The Experts below are selected from a list of 39369 Experts worldwide ranked by ideXlab platform
Ki-ahm Lee - One of the best experts on this subject based on the ideXlab platform.
-
Parabolic Harnack inequality of viscosity solutions on Riemannian manifolds
Journal of Functional Analysis, 2014Co-Authors: Soojung Kim, Ki-ahm LeeAbstract:Abstract We consider viscosity solutions to nonlinear uniformly parabolic equations in nondivergence form on a Riemannian manifold M with the sectional curvature bounded from below by − κ for κ ≥ 0 . In the Elliptic Case, Wang and Zhang [24] recently extended the results of [5] to nonlinear Elliptic equations in nondivergence form on such M , where they obtained the Harnack inequality for classical solutions. We establish the Harnack inequality for nonnegative viscosity solutions to nonlinear uniformly parabolic equations in nondivergence form on M . The Harnack inequality of nonnegative viscosity solutions to the Elliptic equations is also proved.
-
Parabolic Harnack inequality of viscosity solutions on Riemannian manifolds
arXiv: Analysis of PDEs, 2013Co-Authors: Soojung Kim, Ki-ahm LeeAbstract:We consider viscosity solutions to nonlinear uniformly parabolic equations in nondivergence form on a Riemannian manifold $M$, with the sectional curvature bounded from below by $-\kappa$ for $\kappa\geq 0$. In the Elliptic Case, Wang and Zhang \cite{WZ} recently extended the results of \cite{Ca} to nonlinear Elliptic equations in nondivergence form on such $M$, where they obtained the Harnack inequality for classical solutions. We establish the Harnack inequality for nonnegative {\it viscosity solutions} to nonlinear uniformly {\it parabolic equations} in nondivergence form on $M$. The Harnack inequality of nonnegative viscosity solutions to the Elliptic equations is also proved.
Stéphane Mischler - One of the best experts on this subject based on the ideXlab platform.
-
uniqueness and long time asymptotic for the keller segel equation the parabolic Elliptic Case
Archive for Rational Mechanics and Analysis, 2016Co-Authors: Giani Egana Fernandez, Stéphane MischlerAbstract:The present paper deals with the parabolic–Elliptic Keller–Segel equation in the plane in the general framework of weak (or “free energy”) solutions associated to initial datum with finite mass M, finite second moment and finite entropy. The aim of the paper is threefold: (1) We prove the uniqueness of the “free energy” solution on the maximal interval of existence [0,T*) with T* = ∞ in the Case when M ≦ 8π and T* 8π. The proof uses a DiPerna–Lions renormalizing argument which makes it possible to get the “optimal regularity” as well as an estimate of the difference of two possible solutions in the critical L4/3 Lebesgue norm similarly to the 2d vorticity Navier–Stokes equation. (2) We prove the immediate smoothing effect and, in the Case M < 8π, we prove the Sobolev norm bound uniformly in time for the rescaled solution (corresponding to the self-similar variables). (3) In the Case M < 8π, we also prove the weighted L4/3 linearized stability of the self-similar profile and then the universal optimal rate of convergence of the solution to the self-similar profile. The proof is mainly based on an argument of enlargement of the functional space for semigroup spectral gap.
-
Uniqueness and Long Time Asymptotic for the Keller–Segel Equation: The Parabolic–Elliptic Case
Archive for Rational Mechanics and Analysis, 2015Co-Authors: Egaña Fernández, Stéphane MischlerAbstract:The present paper deals with the parabolic–Elliptic Keller–Segel equation in the plane in the general framework of weak (or “free energy”) solutions associated to initial datum with finite mass M, finite second moment and finite entropy. The aim of the paper is threefold: (1) We prove the uniqueness of the “free energy” solution on the maximal interval of existence [0,T*) with T* = ∞ in the Case when M ≦ 8π and T* 8π. The proof uses a DiPerna–Lions renormalizing argument which makes it possible to get the “optimal regularity” as well as an estimate of the difference of two possible solutions in the critical L4/3 Lebesgue norm similarly to the 2d vorticity Navier–Stokes equation. (2) We prove the immediate smoothing effect and, in the Case M < 8π, we prove the Sobolev norm bound uniformly in time for the rescaled solution (corresponding to the self-similar variables). (3) In the Case M < 8π, we also prove the weighted L4/3 linearized stability of the self-similar profile and then the universal optimal rate of convergence of the solution to the self-similar profile. The proof is mainly based on an argument of enlargement of the functional space for semigroup spectral gap.
-
Uniqueness and long time asymptotic for the Keller-Segel equation: The parabolic-Elliptic Case
arXiv: Analysis of PDEs, 2013Co-Authors: Fernandez Giani Egana, Stéphane MischlerAbstract:The present paper deals with the parabolic-Elliptic Keller-Segel equation in the plane in the general framework of weak (or ''free energy") solutions associated to initial datum with finite mass $M$, finite second moment and finite entropy. The aim of the paper is threefold: (1) We prove the uniqueness of the ''free energy" solution on the maximal interval of existence $[0,T^*)$ with $T^*=\infty$ in the Case when $M\le8\pi$ and $T^* 8\pi$. The proof uses a DiPerna-Lions renormalizing argument which makes possible to get the ''optimal regularity" as well as an estimate of the difference of two possible solutions in the critical $L^{4/3}$ Lebesgue norm similarly as for the $2d$ vorticity Navier-Stokes equation. (2) We prove immediate smoothing effect and, in the Case $M < 8\pi$, we prove Sobolev norm bound uniformly in time for the rescaled solution (corresponding to the self-similar variables). (3) In the Case $M < 8\pi$, we also prove weighted $L^{4/3}$ linearized stability of the self-similar profile and then universal optimal rate of convergence of the solution to the self-similar profile. The proof is mainly based on an argument of enlargement of the functional space for semigroup spectral gap.
K. Seddighi - One of the best experts on this subject based on the ideXlab platform.
-
Two Parameter Asymptotic Spectra in the Uniformly Elliptic Case
Results in Mathematics, 1997Co-Authors: P. A. Binding, P. J. Browne, K. SeddighiAbstract:In this article we study the abstract two parameter eigenvalue problem % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC % vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz % ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbb % L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe % pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam % aaeaqbaaGceaqabeaacqWGubavdaWgaaWcbaGaeGymaedabeaakiab % dwha1naaBaaaleaacqaIXaqmaeqaaOGaeyypa0ZaaeWaaeaarmqr1n % gBPrgitLxBI9gBaGGbaiab-T7aSnaaBaaaleaacqWFXaqmaeqaaOGa % emOvay1aaSbaaSqaaiabigdaXiabigdaXaqabaGccqGHRaWkcqWF7o % aBdaWgaaWcbaGae8NmaidabeaakiabdAfawnaaBaaaleaacqaIXaqm % cqaIYaGmaeqaaaGccaGLOaGaayzkaaGaemyDau3aaSbaaSqaaiabig % daXaqabaGccqGGSaaltCvAUfeBSn0BKvguHDwzZbqehiuy0fMBNbac % haGaa4hiaiaa+bcacaGFGaWaauWaaeaacqWG1bqDdaWgaaWcbaGaeG % ymaedabeaaaOGaayzcSlaawQa7aiabg2da9iabigdaXaqaaiabdsfa % unaaBaaaleaacqaIYaGmaeqaaOGaemyDau3aaSbaaSqaaiabikdaYa % qabaGccqGH9aqpdaqadaqaaiab-T7aSnaaBaaaleaacqWFXaqmaeqa % aOGaemOvay1aaSbaaSqaaiabikdaYiabigdaXaqabaGccqGHRaWkcq % WF7oaBdaWgaaWcbaGae8NmaidabeaakiabdAfawnaaBaaaleaacqaI % YaGmcqaIYaGmaeqaaaGccaGLOaGaayzkaaGaemyDau3aaSbaaSqaai % abikdaYaqabaGccqGGSaalcaGFGaGaa4hiamaafmaabaGaemyDau3a % aSbaaSqaaiabikdaYaqabaaakiaawMa7caGLkWoacqGH9aqpcqaIXa % qmaaaa!8AC9! $$\begin{gathered} T_1 u_1 = \left( {\lambda _1 V_{11} + \lambda _2 V_{12} } \right)u_1 , \left\| {u_1 } \right\| = 1 \hfill \\ T_2 u_2 = \left( {\lambda _1 V_{21} + \lambda _2 V_{22} } \right)u_2 , \left\| {u_2 } \right\| = 1 \hfill \\ \end{gathered}$$ where, in the Hilbert spaces H_j, T_j is self-adjoint, bounded below and has compact resolvent, and V_jk are self-adjoint bounded operators, (−1)^j+kV_jk >> 0, j, k = 1, 2. An eigenvalue λ for this problem is a point in R^2 satisfying both equations. Under appropriate conditions, the eigenvalues λ^n = (λ_1 ^n, λ_2 ^n) are countable and in R^2. We aim to describe the set of limit points of λ^n/∥λ^n∥, as ∥λ^n∥ → ∞, in terms of the V_jk.
-
Two Parameter Asymptotic Spectra in the Uniformly Elliptic Case
Results in Mathematics, 1997Co-Authors: P. A. Binding, P. J. Browne, K. SeddighiAbstract:In this article we study the abstract two parameter eigenvalue problem $$\begin{gathered} T_1 u_1 = \left( {\lambda _1 V_{11} + \lambda _2 V_{12} } \right)u_1 , \left\| {u_1 } \right\| = 1 \hfill \\ T_2 u_2 = \left( {\lambda _1 V_{21} + \lambda _2 V_{22} } \right)u_2 , \left\| {u_2 } \right\| = 1 \hfill \\ \end{gathered}$$ where, in the Hilbert spaces Hj, Tj is self-adjoint, bounded below and has compact resolvent, and Vjk are self-adjoint bounded operators, (−1)j+kVjk >> 0, j, k = 1, 2. An eigenvalue λ for this problem is a point in R2 satisfying both equations. Under appropriate conditions, the eigenvalues λn = (λ1 n, λ2 n) are countable and in R2. We aim to describe the set of limit points of λn/∥λn∥, as ∥λn∥ → ∞, in terms of the Vjk.
Soojung Kim - One of the best experts on this subject based on the ideXlab platform.
-
Parabolic Harnack inequality of viscosity solutions on Riemannian manifolds
Journal of Functional Analysis, 2014Co-Authors: Soojung Kim, Ki-ahm LeeAbstract:Abstract We consider viscosity solutions to nonlinear uniformly parabolic equations in nondivergence form on a Riemannian manifold M with the sectional curvature bounded from below by − κ for κ ≥ 0 . In the Elliptic Case, Wang and Zhang [24] recently extended the results of [5] to nonlinear Elliptic equations in nondivergence form on such M , where they obtained the Harnack inequality for classical solutions. We establish the Harnack inequality for nonnegative viscosity solutions to nonlinear uniformly parabolic equations in nondivergence form on M . The Harnack inequality of nonnegative viscosity solutions to the Elliptic equations is also proved.
-
Parabolic Harnack inequality of viscosity solutions on Riemannian manifolds
arXiv: Analysis of PDEs, 2013Co-Authors: Soojung Kim, Ki-ahm LeeAbstract:We consider viscosity solutions to nonlinear uniformly parabolic equations in nondivergence form on a Riemannian manifold $M$, with the sectional curvature bounded from below by $-\kappa$ for $\kappa\geq 0$. In the Elliptic Case, Wang and Zhang \cite{WZ} recently extended the results of \cite{Ca} to nonlinear Elliptic equations in nondivergence form on such $M$, where they obtained the Harnack inequality for classical solutions. We establish the Harnack inequality for nonnegative {\it viscosity solutions} to nonlinear uniformly {\it parabolic equations} in nondivergence form on $M$. The Harnack inequality of nonnegative viscosity solutions to the Elliptic equations is also proved.
P. A. Binding - One of the best experts on this subject based on the ideXlab platform.
-
Two Parameter Asymptotic Spectra in the Uniformly Elliptic Case
Results in Mathematics, 1997Co-Authors: P. A. Binding, P. J. Browne, K. SeddighiAbstract:In this article we study the abstract two parameter eigenvalue problem % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC % vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz % ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbb % L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe % pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam % aaeaqbaaGceaqabeaacqWGubavdaWgaaWcbaGaeGymaedabeaakiab % dwha1naaBaaaleaacqaIXaqmaeqaaOGaeyypa0ZaaeWaaeaarmqr1n % gBPrgitLxBI9gBaGGbaiab-T7aSnaaBaaaleaacqWFXaqmaeqaaOGa % emOvay1aaSbaaSqaaiabigdaXiabigdaXaqabaGccqGHRaWkcqWF7o % aBdaWgaaWcbaGae8NmaidabeaakiabdAfawnaaBaaaleaacqaIXaqm % cqaIYaGmaeqaaaGccaGLOaGaayzkaaGaemyDau3aaSbaaSqaaiabig % daXaqabaGccqGGSaaltCvAUfeBSn0BKvguHDwzZbqehiuy0fMBNbac % haGaa4hiaiaa+bcacaGFGaWaauWaaeaacqWG1bqDdaWgaaWcbaGaeG % ymaedabeaaaOGaayzcSlaawQa7aiabg2da9iabigdaXaqaaiabdsfa % unaaBaaaleaacqaIYaGmaeqaaOGaemyDau3aaSbaaSqaaiabikdaYa % qabaGccqGH9aqpdaqadaqaaiab-T7aSnaaBaaaleaacqWFXaqmaeqa % aOGaemOvay1aaSbaaSqaaiabikdaYiabigdaXaqabaGccqGHRaWkcq % WF7oaBdaWgaaWcbaGae8NmaidabeaakiabdAfawnaaBaaaleaacqaI % YaGmcqaIYaGmaeqaaaGccaGLOaGaayzkaaGaemyDau3aaSbaaSqaai % abikdaYaqabaGccqGGSaalcaGFGaGaa4hiamaafmaabaGaemyDau3a % aSbaaSqaaiabikdaYaqabaaakiaawMa7caGLkWoacqGH9aqpcqaIXa % qmaaaa!8AC9! $$\begin{gathered} T_1 u_1 = \left( {\lambda _1 V_{11} + \lambda _2 V_{12} } \right)u_1 , \left\| {u_1 } \right\| = 1 \hfill \\ T_2 u_2 = \left( {\lambda _1 V_{21} + \lambda _2 V_{22} } \right)u_2 , \left\| {u_2 } \right\| = 1 \hfill \\ \end{gathered}$$ where, in the Hilbert spaces H_j, T_j is self-adjoint, bounded below and has compact resolvent, and V_jk are self-adjoint bounded operators, (−1)^j+kV_jk >> 0, j, k = 1, 2. An eigenvalue λ for this problem is a point in R^2 satisfying both equations. Under appropriate conditions, the eigenvalues λ^n = (λ_1 ^n, λ_2 ^n) are countable and in R^2. We aim to describe the set of limit points of λ^n/∥λ^n∥, as ∥λ^n∥ → ∞, in terms of the V_jk.
-
Two Parameter Asymptotic Spectra in the Uniformly Elliptic Case
Results in Mathematics, 1997Co-Authors: P. A. Binding, P. J. Browne, K. SeddighiAbstract:In this article we study the abstract two parameter eigenvalue problem $$\begin{gathered} T_1 u_1 = \left( {\lambda _1 V_{11} + \lambda _2 V_{12} } \right)u_1 , \left\| {u_1 } \right\| = 1 \hfill \\ T_2 u_2 = \left( {\lambda _1 V_{21} + \lambda _2 V_{22} } \right)u_2 , \left\| {u_2 } \right\| = 1 \hfill \\ \end{gathered}$$ where, in the Hilbert spaces Hj, Tj is self-adjoint, bounded below and has compact resolvent, and Vjk are self-adjoint bounded operators, (−1)j+kVjk >> 0, j, k = 1, 2. An eigenvalue λ for this problem is a point in R2 satisfying both equations. Under appropriate conditions, the eigenvalues λn = (λ1 n, λ2 n) are countable and in R2. We aim to describe the set of limit points of λn/∥λn∥, as ∥λn∥ → ∞, in terms of the Vjk.