguyen Walter A. Strauss
Kinetic & Related Models 2015, 8(1): 153-168 doi: 10.3934/krm.2015.8.153
We establish the global-in-time existence and uniqueness of classical solutions to the ``one and one-half'' dimensional relativistic Vlasov--Maxwell systems in a bounded interval, subject to an external magnetic field which is infinitely large at the spatial boundary. We prove that the large external magnetic field confines the particles to a compact set away from the boundary. This excludes the known singularities that typically occur due to particles that repeatedly bounce off the boundary. In addition to the confinement, we follow the techniques introduced by Glassey and Schaeffer, who studied the Cauchy problem without boundaries.
keywords: magnetic confinement global smooth solution Vlasov-Maxwell system boundary conditions.
Erratum to: Global magnetic confinement for the 1.5D Vlasov-Maxwell system
Toan T. Nguyen Truyen V. Nguyen Walter A. Strauss
Kinetic & Related Models 2015, 8(3): 615-616 doi: 10.3934/krm.2015.8.615
keywords: global smooth solution boundary conditions. magnetic confinement Vlasov-Maxwell system

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