Browsing by Author "Kiselev, A"
Now showing 1 - 7 of 7
- Results Per Page
- Sort Options
Item Open Access CHEMOTACTIC REACTION ENHANCEMENT IN ONE DIMENSION(Communications in Mathematical Sciences, 2024-01-01) Gong, Y; Kiselev, AChemotaxis, which involves the directed movement of cells in response to a chemical gradient, plays a crucial role in a broad variety of biological processes. Examples include bacterial motion, the development of single-cell or multicellular organisms, and immune responses. Chemotaxis directs bacteria’s movement to find food (e.g., glucose) by swimming toward the highest concentration of food molecules. In multicellular organisms, chemotaxis is critical to early development (e.g., movement of sperm towards the egg during fertilization). Chemotaxis also helps mobilize phagocytic and immune cells at sites of infection, tissue injury, and thus facilitates immune reactions. In this paper, we study a PDE system that describes chemotactic processes in one dimension, which may correspond to a thin channel, the setting relevant in many applications: for example, spermatozoa progression to the ovum inside a Fallopian tube or immune response in a blood vessel. Our objective is to obtain qualitatively precise estimates on how chemotaxis improves reaction efficiency, when compared to purely diffusive situation. The techniques we use to achieve this goal include a variety of comparison principles and analysis of mass transport for a class of Fokker-Planck operators.Item Open Access Global regularity for 1D Eulerian dynamics with singular interaction forces(2017-12-18) Kiselev, A; Tan, CThe Euler-Poisson-Alignment (EPA) system appears in mathematical biology and is used to model, in a hydrodynamic limit, a set agents interacting through mutual attraction/repulsion as well as alignment forces. We consider one-dimensional EPA system with a class of singular alignment terms as well as natural attraction/repulsion terms. The singularity of the alignment kernel produces an interesting effect regularizing the solutions of the equation and leading to global regularity for wide range of initial data. This was recently observed in the paper by Do, Kiselev, Ryzhik and Tan. Our goal in this paper is to generalize the result and to incorporate the attractive/repulsive potential. We prove that global regularity persists for these more general models.Item Open Access Global Regularity for the Fractional Euler Alignment System(Archive for Rational Mechanics and Analysis, 2017-10-22) Do, T; Kiselev, A; Ryzhik, L; Tan, C© 2017 Springer-Verlag GmbH Germany We study a pressureless Euler system with a non-linear density-dependent alignment term, originating in the Cucker–Smale swarming models. The alignment term is dissipative in the sense that it tends to equilibrate the velocities. Its density dependence is natural: the alignment rate increases in the areas of high density due to species discomfort. The diffusive term has the order of a fractional Laplacian (Formula presented.). The corresponding Burgers equation with a linear dissipation of this type develops shocks in a finite time. We show that the alignment nonlinearity enhances the dissipation, and the solutions are globally regular for all (Formula presented.). To the best of our knowledge, this is the first example of such regularization due to the non-local nonlinear modulation of dissipation.Item Open Access Hitting time of Brownian motion subject to shear flow(Involve, 2022-01-01) Chouliara, D; Gong, Y; He, S; Kiselev, A; Lim, J; Melikechi, O; Powers, KThe 2-dimensional motion of a particle subject to Brownian motion and ambient shear flow transportation is considered. Numerical experiments are carried out to explore the relation between the shear strength, box size, and the particle’s expected first hitting time of a given target. The simulation is motivated by biological settings such as reproduction processes and the workings of the immune system. As the shear strength grows, the expected first hitting time converges to the expected first hitting time of the 1-dimensional Brownian motion. The dependence of the hitting time on the shearing rate is monotone, and only the form of the shear flow close to the target appears to play a role. Numerical experiments also show that the expected hitting time drops significantly even for quite small values of shear rate near the target.Item Open Access Illposedness of C2 Vortex Patches(Archive for Rational Mechanics and Analysis, 2023-06-01) Kiselev, A; Luo, XIt is well known that vortex patches are wellposed in C1,α if 0 < α< 1 . In this paper, we prove the illposedness of C2 vortex patches. The setup is to consider the vortex patches in Sobolev spaces W2,p where the curvature of the boundary is Lp integrable. In this setting, we show the persistence of W2,p regularity when 1 < p< ∞ and construct C2 initial patch data for which the curvature of the patch boundary becomes unbounded immediately for t> 0 , though it regains C2 regularity precisely at all integer times without being time periodic. The key ingredient is the evolution equation for the curvature, the dominant term in which turns out to be linear and dispersive.Item Open Access Suppression of chemotactic blowup by strong buoyancy in Stokes-Boussinesq flow with cold boundary(Journal of Functional Analysis, 2024-10-01) Hu, Z; Kiselev, AIn this paper, we show that the Keller-Segel equation equipped with zero Dirichlet Boundary condition and actively coupled to a Stokes-Boussinesq flow is globally well-posed provided that the coupling is sufficiently large. We will in fact show that the dynamics is quenched after certain time. In particular, such active coupling is blowup-suppressing in the sense that it enforces global regularity for some initial data leading to a finite-time singularity when the flow is absent.Item Open Access The α-SQG patch problem is illposed in C2,β and W2,p(Communications on Pure and Applied Mathematics, 2024-01-01) Kiselev, A; Luo, XWe consider the patch problem for the α-(surface quasi-geostrophic) SQG system with the values α = 0 and (Formula presented.) being the 2D Euler and the SQG equations respectively. It is well-known that the Euler patches are globally wellposed in non-endpoint C2,β Hölder spaces, as well as in W2,p, (Formula presented.) spaces. In stark contrast to the Euler case, we prove that for (Formula presented.), the (Formula presented.) -SQG patch problem is strongly illposed in every (Formula presented.) Hölder space with (Formula presented.). Moreover, in a suitable range of regularity, the same strong illposedness holds for every W2,p Sobolev space unless p = 2.