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Global well-posedness and long-time asymptotics of a general nonlinear non-local Burgers Equation
Analysis of PDEs · Submitted 7 Dec 2021
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We construct global classical solutions starting from smooth positive data, and global weak solutions starting from positive data in $L^\infty$.
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The abstract, sentence by sentence
This paper is concerned with the study of a nonlinear non-local equation that has a commutator structure. The equation reads $\partial_t u-F(u) (-\Delta)^{s/2} u+(-\Delta)^{s/2} (uF(u))=0$, $x\in \mathbb{T}^d$, with s $\in$ (0, 1]. We are interested in solutions stemming from periodic positive bounded initial data. The given function $F \in C^\infty (R^+)$ must satisfy $F' > 0$ a.e. on (0, +$\infty$). For instance, all the functions $F (u) = u^n$ with n $\in$ N * are admissible non-linearities. We construct global classical solutions starting from smooth positive data, and global weak solutions starting from positive data in $L^\infty$. We show that any weak solution is instantaneously regularized into $C^\infty$. We also describe the long-time asymptotics of all solutions. Our methods follow several recent advances in the regularity theory of parabolic integro-differential equations.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | This paper is concerned with the study of a nonlinear non-local equation that has a commutator structure. | background 0.64 | 0.37 | 0.03 | asserted 0.91 |
| s1 | The equation reads $\partial_t u-F(u) (-\Delta)^{s/2} u+(-\Delta)^{s/2} (uF(u))=0$, $x\in \mathbb{T}^d$, with s $\in$ (0, 1]. | background 0.77 | 0.25 | 0.03 | asserted 0.94 |
| s2 | We are interested in solutions stemming from periodic positive bounded initial data. | background 0.66 | 0.21 | 0.02 | asserted 0.84 |
| s3 | The given function $F \in C^\infty (R^+)$ must satisfy $F' > 0$ a.e. on (0, +$\infty$). | background 0.71 | 0.20 | 0.02 | asserted 0.99 |
| s4 | For instance, all the functions $F (u) = u^n$ with n $\in$ N * are admissible non-linearities. | background 0.75 | 0.21 | 0.02 | asserted 0.98 |
| s5 | We construct global classical solutions starting from smooth positive data, and global weak solutions starting from positive data in $L^\infty$. | result 0.77 | 0.97 | 0.46 | asserted 1.00 |
| s6 | We show that any weak solution is instantaneously regularized into $C^\infty$. | result 0.98 | 0.95 | 0.19 | asserted 1.00 |
| s7 | We also describe the long-time asymptotics of all solutions. | result 0.64 | 0.95 | 0.21 | asserted 0.99 |
| s8 | Our methods follow several recent advances in the regularity theory of parabolic integro-differential equations. | method 0.95 | 0.15 | 0.02 | asserted 0.99 |
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Original abstract
This paper is concerned with the study of a nonlinear non-local equation that has a commutator structure. The equation reads $\partial_t u-F(u) (-\Delta)^{s/2} u+(-\Delta)^{s/2} (uF(u))=0$, $x\in \mathbb{T}^d$, with s $\in$ (0, 1]. We are interested in solutions stemming from periodic positive bounded initial data. The given function $F \in C^\infty (R^+)$ must satisfy $F' > 0$ a.e. on (0, +$\infty$). For instance, all the functions $F (u) = u^n$ with n $\in$ N * are admissible non-linearities. We construct global classical solutions starting from smooth positive data, and global weak solutions starting from positive data in $L^\infty$. We show that any weak solution is instantaneously regularized into $C^\infty$. We also describe the long-time asymptotics of all solutions. Our methods follow several recent advances in the regularity theory of parabolic integro-differential equations.
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