In this paper, we study the Gevrey class regularity for solutions of the spatially homogeneous Landau equations in the hard potential case and the Maxwellian molecules case.
In this paper, we investigate the local well-posedness, ill-posedness, and Gevrey regularity of the Cauchy problem for Mildly Nonlinear and Non-Boussinesq case-(ABC) system. The local well-posedness of the solution fo...In this paper, we investigate the local well-posedness, ill-posedness, and Gevrey regularity of the Cauchy problem for Mildly Nonlinear and Non-Boussinesq case-(ABC) system. The local well-posedness of the solution for this system in Besov spaces B p,r s 1 × B p,r s with 1≤p,r≤∞ and s>max{ 1 1 p , 3 2 } was firstly established. Next, we consider the continuity of the solution-to-data map, i.e. the ill-posedness of the solution for this system in Besov space B p,∞ s 1 × B p,∞ s was derived. Finally, the Gevrey regularity of the system was presented.展开更多
In this paper, we show the existence and regularity of mild solutions depending on the small initial data in Besov spaces to the fractional porous medium equation. When 1 < <em>α</em> ≤ 2, we prove gl...In this paper, we show the existence and regularity of mild solutions depending on the small initial data in Besov spaces to the fractional porous medium equation. When 1 < <em>α</em> ≤ 2, we prove global well-posedness for initial data <img src="Edit_b7b43d4c-00d8-49d6-9066-97151fb5c337.bmp" alt="" /> with 1 ≤ <em>p</em> < ∞, 1 ≤ <em>q</em> ≤ ∞, and analyticity of solutions with 1 < <em>p</em> < ∞, 1 ≤ <em>q</em> ≤ ∞. In particular, we also proved that when <em>α</em> = 1, both <em>u</em> and <img src="Edit_a5af0853-8adc-4a08-b8a2-b9a70ea0f409.bmp" alt="" /> belong to <img src="Edit_03a932cc-aa58-4568-83ad-f16416cc7b71.bmp" alt="" />. We solve this equation through the contraction mapping method based on Littlewood-Paley theory and Fourier multiplier. Furthermore, we can get time decay estimates of global solutions in Besov spaces, which is <img src="Edit_083986e9-4e1c-4494-ac5d-a7d30a12df97.bmp" alt="" /> as <em>t</em> → ∞.展开更多
In this paper, we study the existence and long-time behaviour of the solutions for the multidimensional Kolmogorov-Spiegel-Sivashinsky equation. We first show the existence of the global solution for this equation, an...In this paper, we study the existence and long-time behaviour of the solutions for the multidimensional Kolmogorov-Spiegel-Sivashinsky equation. We first show the existence of the global solution for this equation, and then prove the existence of the global attractor and establish the esti- mates of the upper bounds of Hausdorff and fractal dimensions for the attractor. We also obtain the Gevrey class regularity for the solutions and construct an approximate inertial manifold for the system.展开更多
The Navier-Stokes-α equations subject to the periodic boundary conditions are considered. Analyticity in time for a class of solutions taking values in a Gevrey class of functions is proven. Exponential decay of the ...The Navier-Stokes-α equations subject to the periodic boundary conditions are considered. Analyticity in time for a class of solutions taking values in a Gevrey class of functions is proven. Exponential decay of the spatial Fourier spectrum for the analytic solutions and the lower bounds on the rate defined by the exponential decay are also obtained.展开更多
The authors show the Gevrey class regularity of the solutions for the two-dimensional Newton-Boussinesq Equations. Based on this fact, an approximate inertial manifold for the system is constructed, which attracts ...The authors show the Gevrey class regularity of the solutions for the two-dimensional Newton-Boussinesq Equations. Based on this fact, an approximate inertial manifold for the system is constructed, which attracts all solutions to an exponentially thin neighborhood of it in a finite time.展开更多
In this work we prove the weighted Gevrey regularity of solutions to the incompressible Euler equation with initial data decaying polynomially at infinity. This is motivated by the well-posedness problem of vertical b...In this work we prove the weighted Gevrey regularity of solutions to the incompressible Euler equation with initial data decaying polynomially at infinity. This is motivated by the well-posedness problem of vertical boundary layer equation for fast rotating fluid. The method presented here is based on the basic weighted L;-estimate, and the main difficulty arises from the estimate on the pressure term due to the appearance of weight function.展开更多
文摘In this paper, we study the Gevrey class regularity for solutions of the spatially homogeneous Landau equations in the hard potential case and the Maxwellian molecules case.
文摘In this paper, we investigate the local well-posedness, ill-posedness, and Gevrey regularity of the Cauchy problem for Mildly Nonlinear and Non-Boussinesq case-(ABC) system. The local well-posedness of the solution for this system in Besov spaces B p,r s 1 × B p,r s with 1≤p,r≤∞ and s>max{ 1 1 p , 3 2 } was firstly established. Next, we consider the continuity of the solution-to-data map, i.e. the ill-posedness of the solution for this system in Besov space B p,∞ s 1 × B p,∞ s was derived. Finally, the Gevrey regularity of the system was presented.
文摘In this paper, we show the existence and regularity of mild solutions depending on the small initial data in Besov spaces to the fractional porous medium equation. When 1 < <em>α</em> ≤ 2, we prove global well-posedness for initial data <img src="Edit_b7b43d4c-00d8-49d6-9066-97151fb5c337.bmp" alt="" /> with 1 ≤ <em>p</em> < ∞, 1 ≤ <em>q</em> ≤ ∞, and analyticity of solutions with 1 < <em>p</em> < ∞, 1 ≤ <em>q</em> ≤ ∞. In particular, we also proved that when <em>α</em> = 1, both <em>u</em> and <img src="Edit_a5af0853-8adc-4a08-b8a2-b9a70ea0f409.bmp" alt="" /> belong to <img src="Edit_03a932cc-aa58-4568-83ad-f16416cc7b71.bmp" alt="" />. We solve this equation through the contraction mapping method based on Littlewood-Paley theory and Fourier multiplier. Furthermore, we can get time decay estimates of global solutions in Besov spaces, which is <img src="Edit_083986e9-4e1c-4494-ac5d-a7d30a12df97.bmp" alt="" /> as <em>t</em> → ∞.
文摘In this paper, we study the existence and long-time behaviour of the solutions for the multidimensional Kolmogorov-Spiegel-Sivashinsky equation. We first show the existence of the global solution for this equation, and then prove the existence of the global attractor and establish the esti- mates of the upper bounds of Hausdorff and fractal dimensions for the attractor. We also obtain the Gevrey class regularity for the solutions and construct an approximate inertial manifold for the system.
文摘The Navier-Stokes-α equations subject to the periodic boundary conditions are considered. Analyticity in time for a class of solutions taking values in a Gevrey class of functions is proven. Exponential decay of the spatial Fourier spectrum for the analytic solutions and the lower bounds on the rate defined by the exponential decay are also obtained.
文摘The authors show the Gevrey class regularity of the solutions for the two-dimensional Newton-Boussinesq Equations. Based on this fact, an approximate inertial manifold for the system is constructed, which attracts all solutions to an exponentially thin neighborhood of it in a finite time.
基金supported by NSF of China(11422106)the NSF of China(11171261)+1 种基金Fok Ying Tung Education Foundation(151001)supported by“Fundamental Research Funds for the Central Universities”
文摘In this work we prove the weighted Gevrey regularity of solutions to the incompressible Euler equation with initial data decaying polynomially at infinity. This is motivated by the well-posedness problem of vertical boundary layer equation for fast rotating fluid. The method presented here is based on the basic weighted L;-estimate, and the main difficulty arises from the estimate on the pressure term due to the appearance of weight function.
